The Graceful Tree Conjecture

TOWARDSTHE GRACEFULTREE CONJECTURE

Frank Van Busse1

A thesis submitted in conformity with the requirements for the degree of Masters of Science Graduate Department of Computer Science University of Toronto

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Tùtvards the Graceful Tree Conjecture

Frank \an Busse1 Slasters of Science Graduate Depart ment of Computer Science University of Toronto 2000

In this thesis we preçent several results about graceful and near-graccful tabellings of trees.

Irarious modifications of the conditions for graceful and bipartite graceful labellings are

probed; we conjecture that much stronger assumptions about graceful labellings than previously considered can be appIied to al1 trees, and ive provide evidence towards these conjectures. The most significant result of this thesis concerns a strengrhening of the graceful labelling condition which requires that the centre vertes of' a tree be assigned the label

0: labellings tvhich satisfy this extra constraint are called O-centred grcrc~ful. Empirical results are presented n-hich indicate t hat non-0-centred graceful trees consticute a very small and easilj- charactcrized subset of diameter 4 trees; it is shon-n that for al1 orders rio rriernber of this subset has a O-centred graceful labelling, and that al1 other trees

üf diameter 5 -1 do have one. These results are estended to O-labellings of arbitrary verlices via the notion of O-ubiquitousiy grace/ul trees; similar results (both empirical and theoretical) are obtained with respect to these. -As is-ell, some bûunds are presented for certain rclaxed graceful Iabellings of trees, ernpirical data concerning the prevalence of bipartite graceful trees is analyzed, and a n-eakened foim of bipartite labelling called localiy bipartite is investigated. Acknowledgements

I would like to thank my supervisor, Mike hlolloy, who is not only a mathematician of the first rank, but an exceptional teacher and advisor. That this thesis ever made it safely into port is due primarily to his judgement and guidance-

1 would as well like to thank my second reader, Alesander Rosa, for his suggestions

and advice, and of course for the Graceful Tree Conjecture itself. Throughout my research

for this thesis his body of work on the GTC mas an inspiration and template. Many thanks to my friends and colleagues at the department of Computer Science here at the University of Toronto, and especially to Ioannis Papoutsakis, from whom 1 first learned about the Graceful Tree conjecture, and u-ho has ahays exhorted me to look for the essential reasons behind things. Thanks also to Joe, Leslie, Graham, and the rest of the gang at The Room 338 bistro, where bits and pieces of this thesis fell into place between discussions of quantum

tunneling or art history, garnes of fuzbol, and pints, while Deep Purple played on the jukebos. 1 cannot express the gratitude I feel toward my parents, Jack and Bernadine Van

Bussel, for thcir patience and support throughout my meandering career. Thanks as well to mu brother George, for his encouragement and assistance through the years.

This tbesis is dedicated to my "little bl~ebell'~,Zeina Khan.

iii Contents

Introduction 1

1 Historical Background 4

1.1 Graph Labelling Problems ...... 4

1.1.1 19th century and early 20th century antecedents ...... 4

1.1.2 Bandwidth and related problems ...... 6

1.1.3 Xnalogs outside of graph theory ...... 9

1.2 The Graceful Tree Conjecture ...... 13

1.2.1 Ringel's conjecture ...... 13

1.2. 2 General developmcnts in graceful labclling ...... 17

1.2.3 LVork towards the Graceful Tree Conjecture ...... 21

1.2.4 Tree product constructions ...... 23

1.3 Graceful (and a few not-so-graceful) graphs ...... 28

2.4 irariations on a Graceful Theme ...... 32

1.Near graceful labeilings ...... 32

1.42 Relaxations of the graceful distinctness conditions ...... 35

1.4.3 Harrnonious and additive labellings ...... 36

1.1.4 Edge first labellings ...... 39

1.4.5 Total labellings ...... 10

1.4.6 Other types of labellings ...... 42 1.5 The graceful tree conjecture today ...... 42 1-5-1 Rosa and ~iraii(1995) ...... 43 1.5.2 Aldred and Mackay (1998) ...... 41 1.5.3 Broersma and Hoede (1999) ...... 45

2 Relaxed Graceful Labeliings of Trees 46

2.1 -4 schematization of relaved graceful labellings ...... 46 2.2 Range-Relaxed Graceful Labellings ...... 17 2.3 Vertes-Relaxed Graceful Labellings ...... 54

3 Bipartite and Locally Bipartite Graceful Labellings 63 3.1 Bipartite graceful labellings ...... 63 3.1.1 Useful fcatures of bipartite labellings ...... 63 3.1.2 Prohlcms with fully bipartite labellings ...... 66 3.2 Some empirical results ...... 67

3.2.1 Estent of bipartite and locally bipartite graceful trees ...... 67 3.2.2 Estent of bipartite and locally bipartite graceful labellirigs .... 59 3.2.3 Other empirical results ...... 73 3.3 -4 recursive construction for graceful tree labellings ...... 74 3.4 Some explicit constructions for trees which are not bipartite graceful ... 83 3.4.1 -4 construction for k-cornets ...... 83

3-42 A construction for certain lobsters ...... 90

4 O-Centred and rn-Edge-Centred Gracefulness 99 4.1 O-centred and m-edge-centred graceful labellings ...... 99 4.1.1 Gracefully assigning the O label ...... 99 4.1.2 Some empirical results ...... 101

4.1.3 Two t heorems about m-edge-centred labellings ...... 105 4.2 O-centred graceful trees of diameter 4 ...... 109 1.2.1 Trees of diameter 4 with centre degree 2 - . . . - ...... 109

4-22 Ottier trees of diameter 4 ...... - ...... - . - . 112 . 4.3 Trees that are "ubiquitously'' graceful ...... - . . . . . - . . . . - 133 4.3.1 O-ubiquitously graceful trees ...... - . . . . - - . . - 133

4.3.2 Further empiricai results ...... - ...... 138

5 Conclusions and Future Work 141

5.1 Some open problems ...... - . . . - ...... 142 3.2 Working towards the graceful tree conjecture ...... - ...... 145

Appendix 149

Bibliography 154 List of Tables

3.1 Bipartite graceful and non-BG trees of order n ...... 68 3.2 Trees which are not bipartite graceful, by order n and diameter d .... 69

3.3 Locaily bipartite and bipartite graceful labellings for trees of order n . . 70 3.4 Trees of order n without bipartite graceful, LB(l), or either labelling . . 82

1.1 Non-O-centred and non-m-edge-centred graceful trees of order n ..... 103

4.2 Xon-bipartite graceful vs. non-O-CBG and non-m-ECBG trees ...... 105

4.3 O-ubiquitously graceful trees ...... 134

4.4 m-edge-ubiquitously graceful t rees ...... 138

4 5 ubiquitously graceful trees ...... 139

vii List of Figures

1.1 Cayley diagram for the group D3...... 5

1.2 Representations of bandwidth labellings...... 7

1.3 Tree from figure 1.2 with optimal cutwidth embedding ...... 9

1.4 Two rulers measuring 1~.3.4.5.6.7.5.9.10.11.12.13.16.17...... 12

1.5 Cjdic decomposition of K9 into a tree of size 4 ...... 14

1.6 Canonical a-valuation of a caterpillar...... 15 1.7 The 3-cornet ...... 16

1.8 Graceful labelling of K374...... 16

1.9 Kotzig2sT4...... 23

1.10 Tree product construction ...... 25

1.1 1 -4 graceful labelling of a radially symmetric tree ...... 26

1.13 Graceful Iabelling for the cycle Ci ...... 28 1.13 Graceful labelling of the Dutch 5-windmill ...... 30

1.11 Graceful labelling of the grid Pq x P5...... 31

1.15 -4 graceful labelling of the Peterson graph ...... 32

1. 16 5-graceful IabelIing of the 3-carnet ...... 34 1.17 Cordial labelling of the %cornet ...... 35 1.1S Harmonious labelling of the %cornet ...... 37

1.19 1-sequential labelling of the 3-cornet ...... 41 1.20 h4agic Iabelling of the %cornet ...... 41 2.1 Construction for RRG labelling of C. with n 2 or 3 (mod 4) ...... 48

2.2 Construction for Theorem 2.2 ...... 51

2.3 RRG labelling of a tree using theorem 2.2 construction ...... 52

2.4 French 2-windmill ...... 55 2.5 VRG labelling of tree obtained from depth-first path-decreasing edge labelling ...... 56

2-6 ZI lias degree > 2 ...... 59

2.7 zf has degree 1 ...... 60

'2.8 VRG labelling of a tree using construction from claim 2.6 ...... 61

3.1 Bipartite graceful labelling of a non-caterpillar ...... 66

3.2 Non-locally bipartite labelling of T ...... il 3.3 1U1 = 11'1 or [Li[= IV1 + 1 ...... 72 3.4 The one other tree ...... 73

3.5 Four esceptional trees of order n < 17 ...... 74 3.6 Esample of construction for theorem 3.5 ...... 78

3.7 -4 k-cornet ...... 84

3.8 Construction frorn theorem 3.7 applied to the 9-comet and 10-cornet ... 85

3.9 L(5), the schematized lobster with 5 legs ...... 91

3.10 L~JYIU~of .üi, ~i,and u;i in L(k) ...... 92

3.11 Construction fmm theorem 3.9 applied to L(4) and L(6) ...... 93

3.12 L(4) with labelling fL ...... 95 3.13 Formation of T*from T and L ...... 96

3.14 LB(1) labellings (with tails) for L(3) and L(5) ...... 98

4.1 The srnaIIest counter-esample ...... 101

4.2 First 3 non-0-centred graceful trees ...... 103 4.3 Smallest 3 non-m-edge-centred graceful trees of odd order ...... 104 44 m-edge-centred labelling of diameter 3 tree ...... 106 4.5 m-edge-centred Iabeiling of diameter 5 tree ...... 107

4.6 Labclling for case 1 BI = 1 ...... 105 4.7 O-centred iabelling of diameter 4 tree with 3 branches ...... 1IO

4.8 Initial labeiling for diameter 1 tree with k branches ...... 117

4.9 Two O-centred graceful labeliings of 5 branch trees ...... 123 4.10 -4 O-centred graceful labelhg of a 6 branch tree ...... -... 125

4.11 -4 Cree not covered by any of the cases ...... 129 4.12 O-centred graceful labelling of the &cornet ...... 132 4.13 The first few members of V' ...... 134 Introduction

Let G be a graph on rn edges. \Ve say that the labelling f : V(G)-t {O, 1,. . is

graceful if f and the induced edge labelling g defined by g(uv) = 1 f (u)- /(v)l saçsify the foIlonring properties:

1. f is a 1-1 mapping from V(G)to {O,. . . , m), and

2. g is a bijection from E(G) to (1, . . . ,m).

If a graph G has a graceful labelling, ive say that G itself is graceful.

In the rnid-1960's it was conjectured by Rosa [29] that al1 trees are graceful- This now notorious open problern has been know variously as Rosa's conjecture, conjecture7 or the gracefvl tree conjecture (GTC). While much work on graceful and related labellings has been done since then, progress on the GTC itself has been spotty at bcst; Anton Kotzig [22] has called it a "disease" of graph theory. This thesis explores sorne approaches to tree labelling which may ultirnately be of use in resolving the problem-

In chapter 1 we will take a look at the field of graph labelling and its surrc)unding contcst, and suwey pst work done on the GTC.

Chapter 2 considers relaxations of the conditions for gracefulness given above- In- spired by Rosa and ~irtifi'srecent work on gracesize of trees [31],which consid~redla- bellings satisfying condition 1 but not necessarily condition 2, we look at two othPr basic relaxations: 1. range-relazed graceful labellings, in which both properties are maintained (to an estent) ivithin an espanded range for both vertices and edges, and

2. vertez-relazed graceful labellings, where condition 2 is satisfied but not 1.

The first of these had been looked at in the early/mid 1970's, but not appareutly ~4th respect to trec labellings. In chapter 2 some bounds are given for both RRG and VRG labellings. If T is a tree with n vertices and nt = n - 1 edges, then:

1. T has a range-relaxed graceful iabelling within the range O to 2m.

2 T has a vertes-rela~edgraceful labelling with at lest vertex labels distinct.

If G is a bipartite graph the labelling f of v(G) is said to be bzpartite if for some bipartition of G into the sets (A,B) there esists a constant a such that

This implies that for every pair of vertices u,v such that u E -4 and v E B R-e have f (71) < f (v). Bipartite graceful labellings have also been called a-ualuations. That not al1 trees Iiave a bipartite graceful labelling is a well known fact; nevertheless, they have bcen inuch used in work on the GTC (e.g. in [31]) because of convenient properties they possess. In chapter 3 a weakening of the bipartite labelling condition mhich we will cal1 locally bipartite is considered; as the name implies, we only require f (u) < f (v) if the edge UV is in E(G). In the chapter some empirical results concerning bipartite and Iocally bipartite graceful trees are discussed, the most provocative being the confirmation up to order 19 that al1 trees have a locally bipartite gracefd labelling. As well, a general coiistruction is given for gracefully labelling a tree T if it can be divided into subtrees Tl and T2 which have locally bipartite graceful labellings satisfying certain conditions; and two infinite classes of trees which are known to not be bipartite graceful are shown to be locally bipartite graceful. Chapter 4 look at some empirical and theoretical results concerning graceful labcllings of trees with respect to the assignment of specsed Iabets to specified vertices and/or edges. The main focus is on where in a tree the label O can be assigned gracefully.

The following are defined:

1. k-centred labellings. The labelling f of the tree T is k-centred if for T's centre

vertes u we have f (c) = k. If T has odd diameter then either endpoint of the centre edge is acceptable.

2. k-ubiqnitously graceful trees. The tree T is k-ubiquitously graceful if for every vertex

v E l'(T) there is a graceful labelling of T such that f (v) = k. If T is k-ubiquitously graceful for al1 k in the range {O,. . . , IE(T)I}then T is simply ubiquitously graceful.

k-edge-centred labellings and k-edge-ubiquitously graceful trees are defined sirnilarly witti respect to edge labellings. Some results are obtained for trees of size m which do not have m-edge-centred graceful labellings, but the bulk of the chapter is devoted

to two very intriguicg empirical findings concerning non-0-centred graceful and non-0- ubiquitously graceful trees. The first of these findings suggests that the only trees which do not possess O-centred graceful labellings are a very small and easily characterized subsct of diameter 4 trees; this subset is denoted as V. The second suggests that al1 trecs which are not O-ubiquitously graceful are generated by trees in D in an exceedingly simple way; the set of trees generated in this manner is denoted as 23'. A7e are able to confirm these findings rigourously to a certain extent: it is shom that al1 trees of diameter 4 that are not in V are O-centred graceful, and ail trees of diarneter 4 that are not in V' are O-ubiquitously graceful. Throughout this thesis terminology that is specific to graph labelling will be defined in full whcre it first appears; for terms and definitions from general graph theory the readcr is referred to West [38]. Chapter 1

Hist orical Background

1.1 Graph Labelling Problems

IVhile labelled graphs have been an object of study for as long as graphs have? graph labelling is a relatively young sub-field of graph theory. Problems in this area are concerned wi t h the interaction between suitably defined vertes and edge labellings. Typically, \are are given a graph G and a set S which supports some kind of mathematical operation

(in pactise, S is invariably a subset of the integers or an abelian group), plus a lalelling

(or nurnbehg, valuution) of the vertices f : V(G) i S and a labelling of the edges g : E(G) + S such that the value of g(uv) is a function of f (IL)and f (v). I\'e want to know: wlien does the edge labelling g possess some other desired property? Occasionaliy, one dlsee the obvious variations on this scheme, where for esample the edges receive t Iie primary labelling and the vertices receive the induced labelling.

1.1.1 19th century and early 20th century antecedents

\Yhile the gracefuI tree conjecture is certainly the rnost notorious graph labelling problem, it is not the first; nor is the idea of mathematically interrelated vertex and edge labellings al1 that new. Probably more than any other 19th century mathematician, Arthur Cayley \vas instrumental in making graph theory a legitimate field of endeavour for mathemati-

cians, n-ith his work on trees and his involvement in publicizing the 4-colour conjecture. The set of graphs which bear his name, Cayley diagram (directed) and Cayley graphs (undirected), possess what are similiar to "equitable" labellings (see section 4); these arose from his work in group theory, as a relatively efficient way of representing finite and countably infinite groups:

Definition 1.1 [7]Let H be a group with a set of generators X. G(H!.Y), the Cayley diagram for H, is a directed rnultigraph where the vertices are identzfied wïth the elements

of H; an edge exists Getween vertices a and 6 if and only if as = b for some generator x.

Figure 1.1 shows a Cayley diagram for the dihedral group of order 3.

Figure 1.1: Cayley diagram for the group D3.

If the direction is ignored (or the edges are labelled with a generator and its inverse) the diagram is called a Cayley graph, and denoted simply G(H,X). That X be a gen- erating set is not always crucial in this case, since rnuch of the underlying structure and symmetry will still be revealed if X is not too large or too small. Cayley Diagrams and graphs are useful for, among other things, determining whether two distinct representations are of the same or conjugate group elernents, and whether two groups are isomorp hic.

Schrier diagrams are an estension of Cayley diagrams in a fairly straightfomard Ray.

In the Schrier diagram -4 mod B, where -1 is a group and B is a subgroup of -4, we have the vertices identified with the right cosets of B, and the edges labelled with the elements of S: an arbitras- subset of A. An edge HA' exists if and only if right multiplication by s E S sends the coset H to the coset K. .-Zn interesting variation on the above is the Paley (or quadratic residue) graph P,.

Here the vertex set is IFq, a finite field of order q, and an edge xy esists if x - y is a quadratic residue. If q = 1 (mod 4), zy esists when yx exists and the graph can be trcated as undirected; in that case it is in fact the Cayley graph G({F,, +), Q,), where Q, is the set of quadratic residues. If q = 3 (mod 1) the graph is directed and is known as a Paley tournament. Paley himself, in the mid 19307s, used these to help in the cor~structionof Hadamard matrices. They are of interest in graph theory because in the

1 (mod 4) case t hey have a great deal of resernblance to a typicai random graph of order q tvhcre the probability of an edge is 1...

1.1.2 Bandwidt h and related problems

The esamples of the previous subsection could be seen as instances of the graph labelling schema in reverse: the induced function defines whether or not an edge exists in the first place. The first bona fide graph labelling problem is the bandwidth problem, which is concerned with minimizing the difference between the labels of adjacent vertices.

Bandwidth was originally studied in the 19507s,in relation to matrices with al1 non-zero elements lying within a narrow band about the main diagonal. Other applications have becn in network addressing, circuit layout, and design of error-correcting codes subject to minirnizing the maximum error. Definition 1.2 [13] Let G be a graph with n vertices, and the labelling f : 17(G)+ W a

bijection to (1,2, . . . , n). The bandwidth of the labelling f is defined by:

bf(G)= mau{( f (u)- /(v)l : uv E E(G))

and the bandwidth of G is defined by

b(G) = rnin{bf(G) : f is a labelling of G)

Another way the problem is often posed is to try to identifj- the vertices of the target graph G with those of a path P of the same order. Then the bandwidth of the labelling

is the maximum distance in P between vertices adjacent in G. In figure 1.2 we show both an optimal labelling and optimal embedding for a tree with bandwidth 5.

Figure 1.2: Representations of bandwidth IabeHings.

This can be generalized by making f a function from V to sets of k-tuples. The problem is then seen as minimizing the distance between adjacent vertices in G when they are embedded into a k-dimensionai grid graph. There have been many results for particular graphs or graphs related to particular applications. In general, bandnidth is affected by the density of the target graph, and the

presence of subgraphs that have topological similarities to complete binary trees. Two

ver? nice general bounds on 1-dimensional bandwidth (essentially folklore theorems) are:

Theorem 1.3 [13] If G is a graph on n vertices with diameter d:

Theorem 1.4 [13] If IS zs the maximum degree of G:

The problem of detennining the bandnidth of an arbitrary graph is NP-complete

(Papadimit riou, 1976). Some work has gone into devising approximation algorithms.

Bandwidth, like gracefulness, has spawned a number of variants. In al1 of these, the primary labelling of the vertices is the same as for bandwidth:

Topological bandwidth: G' is a refinement of G if G' is obtained from G by a finite nurnber of edge subdivisiens. Topological bandwidth b'(G) is then defined by:

b*(G)= min{b(Gt) : G' is a refinement of G}

Topolo@cal bandwidth cornes up in VLSI design optimization. Min-sum: here of course we want to minirnize the totaI edge differences. The min- sum of the graph G is denoted s(G):

s(G) = min{ 1 f (u) - f (v)l : f is a labelling of G) uvf E(G) Cutwidth: essentially, ure want to minimize the nurnber of edges passing "over" a vertes (Le. as if they where arranged in a line). The cutwidth of a labelling f of the graph G is defined by: CHAPTER1. HISTORICALBACKGROUND

and the cutwidth of the graph G is then defined by:

c(G)= min{cf(G) : f is a labelling of G)

For esample, the tree in figure 1.3 has cutwidth c(T) = 4:

Figure 1.3: Tree from figure 1.2 with optimal cutwidth embedding.

-411 of these problems have beeri shown to be NP-complete for arbitary graphs. hlin-

surn and cutwidth do, however? have polytime algorithms for trees.

1.1.3 Analogs outside of graph theory

Kow ive will tum our attention briefly to two areas outside of graph theory which have had

a large influence on the manner and direction graph IabeIIing has taken: combinatorial design theory and number theory.

Difference sets

Gracefulness depends upon the distinctness of the differences between arbitrary pairs in a set of numbers, so it should be no surprise that combinatorial design theoy underlies much of the work done on grâceful and related labellings; among design theorists the study of difference sets has a history going back to the 1850's.

Definition 1.5 [Z] A (v,k, A) difference set mod v (or cyclic (v,k, A) difference set) zs a set D = {di,. . . , dk) of distinct elements of Z, such that each non-zero z E Zv can be expressed in the form z 4 - d, (rnod v) in ezactly X different ways. For esample, (1,3,4) is a (7,3,1) difference set, since: 1 = 2 - 1, 2 = 4 - 2, 3 = 4 - 1,

4 = 1 - 4: 5 = 2 - 1, and 6 = 1 - 2 (al1 rnod 7). An obvious graph analog would be a labelling of the complete graph on k vertices with f : V + {O, . . . ,u - l} and g(u,2;) r f (u) - f (v) (mod v); every edge is considered turice, once in each direction. Diffcrence sets first arose in the study of symmetric block designs. -4 (u, k, A) block design is a collection of k-subsets (called blocks) from a set S of size v, such that each pair of elements in S occurs in exactly X of the blocks; the design is called syrnmetric if evcry pair of blocks intersect in exactly X of the elements. If D is a (v:k, A) difference set rnod v, we can denote D + a = {dl + a,. . . ,dk + a) (al1 rnod u) as a translate of D. AI1 translates of a difference set are difference sets themselves, and the collection of translates {D:D + 1,. . ., D + (v - 1)) are the blocks of a symmetric (v, k, A) design. Difference sets can be generalized in two ways:

1. BJ. use of groups other than Z,. The same Paley mentioned earlier invented his namesake graph in order to construct quadratic residue difference sets over finite

fields; in this way he n-as able to show the exisitence of Hadamard matrices of order

q + 1, where q G 3 (mod 4) is a prime power.

2. Difference systems. -4 (u, k,A) difference system is a collection of Dl:.. . : DLk- subsets of an additive abelian group S of size v, such that from the differences arising within the Di ive obtain each non-zero element of S esactIy X times. The sets Dineed not be disjoint. This of course allows construction of block designs ivhere v > k2. X lurtber generalization alloi's the sets to be of different sizes.

Bcyond the family resemblance, there are direct applications of this concept to graceful Iabelling problems. A perfect system of d2gerence sets is a slight modification of the basic difference s!-stem: we require that the absolute value of the differences, as opposed to the differences (mod v), cover al1 integer values from k to u + k - 1. These have been used to construct (or give non-existence prook for) graceful labellings of graphs consisting of several copies of complete graphs of various sizes sharing vertices in some rnanner (e-g. windmill graphs are in this category).

Difference bases and additive number theory

Very sirnilar in intent, though somewhat different in style, is the part of number theory

that deals with covering sets of integers with sums or differences. From here as well graph labclling haç inherited many useful results. itrfiile no doubt speculation in this vein is as old as number theory itself, the work

hcre realiy starts in the 1930's and 40%. 4 difference baszs with respect to n is a set of

integers al,al,. . . , ak such that every positive integer 5 n can be represented as ai - aj for some pair i, j 5 k. Note that this does not preclude any of the ai being greatcr than n, or any of the differences appearing more than once. X restricted diference basis is a set al: . . . , ak with the above property, plus the constraint:

Kote that this necessitates some duplicate differences for n > 4. Tlic major problem of interest here for number theorists was that of establishing thc minimum nurnber of elements required for each kind of basis (we will denote these minimums as k(n) and [(n)for unrestricted and restricted difference bases respectively) - Tablcs of k(n) and C(n) for not-so-large values of n have been compiled, and we have fairly good bounds on both for al1 n: utilizing an earlier theorem concerning niodular versions of these difference bases, Rédei and Rényi (1949) prowd that k2(n) Lim - esists, Iying somewhere between 25 and 2;; somewhat later it u7as shown that the limit for existed as well, between about 21 and 3f. In the 1970's Solomon Golornb [17, 51 made direct use of the latter result to obtain bounds on the size of the largest graceful subgraph of K,. As reported by Golomb, Erdos around the same time gave a bound for the range of the near-graceful labelling of K, that was linear in the number of edges;

whilc it is not likely that he made use of Rédei and Rényi's bound on k(n)directly, there is a good chance that his proof for K, depends on the same earlier theorem that their proof for k(n) does. We will look more closely at this in chapter 2.

One way of envisioning the problern which has had a great deal of appeal is called

the mler model: we want to put the minimum number of notches on a metal bar of

length n such that ive can measure al1 integral distances up to n (or alternatively, we

want to put a maximal number of notches on the bar such that al1 the rneasurabie

distances are distinct). Non-identical rulers which measure the same set of distances are called homometric, and have been the object of some study in their own right: for

certain natural applications of the ruler model pairs of homometric rulers cause unwanted ambiguity. It \iras hoped at one time (circa 1939) that no two minimal rion-redundant

riilers could be homometric; this was (in a reversal of the trend we have seen so far) shown to be false by Colomb and Bloom, by way of near-graceful labellirigs of cornpletc graphs. Figure 1.1 gives an esample of such a pair of homometric rulers.

Figure 1.4: Two rulers measuring 172,3,4,5,6,7,8,9,10,11,l2,13,16,17.

Additive number theory deals with modular versions of these problems. Here we are looking for minimal sets of numbers {ai,. . ., ak} which can generate al1 non-zero elements in Zn via the sums + aj (mod n), and maximai sets such that al1 the sums generated are distinct. These sets are respectit-ely the restricted and unrestricted additive bases. Like difference bases, additive bases have been an object of study since the 307s, under various names (e-g. the postage stamp problem) .

The harmonious graph labelling scheme (see section 4) was devised by Graham and Sloane in 1980 as a direct extension of additive bases; the problern of finding minimal bases then reduces to that of finding harmonious or near-harmonious Iabellings of com- plcte graphs. This !vas no doubt inspired in part by work done on graceful labcling, since one of the first things the authors do is check the harmoniousness of various classes of graphs for mhich gracefulness or non-gracefulness had been determined. We will deal more with harmonious and variant Iabellings in section 4.

1.2 The Graceful Tree Conjecture

1.2.1 Ringel's conjecture

-At a graph tlieory symposium in Smolenice, Czechoslovakia in 1963, the following question was posed by Gerhard Ringel:

Let T be an arbitrary tree with m edges. Can K2m+Lcan be decomposed into 2m+ 1 copies of T?

We still don't know the answer to that one, but the story of the graceful tree conjecture begins here; at the next symposium in the series (Rome: 1966) Alexander Rosa showed that Ringel's conjecture is true if what we now know as the GTC is true. In his reduction Rosa gave four different labelling schemes, each one including the previous ones within its definition:

1. a-valuation: what we have defined in the introduction as a bipartite graceful labelling; 3. ,B-valuution: a graceful labelling proper;

3. a-valuation: the edge labelling must satisfy the same conditions as a ,O-duation, but the vertes range is rela~edto {O,. . . ,2m), where m is the number of edges of the graph;

4. p-valuation: the vertes labelling is as above, and the range of the edge labels is relaxed to {l,. . . ,2m}, under the condition that either label i or label 2m + 1 - i are used: but not both.

The reduction of Ringel's conjecture to the graceful tree conjecture then follows from the fact that K2,+1 can be cyclically decomposed into a copy of any graph G with m cdges if and only if G has a pvaluation. As a bonus, if a graph G has an a-vaiuation, then for al1 p > 1 can be decomposed into copies of G as well.

Figure 1.5: Cyclic decomposition of 1(9 into a tree of size 4.

It sliould be noted that while Ringel's conjecture can be proven using the somewhat weaker a- or pvaluations, relatively little indcpendent attention has been given to either of these labellings. This is perhaps for the obvious reasons that the P-valuation was the most straightfonvard of the four giwn labellings, and the a-valuation the strongest.

It is worth dwelling a moment on the paper presented, "On certain valuations of the vertices of a graph", since it not only defines the problem, but also what are still some of the broadest and most significant results on the gracefulness graphs in general as well as trees. -4 selection are presented here. \ire define the caterpillar as the tree obtained from a path by adding any number of pendant edges attached to each vertex.

Theorem 1.6 [29] 411 caterpillars are graceful, and in fact are a-gracefd

=Zn esample of the canonicai caterpillar construction is giwn in figure 1.6; this labelling a1ways results in a bipartite graceful labelling.

Figure 1.6: Canonical a-valuation of a caterpillar.

Theorem 1.7 [29] Not all trees have a-ualuations.

The smallest offender is the 3-cornet, the tree obtained by subdividing eïery edge of a 3-star:

Theorem 1.8 [29] A11 trees with < 16 edges have a 8-valuotion.

The following is still the most general statement we can make with respect to graceful labellings of graphs; it follows from the fact that the sum of the edges on any closed circuit must be equivalent to O (mod 2): Figure 1.7: The 3-cornet.

Theorem 1.9 (291 IfG is un Eulerian graph wzth m 1 or 2 (mod 4) edges, it has no fi-ualuation.

Theorem 1.10 [29] Any cycle satisfying the above parity condition does have a P- valualion. if m G O (mod 4), it has an a-valuution as wefl.

Frorn the nest thcorern it follon~sthat every tree is a spanning tree of a bipartite graceful graph:

Theorem 1.11 [29] The complete bipartite graph Km,, has an a-oaluation.

Figure 1.8: Gracefut labelling of K3,+

.A number of other theorems are provided without proof; those of most relevance to tiiis thesis (concerning path labellings, and trees nithout a-valuations) are revisited by Rosa and others in the years following, so we will postpone full statement of these results until Our chronology catches up. 1.2.2 General developments in graceful labelling

The graceful tree conjecture generated much interest over the next several years, and

by the mid 19'70's graceful graphs and labellings had even made it to the mathematical recreation pages of Sczentific American; at the end of the decade there were, by some estimates, something like 100 papers on the topic. The man probably most responsible

for the popularization of graceful labelling problems was Solomon Golomb, whom you will remember from the previous section- It \vas be who coined the term "graceful" itself,

and he brought the general idea of gracefuIness out from undcr the shadow of Ringel's conjecture.

Some foiklore constructions

Before ive go too much further, ive should look at several constructions which have becorne

part of the graceful labeller's repertoire. These al1 appear to have been in place very early on, probably before 1970.

Definition 1.12 Let G be a graph on rn edges vrith the labelling f. Then the comple-

mentary labelling f ofG ts defined 6y:

If f is graceful, so is f (the induced edge labelling remains unchanged). Hence, if we have a graceful labelling of G where the vertes v is assigned O, nre also have one where it is assigned m. Furthermore, it should be noted that if f is bipartite, f is as well.

Definition 1.13 Let G be a bipartite graph on m edges with bipartition (-4, B), and let f be a bipartite iabelling of G such that f -l (n)E B and min{ f (B)} = p. j, the reverse labelling of f , is defined by: If f is a bipartite graceful labelling, so is j. Hence if a tree T aith bipartition (A, B) has a bipartite graceful labelling where u E -4 is assigned the label O, T also bas one where L. is assigned 1-4 1 - 1 (and likewise for m and ,B = 1 Al). The labelling is called the

"reverse7' of f because it reverses the order of the edge labels i-e. for UV E E(G) we have 1.m - ml = m + 1 - If (4- f(4I- Let Gt and G2 be vertex disjoint graphs with distinguished vertices vl and v2 respectively. The arnalgamation of Tland T2 at (vl,v2) is the graph obtained by identifying ul with v?. From the following theorem we have that if two trees have appropriate graccful labellings their amaigamation is graceful; the construction used in the theorem will be referred to in the rest of this thesis as the canonical amalgamation construction.

Theorem 1.14 Let Tland T2 be vertex disjoint trees with dislinguished vertices 2;1 and t.2 respectively, Let (-4, B),vl E -4, be the b2partition of Tl, and let Tlo T2 denote the amalgamation of Tl and T7 at (vl> v?). Assume Tlhas a bipartite graceful labelling f1 luith fl(ul)= 0, and that T2has a graceful labellzng f2 with f2(4 = O. We then have a gr-uceful labelling f of TiO T2 defined by:

-4s we can se,, the labelling of T2 is merely shifted up by a constant, so the edge labels generated by f2 remain unchanged. The effect of f on fI is to "stretch" the labelling fi so as to accomodate this shifted f2; the reverse labelling of fl is used in order to get the labels of ul and 712 to coincide. It should be noted that if the labelling f2 is bipartitely graceful, f will be as well. This construction extends readily to one in which the larger tree is formed by adding an edge between disjoint smaller trees. Applications of graceful labellings

While graceful labelling does look like the ultimate toy problem, people have corne up with some applications for it. Most of these are in the reaim of error correcting codes or radar systems. Golomb more than anyone has stressed the usefulness of graceful labellings [SI; his list of applications include:

Resolving arnbiguities in X-Ray crystallography

radar pulse codes

missile guidance codes

convolution codes

synch set codes

It should be noted that these are al1 in fact applications of the ruler mode1 for difference bases rneritioned above; in the graph labelling version, they al1 make use of near-graceful labellings of li',. In a similiar vein, Harary and Hsu [21] invent a relasation of graceful labclling which they cal1 node graceful, in order to give labellings to very sparse graphs; a possible application of this graceful variant is in the construction of radar sequences.

Applications of proper graceful labellings are not as common. The most interesting is due to Bermond [A], who reduced a problem in the placement of antennae in radio astronomy to the graceful labelling of French nlndmill graphs.

The number of graceful graphs

From Sheppard [33] we have that the number of gracefully labelled graphs with m edges is m! This is sirnply a matter of choosing for each edge label k one of the m - k + 1 pairs of numbers (2, i + k} that generate it. Note that this in fact gives an edge-first construction for building an arbitrary gracefd graph; one could even randomize it. In the same place he establishes that the number of bipartitely graceful labeiied graphs (whidi lie calls balanced graphs) on n edges is:

if m is even

mtll . -1 + T- 2 -' if m is odd.

For the number of gracefully labelled trees ive have no such simple formula, nor apparently any usable recurrence or livable approximation; these numbers, however, have been cülculated (with some difficulty) for orders n 5 17 [35].

Why is (or isn't) my graph graceful?

One of the most frustrating aspects of the graceful tree problem is the lack of progress towards finding general necessary and sufficient conditions for gracefulness. Almost al1 results pcrtain to very specific subclasses of graphs, since almost al1 results are based on algebraic constructions which lean heaviiy on some predictability in the graph's structure. More general properties, such as planarity, bipartiteness, chordality, and bounded degree, do not seern to play any decisive role here: for al1 of these we have both graceful and non-graceful graphs.

However, as more work has been done on graceful labellings a certain intuition has developed about what is and is not graceful. These have entered the folklore as the "reasons" m-hy a given graph cannot be graceful. These reasons are generally attributed to Rosa from a journal article of the late 1980's:

1. (trivial) Not enough edges

3. The graph violates a parity condition

3. Too many edges Non-gracefulness proofs being as rare as they are, there is nothing on record to suggest that this list is not complete. However, according to an unpublisked result of Erdos

(reported in Ili]) almost al1 graphs are not graceful.

1.2.3 Work towards the Graceful Tree Conjecture

TIirough the 1970's and 1980's lvork on the graceful tree conjecture continued, largely aloug two lines. In the first, attention mas given to proving that certain easily characterized subclasses of trees were graceful, or to establishing whether stronger conditions

such as bipartite gracefulness held or not; in the second the focus was on building infinite

classes of trees recursively by use of various Yree product" constructions. These latter constructions are discussed in the nest subsection, while here we will look at progress

made on "named" trees, starting with several theorems from Rosa's 1967 paper originally stated without proof and later revisited.

Theorem 1.15 1301 (Rosa 1977) Let n be any positive integer. For any vertez v in the

path P, there is a graceful labefling f of P,, such that f (o) = O. Furthemore, if (and

only if) u is not the middle vertex oj Ps there is a bipartite gracefuf labelling f of P, such

that f (v)= 0.

-4brliam and Kotzig continued work on graceful path labellings as an independent problem, seeking for example to get a bound on the number of bipartite graceful labellings of P,. They were able to show that this number grows exponentially with n.

The following two non-esistence theorems for bipartite graceful labellings will corne

up again in chapter 3:

Theorem 1-16 [22] (Huang, Kotzig, Rosa 1982) If T is a tree of diameter 4 then T i~ either a caterpillar or it has no bipartite graceful labelling. Theorem 1.17 [22] (Huang, Kotrzg, Rosa 1982) Let T be a tree with n vertices, al1 of which have odd degree. If n O (mod 4), the tree T' obtained by subdividing euery edge

of T has no bipartite graceful labelling.

In the sarne place Huang, Kotzig, and Rosa prove that al1 trees with 5 4 degree 1 vertices are graceful, as well as al1 trees with one of their bipartition sets having size < 4. They consider the gracefulness of al1 diameter 4 trees, and prove it for some cases.

Some other results from the 1970's and 1980's. Kotzig, starting with Rosa's observation that not al1 trees have bipartite gracefd labellings, showed that "in a certain sense" most trees do have an one: every tree belongs tu an infinite subclass of trees of which only a finite number are not bipartite graceful 1261. The proof used a somewhat complicated reduction to geometry called T-representation. We will return briefly to this result in chapter 3, when ive look at the actual number of non-bipartite graceful trees of order

71 5 1s. To Berrnond [-LI n-e ow the "graceful lobster conjecture", the iobster being the tree obtained bj* adding any number of pendant edges to the degree 1 vertices of a caterpillar. Special cases of this have been shown true by Ng (1986) and Wang, Jin, Lu, and Zhang

(1994). Olive trees are among the most asymmetric of trees, consisting of a root and k branches, the i-th branch being a path of length i. These were shown graceful by Pastel and hynaud (1978).

Grace [18] has given an intuitively nice extension of graceful labelling to infinite graphs (vertex labels distinct, edge labels bijection to the set of positive integers) and a construction for labeiling infinite trees with finite maximum degree this wa.

Before moving on to tree product constructions we will look at one more resuIt due to Iiotzig (who, it should be noted in passing, seems to have been norking on solving decomposition problerns via something Iike gracehil labelling since at least 1965). Pick- ing up on essentially the same idea of an "edge first" construction for gracefui graphs as Sheppard, he attempts in [27] to to characterize the relationship between different gracefully Iabelled trees of the same size. This characterization involves the construction of the super-graph Tm, n-hose vertices are the set of gracefully labelled trees with m cdgcs; the cdge tlt2 of Tm exists if the gracefully labelled tree tl can be obtained from the gracefully labelled tree t2 by changing the endpoints of one of t2's edges. Figure 1.9 gives an esample, T4;the circle on the right contains the cwertex labelling for the trees (which can be considered fked), and the lines between the particular trees are labelled with the value of the edge swapped between them.

Figure 1-9: Kotzig7sTi.

\\'hile valiant, this attack on the problem was not conclusive; Kotzig \vas able to show that a graccful tree could always eschange one edge of due1 for another with the result remaining a graceful tree: but n-hether any gracefully labelled tree can be obtaincd from any othcr gracefully labelled tree by some series of edge eschanges (Le. whether Tm is connected) remained a conjecture.

1.2.4 Tree product constructions

Among the most sustained efforts made towards proving the GTC involves the buildiiig of "tree products" of various kinds. In general, a subgraph of a gaceful graph need not be graceful; this is the closest thing to a recursive scheme that graceful tree labelling has so far admitted.

The most basic tree product is defined as follows: ive are given trees S of order ns and T of order nr, with c* E V(T) a distinguished vertex in T. The product of S and T is then obtained by

1. Setting the mrtices of S in some arbitrary ordering u,, u2,. . . ,u,, ,

2. Taking ns copies of T,again set in an arbitrarily ordering Tl,T?, . . .Tris, with the

vertes .us of T copied to the vertex vy of T,, and

3. ldentifying the vertes ui of S with the vertes vf of Ti.

That is, n-e replace each vertes u of S with a copy of T rooted at v*.

The first graceful labelling construction of these tree products is provided by Stanton and Zarnke [37]. Given two gracefully labelled trees S and T we have a graceful labelling for any of their products (Le. any vertex we choose to distinguish in T); the formula is a simple function of labellings of S and T, the vertex we distinguish in T, and the ordering we adopt for the copies of T. Note: the version given below has been normalized so that the vertes labels fa11 within O and the size of the tree in question (for some reason most of the trec-product people seem to prefer vertex labellings that start from 1):

Theorem 1.18 [37]Let S and T be trees with ns uertices and n~ vertices respectively, and let v* be a distinguished vertex in T. Let S' denote the tree obtained by replacing each dzstinct vertex ui of S with a copy of T rooted v'; the individuai copy of T replacing ui will bc denoted as Ti,i = 1,. . . ,ns, with the uertez u of T copied to the verîez vi of

Ti.If S and T both possess graceful labelizngs fs and fT respectively, then the tree S' has a graceful labelling f' given by:

(ns - fs(ui) + l)nr+fT(v), dist(v,vB) in T iS odd

fs (ui)n~ f~(u) 3 dht(v, u') 2s euen. Figure 1.10: Tree product construction.

-4 dightly mod ified version of this construction gives us a gracefu.1 labelling of the ree obtained from S and T by replacing al1 vertices of S but one (which we will denote by u') with a copy of T rooted at some distinguished vertex u*; if S and T have graceful labellings fs and fT such that fs assigns O or IE(S)I to u' there is a graceful labelling of the quasi-product tree. By this means Stanton and Zamke are able to prove the folloming theorem:

Definition 1.19 The tree T is radially symmetric if al1 vertices the same distance from the centre vertex v of T have the same degree i-e. in its represenlation as a tree rooted at v euery level has the same branching factor.

Theorem 1.20 [37]Al1 radially syrnmetric trees are grnceful.

They do this by applying their modifieci construction recursively, letting the skelton tree S be a star with centre vertex u', and the Ti's be identical radially symmetric

subtrees; replaciug every vertex of S but u' with a copy of T results in the larger radidly syrnmetric tree. The labellings so obtained in fact appear much more structured than

those we get from the general tree product construction; figure 1.11 shows an esample.

Figure 1.11: -4 gracefuI labelling of a radially symmetric tree.

Koh: Rogers, and Tan [25] provide a variation where the trees that are attached to the

vertices of the base tree S need not be copies of a single tree T. Thcy define a beau pair as a pair of gracefully labelled trees Tland Tz with bipartitions (Ai,&), i = {1,2} such that for every edge label k, either both Al and A2 contain the lower labelled endpoint for their respective trees, or they both contain the higher. For two trees to be a beau pair then it is necessary not only that they have the same order, but they must have the same split into bipartitions. They show that for the formula given in Stanton and Zârnke theorem to work it is sufficient that the trees al1 have the same order, and those attached to the vertices of S Iabelled k and m - k be a beau pair. The subtrees attach thcmselves to the skeleton S via some vertex in each that takes the same label. That al1 the subtrees need be of the same order is still a somewhat onerous condition, and they devote some time to trying to find u~orkaroundsby considering certain "compatible collections" of different sized trees. Beau pairs will corne up again in chapter 3. Chung and Hwang [12] define a graph G as rotatable gmceful if for every vertex v E V(G)there is a graceful labelling f of G such that f (v) = O. They show that if S and T are rotatable graceful trees, t hen a rotatabie graceful tree is produced by attaching a copy of T at some fised vertes of T to every vertex of S. They are of course aware of Stariton and Zarnke's work, but not apparently that given the definition of rotatable graceful their result is already implied by Stanton and Zarnke's original construction (if

Ive want vertes vi in subtree Ti to have label O, we just find labellings of S and T such

that both ui E S and u' E T are labelled O). Grace [19] came upon essentially the same idea as Koh, Rogers, and Tan, though with an entirely different construction using adjacency matrices; he restricts his subtrees to those having bipartite graceful labellings (any two with the same size bipartitions can be a beau pair).

Jin et al [23,24] give a pair of constructions which result in trees having at least two isornorphic subtrees. Given gracefully labelled trees S and T, the joint sum construction provides a graceful labelling for the tree consisting of S and two copies of T linked by a path going through the O-labelled vertices of each, in the order S-Tl-T2. The radical product construction is more involved; it provides a labelling for the graph obtained by alrnalgamating n copies of a tree T with 2m copies of a tree S at a single vertex; T and

S rnust have the same number of vertices, n and m must both be 2 1, and the identified wrticcs must both have degree 1 and take the label O. Work in this vein carries on. Burzio and Ferrarese [IO] modify the construction of Stanton and Zarnke so as to relax the distinguished vertex constraint of the original. This follows from the observation that if we construct a graceful labelling f of the tree product of S and T according to the Stanton and Zarnke formula, and for the copies and T, of

T we have the vertices v, w of T copied to vil Wi of Tiand vj,wj of Tj respectively, then

That is, in Tiand T, the difference between the labels assigned to the twvo "copies" of any vertes of T is the same as that generated by the copies of the distinguished vertex. In Burzio and Ferrarese's new construction then each adjacent copy still has to be connected by pairs of these copied vertices, but dserent pairs can used as the endpoints of the edges of the original skeleton S, increasing the number of trees we can build from IV(S)(- (V(T)1 to ll/-(~)ll~(~)l-~.The original Stanton and Zmke formula is modified slightly so as to rernove reference to the distinguished vertex (instead, the cases go by T's bipartition into

sets -AT and Br).As with Stanton and Zarnke's construction, another minor modification allows us to resenre one vertex of S which n-il1 not be replaced by a copy of T; from this

it follows as a corollary that the subdivision graph of a graceful tree is graccful.

1.3 Graceful (and a few not-so-graceful) graphs

Progress in gracehl labellings outside of the graceful tree conjecture has been prolific if sornewhat piecemeal; here we take a quick look at what has been done.

-As mentioned earlier, that the cycle C, is graceful if and only if n O or 3 (mod

4) and that the complete bipartite graph Km,is always graceful, were given in Rosa's original papcr, while the non-gracefulness of the complete graph K,, had in a scnse been estaMished before graceful labelling had even been defined.

Figure 1.12: Graceful 1abelIing for the cycle C7.

Cycle variants. Many graphs built out of a single cycle have been looked at, with most found to be graceful. Among them are: the croum, a cycle with a pendant edge attached to each vertex (Fmcht, 1979); the wheel ifin = CC,V Ki(Fmcht, 1979); the heln H,, obtained from wheel by attaching pendant edge to each cycle vertex (Ayel and Favaron, 1982); the web, obtained by linking the outer vertices of a helm with a new cycle and adding a new round of pendant edges (Kang et al, 1996); gear gmphs, obtained from wheel by subdividing each cycle edge (Ma and Feng, 1984); and any cycle with a cliord (Delorme et al, 1980). Al1 these graphs bypass the Eulerian parity condition of theorem

1.9: so the fact that they are al1 graceful should not be too much of a surprise; but note that they al1 furnish examples of graceful graphs with non-graceful induced subgraphs.

The more general case, uyhere we just have a cycle in the middle of some relatively unorganized graph, has not faired so well. Truszczynski has conjectured that al1 unicyclic conncctcd graphs escept C,,, n = 1 or 2 (mod 1), are graceful. He has gotten things underway by showing a cycle with a path starting out of one of its vertices (called by . hma dragon, by others a tadpole) is graceful.

Unions. The union of several disjoint cycies has been the object of much scrutiny, since decompositions of complete graphs into cycles in various combinations is an ongoing concerri. \Ve have from Kotzig (1945) that TC,, r disjoint copies of C,, is graceful for r = 3 and n = 4k, k > 1, but not if r 2 2 and n = 3 or 5. The union of arbitrary cycles Cm + C, has been shown to have a bipartite graceful labelling if and only if m and n are both even and m + n = O (rnod 1) (Xbrham and Kotzig, 1996). The ordinary gracefulness of such graphs when m + n = 3 (rnod 4) has been shown only for certain cases. The union of 3 disjoint cycles has a bipartite graceful 1abelIing if and only if the nimber of vertices ZE O (mod 4) escept in one case: 3C4 (Eshgi, 1997 [14]).

Various other unions of cycles, cornpletc bipartite graphs, small complete graphs and the like have been lookcd at. The union of a cycle and a path has been conjectumd to be graceful if and only if the number of edges is at lest 7 (Frucht and Salinas, 1985); this has been shown for various cases. That the union of 2 or 3 non-star compIete bipartite graphs is gracefui u-as shown by Seoud and Youssef (late 1990's); the union of two or more stars is of course too sparse to be.

Amalgamations. The windrnill lYArn) consists of rn copies of I(, attached at a cornmon vertes. We have frorn Bermond [?] that the Dutch Wndmill cit)(also known as the friendîhip graph) is graceful if and only if t O or 1 (mod 4). K;~)is not graceful for n = 4, m = 2 or 3, but Bermond conjectures that for rn 2 4 it is; this has verified for m 5 22 (Huang and Skiena: 1994). No windmills are graceful for n > 5 (Koh et al, 1980).

Figure 1-13: Graceful labelling of the Dutch 5-windmill.

The arnalgamation of m n-cycles at a common vertex has not so far yielded such general results. \Ve do have that one point union of any two cycles is graceful if the Eule- rian parity condition holds (Bodendiek et al 1975). Rosa has conjectured that triangular cacti (copies of C3 which painvise share common vertices) satis%ng the Eulerian parity condition are also graceful. Only certain cases for this have been shown so far.

Graph products. Grids and other product graphs have been popular because of tlieir very predictable structures. That the n-cube (the product of n copies of K2) is gaceful was shoufn by Kotzig as early as 1965. Other graceful products are: the planar grid Pm x P, (-4charya and Gill, 1978); the subdivision graph of the ladder P, x P' (Kathiresan, 1992); the prism Cm x Pn (Jungreis and Reid 1992) when rn and n are even or m = O (mod 1); Sn x P2, when the Eulerian parity condition is satisfied (hlaheo for n everi, Delorme for n odd, both 1980); Sn x Pm when n is evea, the odd case being still open (Gallian and Jungreis, 1988).

Figure 1-14: Graceful labelling of the grid P4 x P5.

Joins. The join of two graphs G1 with n vertices and Gz with m vertices requires the addition of nm new edges, so Cl and G2 cannot both be very substantial if the result is going to bc graceful. Some work went into showing that the fan P, v KI and similar graphs were graceful, but this was superseded by Grace's result (1983) that the join of any graceful tree and 57, is graceful. Balakrishnan and Sampathkumar were able to esterid this by replacing the tree with any graceful graph G on m edges plus enough isolated vertices to make n - 1 = rn, and thereby prove that every graph is a subgraph of a graceful graph: if, starting with any graph G that has hrras a subgraph you iterate the process k times, you end up with a graph G(') that has Krfkas a subgraph.

Isolated grap hs. Some famous graphs have been given graceful labellings; t hese include the Peterson graph, Grotzsch graph, the Heawood graph, the Herschel graph, and various platonic solids, such as the dodecahedron. Figure 1.15: .A graceful labelling of the Peterson graph.

1.4 Variations on a Graceful Theme

X multitude of new labelling schernes has arisen since 1980, when Graham and Sloane introduced harmonious labellings. Here we take a quick look at some of these. In order keep the definitions short and avoid tedious repetition, we will assume from the outset that G is a graph with n vertices and rn edges, that f is a vertes labelling, and g is an edge labelling.

1.4.1 Near graceful labellings

The most obvious variation on the graceful labelling scheme involves relaxing the conditions placed upon the ranges of the vertex and/or edge labels while otheru-ise holding to the original dcfinition. Relased graceful labellings of this sort have been considered as long as graceful labellings have, Rosa'a o- and pvaluations being two early exarn- ples. The popularizer of graceful graph problems, Solomon Golomb, focused much of Iiis attention on tbese relaxed graceful labellings, particularly in relation to non-graceful graphs such as K,; for these, of course, the question becomes "What is the optimal near graceful labelling?" His own definition of gracefulness proper places it strictly in this context: the graph G on rn edges is graceful if T(G) = m,where r(G) is the minimum range needed to label G without duplicate edge or vertex labels. While interest in this simplest of relaxations peaked in the 197OYs,the idea has resurfaced from tirne to time; for esample, the node-graceful graphs of Harary and Hsu 1.211 employ essentially what is Golornb7sscheme to label very sparse graphs. There are several quite similar variations on gracefulness which relax the conditions only slightly. From Rosa (1988) we have nearly graceful labellings, where the range of the verticcs and cdges are both allowed to go to m + 1, but rn and m + 1 cannot bot11 be an edge label. This minor modification allows us to label graphs that violate the Eulerian parity condition. Alrnost-graceful labeliings were introduced by Moulton a year later; they are csactly the same, escept in that m can not be used as a vertes label (m+ 1 still can).

Frucht's pseudograceful labellings followed in 1990. With these we have corne almost full circle back to graceful labellings: vertices an injection into {O,. . . , m - 1, m i1). edges a bijection to {lo. . . , m).

One of the earliest and most farnous variants is the k-gracefd labelling scheme, n-hich

Ive have from Slater (1982) and Maheo and Thullier (1982) independently. The vertex labelling f is relaxed, in that it is nonr an injection into {O, 1, . . ., m + k - l}, but the range of the allowable edge labels is merely shifted up to {k, . . . ,m+ k - l}. Every graph with a bipartite graceful labelling is k-graceful for al1 k, since ive can add any constant ive want to the labels in the high partition uithout causing in conflicts; but there are graplis which are k-graceful for al1 k and do not have a bipartite graceful labelling. k- gracefuhess is not quite affected by parity the way gracefulness is; we have that C, is k-graceful if and only if

n O or 1 (mod 4), k even, $ "-'2 = 3 (mod 4, k odd, 5

An extension of thk scheme via arithmetic progression is that of a (k, d)-gracefuf Figure 1-16: 5-graceful labelling of the 3-cornet. labelling. This is due to -4charq.a and Hegde (1990). Vertes labels are in the range

{O, . . . : (m- l)d+ k}? while the edge labels rnust take the values {k,k + d, k + 2d, . . . , k -+ (n~- 1)d). As with k-graceful labellings, we have that graptis with bipartite graceful labcllings are (k,d)-graceful for al1 k and d. It haais0 been shown that:

r Every k-graceful graph is (kd,d)-graceful.

r EL-eryconnected (kd:d)-graceful graph is k-graceful.

If a graph is (k,d)-graceful and not bipartite, k 5 (m - 2)d-

-ln interesting modification is due to Gnanajothi (1991). Odd graceful labellings allow vertices to bc labelled with any integers in the range {O, 1,2, . . . ,2712 - l), but the sdges are restrictcd to odd numbers 5 2m - 1. She has shown that al1 graphs with a .bipartite graceful labelling are odd-gracefui, and that a11 odd graceful graphs must be bipartite; she conjectures that al1 trees are odd graceful, and has verified it up to order 10.

The last of the near graceful labellings we look at nere is actually a strengthing of the bipartite graceful labelling. A graph is strongly graceful is it possesses a bipartite graceful labelling with the extra condition that for every vertex v the labels of al1 incident cdges form a sequence of consecutive integers. This labelling cornes from Maheo (1980). -An esample of a strongly graceful labelling is the canonical caterpillar labelling, as in figure 1.6; hence al1 caterpillars are strongly graceful. Bodendiek and Schumacher have shown that they are the only trees with this property.

1.4.2 Relaxations of the graceful distinctness conditions

-4 few generalizations of graceful labelling have appeared where the requirement that the vertes and cdge labels al1 be distinct is replaced with some constraint on how often or

where a duplicate label can be used. The first and most popular of these is Cahit's cordial Iabelling scheme (1987). It is essentially a binary version of graceful labelling: vertices arc given labels O or 1, and edges are assigned the absolute difference of their endpoints,

hence either O or 1 as wel1. The labelling is cordial if, for the vertices and edges taken

scparately, the number of O and 1 labels differ by at most 1. Many products and unions

liavc been shomn to be cordial. as have al1 trees and al1 complete bipartite graphs; but cornpletc graphs of order > 3 are not, nor are Eulerian graphs when the number of edges

E 2 (rnod 4).

Figure 1.17: Cordial labelling of the 3-cornet.

-4 natüral gerieralization of both cordial and graceful labellings, due as well to Cahit (1990), are k-equitabie labellings. The vertex labelling is within the range O to k - 1, with edges as above labelled by the absolute difference between the endpoints. If for al1 pairs of numbers i and j in the stated range the number of times i and j are used as labels differs by at most 1 (vertices and edges again considered separately) the labelling is k-equitable. Hence we have that the properties of being cordiai and Zequitable are equivalent, as are the properties of being graceful and (m+1)-equitable. P, is k-equitable for al1 k, and it has recently been shown that al! trees are 3-equitable [36]. K, is not k-equitable for any k in the range 2 < k < n. Bzgraceful labellings were designed particularty for trees, in order to obtain sornething

similar to a bipartite graceful labelling where none were known to esist. These corne to us from Ringel, Llado, and Serra (1995). Let T be a tree with bipartition into sets A and B. For the vertex labelling f it is required that al1 labels within each of these sets bc distinct integers in {O, . . . ,m}, but duplication is allowed between the sets. The edge

labelling g is defined by:

that is, absolute values are dispensed with because the lower of the tn70 vertes labels

must Iic in A. It has been shomn tliat al1 lobsters and various other trees are bigraceful; al1 trees are conjectured to be. It should be noted that the term bzgraceful has also been

used by Gnanajothi to describe a graceful graph which also has a graceful line graph.

1.4.3 Harrnonious and additive labellings

Harmonious labellings and their variants have by now attracted as much attention as graceful labellings. -4s mentioned above, the original harmonious labellings wre brought forth by Graham and Sloane in 1980 as an additive version of graceful labeilings. The vertes labelling f takes numbcrs in the range {O, . . . , m - 1) (one repetition is allowed for trees, othenvise they should be distinct), and the induced edge labelling g is defined by g(uv)= f (u)+ f (u) (mod m); the graph is harmonious if g is a bijection to {O,. . ., m- 1). It is of course conjectured that al1 trees are harmonious as well as graceful. Among the many known harmonious graphs: C, if and only if n G I or 3 (mod 4); al1 wheels; al1 helms; the grid Pr x P, when r and s do not both equal 2; the prism Cr x P, when r is even or s is odd. Figure 1.18 gives an example of this labelling scheme.

Figure 1-18: Harrnonious labelling of the 3-cornet.

Two slight modifications have been made to this scheme which do not require an exception be made for trees. Elegant labellings, from Chang, Hsu, Rogers (1981) shift

the arithmetic from Z, to Z,,i: vertex labels are in the range {O, . . . , m) and edge labels defined by addition of the endpoints rnod m + 1; these edge labels must be al1 distinct and non-zero for labelling to be elegant. Elegant graphs include: C,,if and only if n O

or 3 (mod 4); al1 fans; al1 paths except P4;and al1 complete bipartite graphs. Felzcitous labellings, like eelgant, allow vertes labels from the integers {O, . . . ,m}, but the induced edgc labelling is defined by addition mod m, as with harmonious; this labelling is in

fact almost identical to harrnonious for trees, but it works on some unharmonious cyclic graphs. Lrarious felicitous graphs: C, except when n E 2 (rnod 1), Kr, if and only if both + and s are greater than 1, cir)for odd r. -4 number of extensions of harmonious labelling which are analogous to k-graceful and (k,d)-graceful labellings have been developed. In al1 of these modular arithmetic is dispensed with in favour of straight addition. Chang, Hsu, and Rogers (1981) have given us strongly c-harmonious labellings, a-here f is in the range {O, . . ., n-l},g(uu) is defined

by f (u) + f (v), and the constraint is that g is a bijection to {cl . . . ,c + m - 1). As with harmonious labellings, for trees we must allow one vertex label to be used twice. Every strongly c-harmonious graph is harmonious, but it is not yet known whether there exist harmonious graphs that are not strongly c-harmonious. Stronglv c-elegant labellings, from the same authors at the same time, are a slight modification of the above scheme where the vertes range is allowed to go to m; these were introduced independently as sequential

labellings by Grace (1982)) and in the case where c = 1 have been called consecutive labellings. Xcharya and Hegde, who wcre responsible for (k, d)-graceful labellings, have also brought us strongly k-indexable and (k, d)-anthmetic labellings (both 1990). For

tlic first of these, the vertex labels are frorn {O,. . ., n - l), and the requirement for the induced edge labels is merely that they al1 be distinct. For the second, the vertex labels

are distinct non-negative integers, while the induced edge labels must be, as with (k,d)- graceful labellings, {k, k + dl k + 2d: . . . , k + (m - I)d). "Strongly k-indesability" is not al1 that strong a property; it can readily be shown that al1 trees and al1 unicyclic graphs possess it.

One obvious generalization of harmonious labellings is to take labels from groups other than Z,. H-harmonious labellings (GaIlian) do just this: if H is any Abelian group of order nL, \lth the group operation *, then G is H-harmonious if it has a vertes labelling

f from the elements of H and an induccd edge labelling g(uû.) = f (u)* f (v) such that g is a bijection to H. As with harmonious labellings, for trees we need to repeat one vertes label. H-elegant labellings are a similar extension of elegant labellings to an arbitrary -Abelian soup H of order rn + 1. -4-cordial labellings, from Hovey (1991) are further a generalization of both harmonious and cordial labellings: the vertex labelling f is from any Abclian group -4 with operation *, g(uv) = f (u)* f (v),and the cordiality condition is satisfied if (with respect to vertices and edges separately) for every pair a, b E -4 the nurnber of times labels a and b are used differ by at most 1. The case where -4 is the cyclic group Zk has been the most studied; the labelling is then called k-cordial. We have: caterpillars k-cordial for al1 k; a11 trees for k = 3,4,5; al1 cycles C, except when k is even and f is an odd integer.

An independent line of additive labellings starts with the sum graphs of Harary (1990). G is a sum graph if there exists an arbitrary set of positive integers S of size n such that tlie vertex labelling f is a bijection to S, and the sum of the labels of every pair of adjacent vertices is an element of S (no distinctness is required here). Given the non- negativity constraint, a necessary condition for G to be a sum graph is that it must have

isolated vertices. Harary defines the sum number s(G) of G as nurnber of vertices which miich be added to G to make it a sum graph; he has shown that for every tree T with

more than 1 vertex, s(T) = 1. With integral sum graphs (1994) he makes the obvious niodification of allowing the set S to contain negative integers; hence some connected graphs, such as C, and the wheel ?Vk (in both cases if and only if n # 4) are integral surn graphs. Mod sum graphs, due to Boland et al (1990) are a modular extension of the

sum graph idea: S is a subset of {Z, . . . , r - l), and we require that for every edge uu the sum of the endpoints mod r is in S.

1.4.4 Edge first labellings

Some people have considered variants where the edge tabelling g is primary and the vertex labelling f is induced. The first of these to appear is LO'S edge-graceful labelling

(1955). The cdge labelling g is a bijection to {1, . . . ,m}, while the vertex labelling f is defined by

the graph G is edge-graceful if f is as well a bijection. Apparently this is not so easy with graphs of even order, but it is conjectured al1 odd order trees are edge-graceful. Line-gracefui labellings (Gnanajot hi, 1991) have an identical definition for the induced vertex labelling, but the edges are alIowed to corne from {O,. . . , n}. If a graph G on m cdges is line-graceful, m f 2 (mod 4); al1 paths, cycles, and stars which satisfy this parity condition can be Iabelled this way. Cahi t has invent ed several edge-firs t variants of his cordial/equi table labelling schemes. IlTith H-cordial labellings (1996) the edges are labelled with either +1 and -1 as equitably as possible; the induced vertex labels, again the simple sum of incident edges, must al1 be equal to fK or -K for some K, and as well be equitably distributed. E-cordial labellings, wliicli Iie developed with Yilmaz (1997) are a more straight-fomard takeon on cordial labellingç. Here edges are assigned either O or 1, and the vertices labelled with tlie sum of incident edges mod 2; the cordial condition on the distribution of O and 1 vertices then applies. No graph mith n G 2 (mod 4) vertices is E-cordial; al1 trees, cycles, complete graphs, and complete bipartite graphs which satisfy this parity condition are.

\Vit h Ek-cordial labellings he estends the E-cordial scheme in the obvious way by taking labels from the set (0, . . ., k - 1) instead of {O, 1).

1.4.5 Total labellings

Labellings of both the vertices and edges simultaneously under some constraint have a longer history than edge-first labellings; work on integer scquences with similar properties has been going on since before the mid-century. From Slater we have simply sequential labelling, where the total labelling f is a bijection from E(G)U V(G)to {1,2,. . . ,n +n} tiiat must satisfy f (ut.) = 1 f (u)- f (v)l for UV E E(G). This \.as subsequently generalized (cc. 1981) to k-sequential labelhg, where the range is shifted up to {k,k + 1, . . . , m + n + k - 1). Note that this preciates Grace's very different use of the term "sequential". \ire have: Kn not 1-sequei~tialfor n > 3; Kn not k-sequential for al1 k, n > 1; stars k-sequential if and only if k divides n; I(,, k-sequential for k = l,r, or s; P, for k = 5 (k even) and k = y- (k odd). Slater and others carried on in this vein witb k-sequentially additive labellings (1983), where the vertices and edges are labelled as above but must satisfy f(uv) = f (u) + f (u). Acharya and Hegde (1990) went on to develop this further with additiuely (k, d)- sequential labellings; for these the vertex/edge labelling must be a bijection to {k, k + d, k + 2d,. . . : k + (m+ n - l)d}. Magic labellings have been around almost as long as graceful labellings, being intro- Figure 1-19: 1-sequential labelling of the 3-cornet.

duced by Kotzig and Rosa in 1970. The labelling f is a bijection from V(G)U E(G)

to (1 2, . . . , m + n} which satisfis the condition that for each edge uv E E(G) f (u)+

f (r)+ f (UV) is constant. This Iabelling has aiso been called edge-magic by Ringel and Llado. Magic graphs includc al1 cornpietc bipartite graplis and al1 cycles, and al1 trees

are conjectured to be rnagic as well. No rnagic Iabelling is possible when we have the folloti-ing combination of conditions: m ewn, m + n 2 (mod 4): and the degree of every vertes odd. An esarnple of a magic labelling is shown in figure 1.20.

Figure 1.20: hlagic labelling of the %cornet.

Antimagic labellings (Hartsfield and Ringel, 1990) reverse the above constraint: f (u)+ f (ô.) + f (UV)must be distinct for each edge UV E E(G). It is conjectured that every connected graph except P2 is antimagic. 1.4.6 Other types of labellings

A11 of the above labellings have made use of either differences or sums in the formulation of t heir constraints. Some intriguing departures from this are:

rn Prime labellings, from Entringer. Vertices are given labels in the range (1,. . . , n}, witb the requirement that any two adjacent vertices are relatively prime. Paths, stars, caterpillars, complete binary tree, and cycles have been shonn to possess this labelling,

rn Verta prime labellings, from Deretsky, Lee, and Mitchem (1991). Edges are labelled aith (1,.. . , m);the greatest common denorninator of al1 edges incident on a vertex must bc 1. This is apparently very easy to do: al1 connected graphs, and al1 forests, are vertex prime-

rn Hamming-graceful labellings, from hllollard, Payan, and Shixïn (1987). Error cor-

rccting codes are often mentioned as an application of grapii tabeliing; this seems to be the one application of coding theory to grapb labelling. The vertex labels are taken from the set of binary m-tuples, and the induced edge labels are the Hamming distance between the endpoints, with the added proviso that they are al1 distinct. -411 trees, al1 graceful graphs, and, in a reversa1 of the usual trend, many K (en = k2 or k2 + 2) have Hamming-graceful labellings.

1.5 The graceful tree conjecture today

In the 1st half decade, while graph labelling activity continues unabated, not a great deal of work has gone directly towards dealing cvith the graceful tree conjecture. This is to be expected: on a fundamental level little progress has been made towards a solution since tbe problcm was first fomulated in 1967. The graceful numbering of a graph involves algebraic complications not amenable to the usual techniques of graph theory, while number theory and design theory do not, as of yet, have tools capable of handling the arbitrariness of structure a tree can possess.

This is not to say all hope has been abandoned, Here we =dl look at three of the more noteworthy recent attempts to find an opening into the problem.

1.5.1 Rosa and ~irifi(1995)

The first of these 1311 relaxes the problem from an existence question to a bounds question;

this approach, as stated in the paper, is inspired by and esplicitly modelled after work done on bandwidth related problems. The requirement that all the induced edge labels be distinct is waived; instead, the maximum number of distinct edge labels a proper vertes labelling can generate is used as a measure of the gracefulness of a graph. If G is a graph with m edges gs(G), the gracesize of G: is defined by

over all vertex labellings f : V(G)-+ (0: . . . , m), where g is the induced edge labelling g(uc) = Ij(u)- f (v)1. The a-size of G a(G)is similarly defined, except in that we are confined to bipartite labellings of G.

Theorem 1.21 [31] For every tree T with rn edges o(T),the a-size of T, satisfies:

This of course is also a lower bound on gs(T),so we know now that all trees are at least 3 graceful. The proof is by induction via the canonical amalgamation construction given in theorem 1.14, hence the restriction to bipartite labcllings. While no duplicate edge labels are created by these constructions, they do require breaking the tree up into parts with a-size large enough to guarantee the bound; therefore much of the proof is devoted to the many special cases that have to be handled. While the upper bound on gs(T) for any tree with m edges is of course m, there are trees witliout bipartite graceful labellings, so there are trees on m edges for which the

a-size is strictly les than m. An upper bound (of sorts) on the a-size of an arbitrary tree is given as weli:

Theorem 1.22 [31] Define cr(m) as the minimum of a(T)ouer a21 trees urith m edges. We then have

This is obtained through an a-optimal labelling of the y-cornet; since it is likely that coniets arc the worst behaved trees in terms of bipartite graccfui labellings, Rosa and

Siran conjecture that Zrn is asymptotically correct value of a(m).This \vas iater verified for al1 trees on n 5 12 vertices and maximum degree 3 [8].

1.5.2 Aldred and Mackay (1998)

\\*hile cornputers have been used before in mork on the GTC (for example, Rosa's venfica- tion up to order 16 iras presumably machine aided), the most ambitious such undertaking so far has been by Aldred and blackay [l],who confirm the conjecture for al1 trees with < 37 vertices. They also confirm that al1 trees with < 26 vertices are harmonious. Efficiency is of the utmost concern here, since there are around T50,000,000 non- isomorphic trees of order 27; hence the paper itself focuses largely on algorithmic aspects of the problem. Their program utilized various heuïistics and random perturbations to obtain a graceful labelling quickly, the basic idea being that a graceful iabelling is generally not too far from a random labelling, in a certain sense.

While their findings are not startling, the idea of using cornputer aided verification as an empirical aid to theoretical research was an inspiration for much of the material in diapters 3 and 4 of this thesis. 1.5.3 Broersma and Hoede (1999)

The hst paper we will look at here reduces the graceful tree conjecture to a similar, somewhat more restricted problem. The authors invent a variation of gracelul labelhg called strongly graceful (note: this tenn has been in use since about 1980 to denote a

different labelling, see section 4 above).

Definition 1.23 [9] If T is a tree on m vertices which hm a perfect matching 61, then

the labelling f is strongly gracefil if

1. f is graceful

2. For euerg edge uu E AI, f (u)+ f (u)= m

They show that the GTC is equivalent to the conjecture that every tree with a perfect matching has a strongly graceful labelling. Several simple transformations and a method of "gowing" larger strongly graceful trees out of smaller ones are also provided. Of particular interest are spzketrees and contrees; a spiketree is a tree with a perfect matching obtained from an arbitrary smaller tree by adding a pendant edge to each vertex, while s contrce is obtained from a larger tree with a perfect matching AI by contracting every edgc in M.

IVhilc reduction of graph labelling problems to problems outside of design theory, additive numbcr t heory, or graph labelling itself are practically non-existent, reductions within graph labelling are not a rarity. The authors concede that we are also no closer to a solution of the GTC in any immediate way, since the new problem is smaller in scope but much harder. The potential of this approach, however, has not yet been fully explored; the more predictable structure ive gain with the assumption of a perfect matching may, for esample, make some interest ing recursive constructions possible. Chapter 2

Relaxed Graceful Labellings of Trees

2.1 A schematization of relaxed graceful labellings

-As ive have çeen in the previous chapter, a plethora of new labelling techniques have been introduced in the last 30 years. hlost of these new labellings are not direct offshoots of graccful labelling, and so have only a limited application to any solution to the graceful trec conjecture itself. In this chaptcr me will Iook at some of the more direct relasations of graceful labellings, and how these might apply to the GTC; the next two chapters will deal wit h certain nTaysthe graceful labelling constraints can be strengthened. IVe will start by reiterating the three salient properties of a graceful labelling. Let G be a grapli with vertex set I7and edge set E, and let m = IEl and n = IVI. The mapping f (L-) i N with the induced rnapping g(E) -t N is graceful if and only if

(*) g is defined by g(uv) = 1 f (u)- f (u)l for al1 UV E E.

(1) f (17) is an injection to {O, 1,. . . , m)

(2) g(E) is a bijection to {1,2,. . . , m)

One could say tliat property (*) is the defining one: it is what makes a labelling graceful at al1 in the first place. As such, any non-trivial modification of property (*) might be considered too strong to be properly called a "relaxation" of gracefulness.

Properties (1) and (2), on the other hand, can be weakened to almost any evtent with the resulting labelling scheme still sharing many important structural features aith graceful labelling. We will t Oerefore consider the following t hree modifications of graceful labelling as being the most elementaq:

1. Edge-refuxed Property (2) is relaxed, in that g need not be a bijection but can be

any mapping from E to (1, . . . ,m).

2. Range-reluxed: Properties (1) and (2) are relaxed, in that the upper bound on f

and g is allowed to be some m' 3 m. g need no longer be a bijection, but rnust stili be an injection.

3. Vertex-relaxed: Property (1) is relaxed, ailowing f to be any mapping from V to {O:. . . 'm}.

Edge-rela~edgraceful labellings are in a sense the most simple and natural, in that ariy proper labelling of the vertices of a graph with numbers in the range {O,. . . ,m) satisfies the requirements. Weno longer ask whether a labelling f is graceful, but rather

'.Han- graceful is it?" The obvious measure of gracefulness in this context is the number of distinct edgc labels, Ig(E)I. Only recentl~however, have arbitrary labellings been looked at in this way; the 1995 results of Rosa and ~iraiion the gracesize of trees [31] are the first t hings that have been done in this vein. -4s the reader will recall from chapter 1, they were able to show that al1 trees on rn edges have an edge-relaxed graceful labelling with at least Brn of the edge labels distinct.

2.2 Range-Relaxed Graceful Labellings

Range-relased graceful labelling, our second scherne, has a much more estensive history than the other two; it has been brought forth and fiddled with under various guises since the graceful labelling problem n.as first popularized by Solomon Golomb in the carly 1970's. Since that time we have seen k-graceful, node-graceful, nearly graceful, dmost graceful and pseudograceful labellings (see chapter 1 for details), which are dl cornprehended in one way or another under the range-relaxed scheme.

Since not al1 IabeIlings in even the most extended range need be range-relaxed gracehl, one could Say that labellings of this sort are "not as relaxed" as their edge-relaxed counterparts. However, as is the case with edge-relaxed graceful labellings, every graph has a range-relaxed graceful labelling in some range; if, for example, the vertices are labdled with distinct powers of 2, the induced edge labels must al1 be distinct. This of course is definitely not the best we cari do-the cycle C, is not graceful when n = 1 or 3 (mod 4), but as the construction in figure 2.1 demonstrates, it always has a RRG labelling (to coin an acronym) in the range {O, . . . , n + l}. From the folloning theorem, due first to Erdos (unpublished) and reported by Colomb, we know that for any arbitrary graph G on m edges there is a RRG labelling in a range polynomial in m.

n+ 1 O

f Edge labels decreasing decreasing by 1 escept by 1 except at marked at marked vertices

3 4 n = 1 (mod 4) n = 2 (mod 4)

Figure 2.1: Construction for RRG labelling of C, nith n = 2 or 3 (mod 4) Theorem 2.1 [17] The complete gmph I(, has a range-relaxed graceful labelling in a range proportional to n'.

Proof: Erdos' original proof seems to be unavailable, but it most probably made

use of a result from Singer [31]stating that if q is prime or the power of a prime there esist q + 1 integers ao, . . . , a, such that the differences 4. - aj represent a complete set of residues modulo q2 + q + 1; such a set of integers is called a perfect dzfference set of order q + 1. There are only (q + 1)2 possible differences, and since q + 1 of these differences must bc of the form ai - ai = 0, the rernaining q2 + q of these must be non-zero and distinct modulo q2 + q + 1. Any such set of integers gives us a range-relaxed graceful labelling of Kn when n = q + 1 in the range n(n - 1) = 2(2). When n is not a prime power plus 1, we can of course take the lowest number grcater than n that is; making use of, for esample, Bertrand's postulate we how that tliere is always a prime in the range 7t - 1 to 2(n - l), and hence a usable n' 5 272 that will allow us to construct the labelling in the range 2 (1)9 8(:). Whilc wc can use more recent results from number theory to bring the range below 3(2), the empirical evidence frorn range-optimal graceful labcllings of I(, for srna11 n seems to suggest that even 2(>;)is not the best we can do. .-Zt any rate, for any connected graph G on rn edges and n \-ertices we cari obtain an RRG labelling of G in the range c(:)- = 0(m2),and the idea that for any such graph ive can obtain a bound linear in m is qiiite plausible. Work in this vein, however, seems to have been limited to considering cornpletc graphs, with not much being done after the early 1970's at ail. Theorem 2.2, the first of the original theorems presented in this thesis, can be seen as a belated next stcp towards addressing the situation by establishing an upper bound for trees.

Theorem 2.2 If T is a tree on m edges, T has a range-relazed graceful labellzng such that the rnuxirnurn vertex label zs strictly less than 2m.

Proof: Let uo be an arbitrary vertex of T, and consider T in its usual representation as a tree rooted at vo; that is, vertices the same distance from vo are drawn on the same Ievel, and edges are not allowed to cross each 0th. We dlassume that the longest path from is leftmost in this representation. Let the number of vertices in this longest

path = 1, let the vertex on this path at level i be denoted as v,, i = 0,1,. .. , 1 - 1, and let thc number of vertices on the i-th level be denoted as ki. The following construction provides a labelling f of V(T)in the range {O, . . . ,2m - 1 + l}.

1. The root vertex 2;0 takes a provisional value a; once the range of values has been

established we will shift a11 labels by a constant such that the lowest label = O. vl:

the leftmost child of vo, is given the label CI + 1.

2. For i > 1 each vertes vi on the leftmost path receives the labelling

3. .A vertes u on the i-th level k places to the right of ui, O 5 k 5 ki - 1, reccivcs the labelling

f (vi)- CE if i is even

f (IIi) + k if i is odd

By the construction al1 vertex labels are distinct, since on even levels they are mono- tonicaIly decreasing as ive go frorn left to right and from top to bottom, while on the odd lcvels they are increasing. Figure 2.2 shows how these two labelling schemes are interlaccd. Likewise, the edge labelling g of E(T)is monotonicaily increasing going from left to right and from top to bottom in the tree:

1. Consider two edges viui+l and wiwiil between vertices on consecutive levels i and i + 1: where uiui+i is left of utiwi+l and i is even. By part (3) of the construction, since edges cannot cross uTehave /(ui)> f (wi)and f (uiC1)< j(~i+~);hence leftmost level i : ki vertices. i is even, x .y path .

f 3 G . 'y-ki+ 1 label= 7~ '-- Y-I '- y-.? . . -

Figure 2.2: Construction for Theorem 2.2

Thc same follows for i odd by an analogous proof.

2. Consider any two edges ~liui+~between levek i and i + 1 and wiilwit2 between levels i + 1 and i + 2 where i is again even. Since the rightmost vertices ri on level i and r,+~on level i + 1 are respectively the minimum and masimum label on levels i and i + 1,

while the Ieftmost differcnce ~(z;~+~v~+~)is a lower bound on the labels of the edges betn-een levels i + 1 and i t 3. Hence \ire have

,Again, this holds for i odd w-ith minor modifications.

Lastly, u-e check that the bound on the range holds. Let fMr,vand fMAX be respectively the minimum and maximum vertex labels generated by the labelling f. In the case where I is even, the highest labelled vertes is the nghtmost on level 1 - 1, and the lowest labelled vertes is the rightmost on level 2 - 2; hence by the construction

G iving us the range:

Figure 2.3: RRG labelling of a tree using theorem 2.2 construction

By choosing for our root vertex one endpoint of a longest path in T, we can obtain a labclling in the range 2m - diameter(T). We note that this bound is tight for paths, and is "good" for long, stringy trees; but for trees of low fixed diameter u7e still have not broken (2 - &)m.There do exist variants of the above construction that in practice seem to always give us a range weil within fm; these, however, assign labels opportunistically rather than monotonically, and have not as of yet given us any rigourous results.

&'hile work on the upper bound remains a priority, range-relaxed graceful labellings have other aspects that may also be worth investigating:

1. -4s rnentioned in section 2.1 of the previous chapter, in order to prove Ringel's conjecture we do not need to show that al1 trees are graceful, merely that al1

trees have a a or p valuation. o and p valuations, as the reader will recaIl, are a special class of range-relased graceful labellings with the vertex labeiling in the

range {O, . . . ,2m}; conditions on what values the edges can take are more stringent. It may be possible to modify the above or sirnilar constnictions to achieve these conditions.

3. IVhilc it is a known fact that a subgraph of graceful graph is not necessarily graceful, if a graph G on m edges is graceful every subgraph of G has a range-relaxed

graceful labelling in range {O,. . . , m};for example, every spanning tree of a graph

C can obtain a range-relaxed labelling in this way. In many (though not all) cases

a range-relaxed graceful labelling of a tree on n vertices can be completed to a graceful labelling of a graph on n vertices by adding edges between appropriate verticcs. The subset of RRG labellings for which this is possible might be worthy of attention.

3. Of direct relesance to the graceful tree conjecture are range-relaxed graceful labellings under structural constraints such as forbidden or required labels. If T is a tree on rn vertices with a graceful labelljng f, removal of any edge from T gives us

two subtrees HI and Hz with "matched" range-relaxed graceful labellings fi and

f2 in the range {O, . . . , m);putting this process in reverse, one strategy Ive can use to gracefully label a tree is to find matched RRG labellings of its subtrees. The canonical amalgamation construction of theorem 1.14 does this implicitly when it stretches and shifts the graceful labellings of the subtrees that are amalgamated; in the next chapter the reader will see the the rangerelaxecl graceful labelling of a subtree used eqlicitIy, as part of the construction for lemma 3.11.

2.3 Vert ex-Relaxed Graceful Labellings

Vertes-rclaued graceful labellings were apparently studied in the 1970's by Bermond and Lehel in connection mith graceful labellings of windrnill graphs (11,but since then there docs not seern to have been any work done with them. The closest recent antecedent is the bigraceful labelling scheme of Ringel, Llado, and Serra (see chapter 1 section 4.2 for a descript ion); other variations of gracefulness which relax distinctness conditions (such as eqiiitable labellings) introduce new constraints as well.

\Vhile in a sense vertes-relased graceful labellings can be thought of as "dual" to cdge-relaxed graceful labellings, in structure a VRG labelling tends to look much more like a graceful labelling; for esample, vertices labelled O and m must be present and adjacent, and rn - 1 adjacent to O or 1 adjacent to rn, and so on. Not al1 graphs have a \'RG labelling-1 have in fact yet to encounter a connected ungraceful graph ahiçh does. \\:hile a full investigation of this must tvait, it is certainly worth looking briefly at bow i-crtes-relaxation fairs against some known classes of ungraceful graphs before going on to the main business of this section:

1. If a graph haç too few edges to be graceful then it is not connected in the first

place. We will find that in this case there generally is a VRG labelling.

2 For Eulerian graphs, the parity condition of theorem 1.9 that the number of edges m = O or 3 (mod 4) applies as much in the VRG case as when al1 vertices are distinct.

3. With dense graphs, allowing duplicate vertex labels offers no adwntage; for a graph to have any VRG labelling that is not simply graceful at least two vertices with disjoint closed neighbourhoods must be present (othemise, we will have two vertices mith the same label both adjacent to some third vertes, or adjacent to each other).

4. Similarly, the presence of any dominant vertices precludes the use of duplicate labels, hence French 3 and 3 vvindrnills are not vertex-relaxed graceful despite not being Eulerian or too dense; for French k-windmills, k 2 4, the only vertex-relaxed graceful labellings are proper graceful labellings.

Figure 2.4: French 2-windmill

It is easy to confirm that al1 trees do have a vertes-relaxed graceful labeliing; we can, for instance, give any tree T on m edges an edge labelling g which "reverse induces" a wx-tcs-relaxed graccful labelling f of T.Whilc the followîng construction will not be used in the main theorem of this section, it is of interest in its own right and cornes up again briefly in the nest chapter. We ni11 define an edge labelling g of a tree T as path-decreasing if for some distinguished vertes u in T the edge labels decrease monotonically along every path out of v, i.e. g(ulu2)> g(u2u3) if dzst(v, u1) < dist(v,u2) for u~u~,u~u3E E(T).

Theorem 2.3 Euery tree haa uertez-relazed graceful labelling with an associated path- decrcasing edge labelling.

Proof: We can choose any arbitrary vertes u of T as the root. -4 path-decreasing edge labelling of T starting from v can be done any number of ways (eg breadth-first or depth-first) such that al1 edge labels are distinct and in the range (1, . . . ,IE(T) 1). The vertes-relaxed graceful labelling f is then defined by:

1. Let f (v) = 0.

2. Let f (u) be the alternating sum (+ - + - ...) of the edges on the path from 2; to u. Since the magnitude of the terms of these sums are decreasing, wve have that no f (u) will be less than O or greater than [E(T)1.

Figure 2.5: VRG labelling of tree obtained from depth-first path-dccreasing edge labelhg

-4s in the previous section: one thing we are concerned with is hotv graceful a particular \vertes-relased graceful labelling f of G is; the rneasure is of course the number of distinct vertes labels, 1 f (V(G))I.For no graph G on rn edges can this nurnber be les thau a: if we have k distinct vertex labeIs then each can be used at most once with any of the others, so m 5 (:) . With trees we cxpect to do much better, and we can: from theorem 2.4 below we have a lower bound that is linear in the number of vertices. It should be rioted that in practice the path-decreasing construction above seems to invariably do better than this bound; it was not, hoivever, as suited to Our purposes as the construction used in the proof of the theorem. Theorem 2.4 Tf T is a tree on n vertices and m = n - 1 edges, T has a vertez-relazed graceful labelling f such the number of distinct vertex labels is stnctiy greater than 5.

Our proof will be by induction establishing a slightly stronger statement, for which we will need a little bit of new machinent. Recall frorn the introduction the définition of a bipartite labelling: the labelling f of G is bipartite if the vertices of G have a bipartition into sets -4 and B such that there e-xists a constant ûr E N satisfying, for every vertes

L7 € -4

f (4 5 a7 and for every vertex u f B

f (4> a-

If G is a tree then a is of course the highest label in the low partition -4, and is equal to 1-41 - 1. \Ve now define a weakened version of bipartite labelting, in that the above condition is applied (with some modifications) not globally but locally; the highest label in the Iow partition as a consequence loses its defining status.

Definition 2.5 Let G be a graph with a labelling f. We say that f is locally bipartite if the vertices of G have a bipartition into sets A and B such that for every vertez v E -4

and for euery vertex u f B

whcr-e N(v) is the open nezghbourhood of v.

\i7e note that, as is the case for bipartite labellings, only bipartite grapbs can have locally bipartite labellings. Locally bipartite graceful labellings are of some interest in thcir own right, hence are the focus of the next chapter. For Our present purposes we are interested in the fact that they have the property that the value of al1 edge labels can

be shifted up by a constant simply by adding that constant to the labels of al1 vertices in the high partition. This is a property inherited fiom fuiiy bipartite labellings; it is

made use of in, for example, the canonical amalgamation construction of theorern 1-14. -A similar construction is used in proving the following daim:

Chim 2.6 Let T be a tree, with the sets A and B a bipartitzon of the vertices of T,

and 1: an arbitraqy vertex in .4. T ahays has a vertex-relaxed gracefvl labekling / vlhich satisfies the folloving properties as well:

1. f is locally bipartite, with B being the high partition

3. The labels of al1 vertices in B are distinct.

Such a labelling udl be denoted as VRG' with respect to u.

Proof: Assume that al1 trees on les that n vertices satisfy the claim, and let T be any trce on n vertices and m edges. For every vertex u in V(T),ive can construct a labclling f that is VRG' with respect to v in the following manner:

CASE: 21 has degree 2 2. Then T can be split at v to form two trees Tl and T2 of orders strictly less than n and greater than 2. Let vl and 212 be the vertices of Tl and 7'' respectively which are identified to form v in T, and let (Al,BI) and (.A2, B2) be bipartitions of Tl and T2 such that vl E Al and u2 E A2.

By our assumption Tl and T2 possess VRG' labellings where ul and uz are assigned the label O, which we will denote fl and f2. If the number of edges in Tl is ml, then the labelling f of T defined by Figure 2.6: v has degree 2 2 is VRG' with respect to v. Since fland f2 are VRG' with respect to u1 and v2 respectively, we have:

2. f1 and f3 are both locally bipartite; addition of a constant to the labels in the high

partition docs not change that fact for f2, nor does amalgamation of the subtrees

at v,so f is as well Iocally bipartite.

3. The edge fabels generated by f in Tl are the same as those generated by fl: 11, . . . .ml}. The edge labels generated by f in T2 are those generated by f2 shifted

up by the constant ml: {ml + 1,. . .,ml + m,}, where rnz = IE(T2)1. Hence f is a vertes relaxed graceful labelling of T.

4. A11 labels given by fl to B1are distinct, as are al1 labeIs given by fi to B2. Since

f2 is Iocally bipartite no vertes in B2 is given the Label O by f2- Hence

Ttierefore a11 Iabels given to BI U B2 by f are distinct.

CASE: u has degree 1. Let w denote the sole neighbour of v. If T is not Pz (uthich definitely satisfies our claim), then w has L 3 1 other neighbours rl, r2, . - . ,rk. Let the trees of T - {w) rooted at each ri be denoted by Ti,their size by mi, and bipartition by

(.Air Bi)with Ti E Ai- Figure 2.7: u has degree 1

Since no Tiis of order > n - 2, by our assumption each has a VRG' labelling with respect to ri which we will denote fi. A labelling f of T which is VRG' with respect to v is then given by:

We vcrify that this is VRG'; in the following Miwill denote the sum mj:

1. f (v) is esplicitly assigned O by the construction.

2. Each fi is locally bipartite; the effect of f on fi is to add the constant i to al1 Iabels of \'(Ti), and the additional constant h.Ii to the labels of the high partition Bi,hence f applied to each Tiis locally bipartite. Since f (ri)is the lowest label in each T,, and f (w) of course the highest in T, f is locally bipartite with respect to T as weil.

3. Since the addition to a constant to al1 vertex labels does not effect the edge labels generated, we have that f applied to each Tigenerates {Mi+ 1,. . . , Ilfi +%); since Air, = O and Mi+i= Mi+ mi, these labels with respect to the Ti'sare distinct and cover (1, . . . ,Mk + ml-}.The edges labels geenerated in the subgraph of T induced by {u, w,rl, . . . ,rk} are, since f (ri)= i, simply {rn - k, m - k + 1, . . . ,n). Alk + rnr is the number of edges of T not in this induced sübgraph, hence f is vertes-relaxed graceful.

4. For each fi we have al1 vertex labels in Bi distinct, with max{fi(Bi)}5 mi and

in{(B)} 1 Hence we have that

Therefore the labels f assigns to u:=, Biare al1 distinct. The maximum of these is at most Mk+ rnk + k = m - 1, so al1 labels in the high partition of T are distinct and f is VRG' with respect to u.

Figure 2.8: VRG labelling of a tree using construction from daim 2.6

Of course. to obtain the bound stated in theorem 2.4, we choose a vertex u in V(T) such that the set A is the smaller one of the bipartition. We can then obtain a labelling / that is VRG' aith respect to v by applying the above construction recursively. -4s mentioned before, the Label O cannot be used for any vertes in B; hence we have

-1s the reader may have noticed, the VRG' labellings we have made use of in the proof are in fact quite similar to Ringel et al's bigraceful labellings mentioned above; the main difference is that VRG' labelhgs do not require that vertes labels in the low partition A rnust al1 be distinct. -4s such, daim 2.6 can as well be regarded as a step towards proving

the still open conjecture that a11 trees are bigraceful. There is of course no direct relation between the vertex-relaued Iabelling of a graph G and any proper graceful labelling of a sub- or super-graph of G. However, for any grapli G with vertes-relaued labelling f we do obtain a gracefully labelled graph G' of order 1 f (lT(G))Iby simply identifying verticcs with the same label. Duplicate edges and self-loops are neïer introduced by such vertes identifications, since we can never have

two like-labelled vertices both adjacent to some third vertex or eacb other. Hence if T Las vertes-relaued graceful labelling f with cn distinct vertex labels, c < 1, ive can obtain a range-relased graceful labelling in the range n - 1 for at lest one tree T' of order cn, by identifying like-labelled vertices and then estracting a spanning tree. In the other direction, while it is possible for every gracefuiIy labelled graph G on rn edges to obtain a vertes-relased gracefully labelled tree T of order rn + 1 (by breaking cycles at certain vertices and retaining the duplicate labels), we cannot always get a vertex- relaxed gracefully labelled tree from a range-relased gracefui labelling of a smaller tree; as stated in the last section, a range-relaued graceful labelling of a tree of order n does not ncccssarily extend to a proper graceful labelling of some graph of order n. Chapter 3

Bipartite and Locally Bipartite Graceful Labellings

In tliis chapter, v-e ni11 explore sorne aspects of the locally bipartite condition defined in

the previous chapter; in particular, we wiiI see that while in some senses locally bipartite

graceful IabeIlings are not as easy to work with as fuIly bipartite graceful IabelIings, the

condition is much more robust, and so may be of some use in future work on the graceful

tree conjecture.

3.1 Bipartite graceful labellings

3.1.1 Useful features of bipartite labellings

Threader will recall Our previous formulations of bipartite and locally bipartite labellings from definition 2.5. An equivalent statement of the definition that brings out the relation betwcen these conditions more strongly is the following:

Alternate definition Let G be a bipartite graph with bzpartition into sets -4 and B. The labelling f of V(G)is:

1. Bipartite if for al1 u E A, v E B: f (u)< f (v) CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 64

2. Locally bipartite if for al1 u E -4, v E B: uu E E(G) * f (u) < f (v)

Much of the work done on graceful 1abeIIings of graphs has in fact been focused on bipartite graceful labellings. The reason, in part, is that they have very convenient properties which, for esample, can be applied to esplicit or recursive constructions. The two that are most often exploited are the following:

(1) Scalability. For both bipartite and locally bipartite graceful labellings, we have tliat the addition of a constant to every vertex in the high partition shifts the values of al1 edge labels up by that constant. This is the property esploited by the canonical amalgamation construction of theorem 1.11, for example. It should be noted that for bipartite labellings this shifting never involves collisions of the vertex labels, whiie for locaIIy bipartite labellings they will occur if the constant applied is too low; howver, as u-e will see when we corne to lemma 3.1, for al1 constants over a certain value the result is a valid range-relaxed graceful labelling. If f is a graceful but not locally bipartite labelling simply adding a constant > O to any subset of the vertes labels wi11 never sindarly result in an RRG Iabelling with the additional properties that the edge labels are arc corisecutive and the highest vertex and edge labels coincide.

(2) Predictable assignment of labels to each partition. If j is any graceful labelling of a bipartite graph G on m edges with bipartition (A, B), ute always know that the labels O and m go to vertices in different partitions, but that is about it. When f is a bipartite labelhg as well, however, we also know that if fd'(0) E -4 then al1 the 1-41 lowest labels in the image of / go to -4. This property is most relevant to graceful labellings of trees: since in that case the vertex labelling as well as the edge labelling is a bijection, we have f(.4) = {O, 1,. . ., 1-41 - 1) fw = WI, IAI + 1, - . , with of course vertices labelled IAI - 1 and IAl adjacent.

From these two properties we are able to deduce such things as the following: CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 65

a) If the graph G has a bipartite graceful labelling f, then G is k-graceful for al1 k 3 O (see chapter 1 section 4.1).

b) If the graph G on m edges has a bipartite graceful labelling f with induced edge labelling g, there is a reverse labelling f of G which is also bipartite graceful,

with the feature that for Ve 'e E(G) the induced edge labelling ij of f satisfies

g(e) = m + 1 - g(e) (see definition 1.13 in chapter 1).

Since property (2) does not apply to Iocally bipartite labellings, neither of the above statcments extends directly to locally bipartite graceful labellings. There are, however, somewhat weaker analogues mhich make use of a little extra information.

Lemma 3.1 Let G be a bipartite graph on m edges, with the bipartition (A,B). Let f be a locally bipartite labelling of G with a being the highest label in the low partition -4, and ,O being the lowest label in the high partition B. If f is as well graceful then:

a) G is k-graceful for al1 k 2 a - fl + 1.

6) The LB-reverse labelting of f, f. defied by

is a fully bzpartite range-relaxed graceful labelling with vertex tabels in the range O

to m +a - 0 + 1. For the induced edge Iabellings g and g off and j respectively we have g(e) = m + 1 - g(e) for e E E (G), so the the cdge labelling g as still a bijection to {l,.., m}.

Proof: Statement (a) follows dircctly from the fact that locally bipartite Iabellings do possess property (1); the lower bound on k is set so as to ensure none of the high labeIs we are shifting collide with any of the unchanged low labels. For (b), we note that this version of the reverse is, when applied to fdybipartite graphs, the complement (see definition 1.12) of the reverse labelling given in definition 1.13; we cannot use definition 1-13 directly here because we cannot assume a = ,6 - 1, but this form of the reverse does not depend upon any such fact and is therefore safe to use. Lemma 3.1 n4l corne into play in section 3; t here ive wïll also look at some of the issues involved in making assumptions about how locally bipartite graceful labellings assign labels to bipartition sets. rn

3.1.2 Problems with fully bipartite labellings

-4s was mentioned in chapter 1, it was known from the start that not al1 trees have a bipartite graceful labelling, wi th the smallest non-caterpillar being also the smallest not bipartitely graceful tree [29]. Furthermore, caterpillars are still the most general class of trees to have been proven bipartitely graceful, and the canonical catepillar labelling as given in figure 1.6 is still the paradigm for ground up (rather than recursive) bipartite graceful labelling constructions. Catcrpillars, however, represent but a minute proportion of al1 trees; nor are they the only trees with bipartite graceful labellings, as we can see from figure 3.1.

Figure 3.1: Bipartite graceful labelling of a non-caterpillar

So the burning question becomes, which trees do have bipartite graceful labellings, and how plentifui are they? Kotzig, as mentioned in chapter 1 section 2.3, made the most direct attack on this problem [26]; he was able to conclude that "most" trees do have one, in the sense that every tree belonged to an inhnite subclass of trees having only a finite number of members nhich nere not bipartitely graceful: if we define TJi) as a copy of the tree T rvith the edge e replaced by a path of length i, then for al1 T and for al1 e E E(T) there is a finite k such that Te(i)has a bipartite graceful labelling for al1 i > k. Work in this vein, however, seems to have stopped here. .A possibly more natural line of attack is to try to find and verify subclasses of trees with no bipartite graceful labeilïng; there at least secrns to be somewhat more ongoing work along these lines. In chapter 1 two theorems due to Huang, Rosa, and Kotzig [22] concerning two fairly broad subciasses of trees were given; since they will come up later in this chapter, we will reiterate them here:

1. (Theorem 1.16) Al1 trees of diameter 4 which are not caterpillars have no bipartite graceful labelling.

2. (Theorem 1.17) If tree T has n = O (mod 4) vertices, each of ahich has odd degree, then the tree obtained by subdividing ewry edge of T has no bipartite graceful labelling.

3.2 Some empirical results

3.2.1 Extent of bipartite and locally bipartite graceful trees

Locally bipartite graceful labellings share one attractive feature rvith fully bipartite graceful labellings, scalability. They also have one attractive feature not shared with bipartite labellings. This is their robustness. To date, no tree has come to light that cannot be given a locally bipartite graceful labelling. Verified: Al1 trees of order n < 19 have a locally bipartite gracefil labelling. The cornputer program used to confirm this at the same time kept track of non- CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 68 caterpillars with bipartite graceful labellings and trees with no bipartite graceful labelling (details about this and other prograrns used in this thesis ail1 be given in the -4ppendL.x).

As we cmsee from table 3.1, while the ratio of caterpillars to trees in generd declines, bipartitely graceful trees are always in a large majority; and the proportion of trees with no bipartite graceful labelling becomes vanishingly small, peaking at about 10% at order 9 and dwindling to about 1% by order 18.

Trees Caterpillars Other BG Non-BG

Table 3.1: Bipartite graceful and non-BG trees of order n

The numbers here flesh out Kotzig's conclusion somewhat spectacularly. In particular, when we look at not bipartitely graceful trees by order and diameter (table 3.2) we find al1 are huddled around the lower diameters; it seems, for example, entirely possible that al1 trees of size rn with diameter > $m have bipartite graceful labellings.

However, it should be noted that while trees that are not bipartitely graceful are rare in relation to trees in general, their numbers do seem to be increasing at an exponential rate with respect to the order, n; 1 would place this around 1.9"-', though the data Table 3.2: Trees nihich are not bipartite graceful, by order n and diameter d ai-ailâble on tliis is admittedly not enough to make any trustworthy extrapolations. For the sake of cornparison, the number of caterpillars of order n cornes in at esactlÿ Y-' +

.I("-~)/~,- and the total nurnber of unlabelled trees of order n at roughly 2.96"/1.87n5/'

3.2.2 Extent of bipartite and locally bipartite graceful labellings

The second set of test runs compared the locally and fully bipartite conditions on a hbelling by labelling basis. Checking the number of bipartite and locally bipartite graceful labellings against the total number of gracefully labelled trees turned up what one would expect: that both bipartite and iocally bipartite labellings decrease in proportion to arbitra- graceful labellings both in general and for most particular trees, with bipartite labellings falling away more quickly. In the range tested one might notice that the proportion of fully bipartite to locally bipartite graceful labellings is similar to the proportion of locaily bipartite to al1 graceful labellings, but in al1 likelihood this bit of symmetry breaks down for n > 11. CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLIKGS 70

n 1 Graceful LB B

able 3.3: Locally bipartite and bipartite gracefu' labellings for Lrew uf vider n

Resdts for some particular trees called for deeper scrutiny, so tests were run to isolate those subclasses of trees for which the al1 graceful labellings were locaily bipartite, and those for which al1 locally bipartite graceful labellings werc fully bipartite. 1. Graceful versus locally bipartite. The class of trees for which al1 graceful labcllings arc locaIly bipartite seems to be simply al1 trees of diameter 5 3; this \vas confirmed up to order 16. Not that this should be any surprise, since small dominating sets of vertices should have a restricting affect on graceful labellings, and large diameters a Iiberating one. PLand P2 each only have one labelling to speak of, while stars only have 2 graceful labellings, which are necessarily both fully bipartite; that al1 trees of diameter 3 must as wcI1 belong to this subclass is confirmed by the following theorem.

Theorem 3.2 If T zs a tree with diarneter 3 then every gracefil labelling of T is locally bipartite.

Proof: Let T be a diameter 3 tree, with the vertices of the central edge being denoted by u and v; al1 other vertices have degree 1 and are adjacent to either u or v. Let f be a labelling of T, let g be the edge labelling induced by f, and let m = IE(T)I. If we ha~eg(uu) = m, then either f (u) = O and f (v)= rn or the other may around; hence the labelling must be locally bipartite whether it is graceful or not. Likewise, if the labels O CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 71

and m are given to degree 1 vertices, f cannot be graceful since these vertices cannot be adjacent. Hence it must be that, if f is graceful but not locaily bipartite, one of u or v take the O or m label, and the (respectively) m or O label is given to an adjacent degree 1 vertes. We can assume, by spmetry with respect to u and v and the existence of the complementary labelling with respect to O and m (see definition 1.12), that f (u) = O and f (z;) = r < m. Let s denote the maximum label given to any vertex adjacent to v; if f is not locally bipartite, then s > r. Any vertices with labels 2 s + 1 are adjacent to u.

Figure 3.2: Non-locally bipartite labelling of T

If s > r, howcver7 f cannot be graceful: no edge pendant on u can have the induced label s, since 1 5 r < s and al1 vertices adjacent to v have labels 5 s; nor can any pendant on u, unless we have two distinct vertices with the label S. Elence if f is graceful it is Iocally bipartite. w

2. Locally versus fully bipartite. -4 more intriguing class of trees are those having no locally bipartite labelling that is not fully bipartite. Al1 trees of diarneter 5 2, of course, belong; another subclass are diameter 4 trees with 2 branches, where the nurnber of degree 1 vertices on each branch are equal or differ by 1 (as shown in figure 3.3). -411 trees in this family have only fully bipartite or non-locally bipartite graceful labellings.

Theorem 3.3 Let T be a tree on rn edges with diameter 4, with the centre vertez w haviny degree 2, and the neighbours of w,u and u, hauzng degree and respectively. CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 72

Figure 3.3: IUI = IVI or ]UI = [VI+ l

Then any locall3 bipartite graceful labelling f of T is fully bipartite.

Proof: -Assume that f is locally bipartite graceful. We begin by noting that the

complementary labelling f given in definition 1.12 will be locally bipartite iff is locally

bipartite, since f (q)< j(c2)implies that rn - f (ul)> m - f (71~); and J will not be fully bipartite if f is not: since in that case the lowest label in the high partition will be less than the highest label in the low partition, and complementation both reverses the order of the particular vertex labels and the "highness" and "lowness" of the partitions. Hence if there is any locally bipartite labding of T which is not fully bipartite, thcrc is one in

which (u, u) are the low partition and one of u or u is assigned the label 0. We can then

assume that this is the case for f; let x E {u,u) be the vertex that is assigned 0, let y

be the other, and let f (y) = r. We will denote the set of degree 1 vertices adjacent to x

as X?and the set of degree 1 vertices adjacent to y as Y.

If r > 1, then to preserve the locally bipartite condition all vertices adjacent to y must

have labels > r, so labels 1 to r - 1 must go to vertices in X. Now if any vertex with label s is in A', then the label s + r must as well go to a vertex in S,since othenvise we would have duplicate edge labels. Hence labels r + 1,. . . ,2r - 1 must go to X, and (eventually) all labels f 0 (mod r). This leaves for the labels of vertices adjacent to y (including w) only non-zero multiples of r. If r > 2 this is impossible, since jjX 1 - lYl1 5 1. If r = 2, both x and y have even labels, so of the 171 + 1 even labels we start with the number of CHAPTER3. BIPARTITEAND LOCALLY BIPARTITEGRACEFUL LABELLINGS 73 even labels we have left for the neighbours of y are:

Hence if f is locally bipartite and graceful, r = 1 and f is fully bipartite. The only other trees of order < 16 for which al1 graceful labellings are either fully bipartite or non-locally bipartite are Pa, P7,Ps, and the tree shown in figure 3.4.

Figure 3.4: The one other tree ...

3.2.3 Other empirical results

Finally, various tests were done to establish sorne sub-conditions that did not affcct the robustness of the locally bipartite graceful labelling scheme, or at least did not affect it too much. Since the case where we want to specify that a certain set of labels is assigned to one of the bipartition sets is very relevant to the subject of the next section, we will iook at some empirical results pertaining to it there; while the next chapter will focus on what happens when we want to specify that a particular label goes to a particular vertes, for general, locally bipartite, and fully bipartite graceful labellings. Of ail the other tests, the most suggestive results related to locally bipartite graceful labellings for which the induced edge IabelIing is path-decreasing; this edge labelling, the reader will remember, was used in the first construction (theorem 2.3) given for vertex- relaxed graceful labellings in the 1st chapter. While it would have been too much to CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 74 hope that al1 trees have some labelling of this sort, out of the roughly 80,000 trees of order n 5 17, only the 4 shown in figure 3.5 do not. This certainly bears investigating at some future point.

Figure 3.5: Four exceptional trees of order n 5 17

3.3 A recursive construction for graceful t ree labellings

Of course, this al1 would be merely of passing interest if ive had no way of applying the locally bipartite condition in any way to gcncral graceful labelling problcms. The snag is that while Iocally bipartite labellings have the same nice scaling properties as fully bipartite labellings, t hey lack the full predictability of bipartite Labellings in terms of assignment of labels to the two partitions. Hence to make explicit use of the locally bipartite condition 1.c will usually have to make some additional assumptions, as was the case in vertex-rela~edgracefui labelling construction from chapter 2. Here we wi11 sce that even with these additional assumptions there are recursive constructions which represent an advance over the previous state of things. The reader will recall tree product constructions which were the subject of section 2.4 in chapter 1: the tree S' is obtained from trees S and T by identifying each vertex of

S with a copy of T rooted at a distinguished vertex us; as stated in theorem 1.18, if S and T are gracefui, so is S', with any choice of v' in T. With a slight modification this CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 75

becomes a construction for labelling radially syrnmetric trees, but not a great many non- syrnmetric trees can be vertes partitioned into induced subtrees which are isomorphic.

This construction ivas considerably broadened by Koh, Rogers, and Tan [25], who noted that its vaIidity depended only in part on the structure of the subtrees, and otherwise on the structure of the graceful labelling they received. For trees Tl and 7'' with bipartitions into (Al,Bi) and (&, 82) and graceful labellings fi and fi,they defined fi and f2 as a beau pair if:

(1) f~(-41)= f2(-&), and similarly fl(Bi) = f2(B2),and

(2) for ulul E E(Ti) and U~U?E E(T2),if It(vI) - fi(ul) = f2(u2)- f2(u2) then either

ul f and u2 f or VI E -a1 and wz E -A2 (note that the absolute values of the differences are not used here; the condition states essentially that for each edge label the higher of the values used to generate it in each labelling must go to vertices lying in "associated" bipartition sets).

They were able to show that the formula used by Stanton and Zarnke to give graceful labellings to tree products works as well with a modified version of such products, where the n vertices of the skelton tree S were identified with not aecessarily isomorphic trees

Tl,. . . , T,. The new constraints took much (but not all) of the onus off of the structure of the subtrees and put it ont0 the particular graceful labellings involved; lett ing fs denote a g-raceful labelling of S and fi denote the graceful labelling of Ti, i = 1,. . . :n, these const raints were:

a) Each subtree Ti must have the same order, n~.

b) For some y E {O,. . . , n - 1) al1 subtrees Tiare attached to the main body via the

vertes vi f V(Ti)which is assigned the label y by fi. These y-labelled vertices take the place of the distinguished vertex v* in the original construction.

c) The subtrees attached to vertices of S which are asigned the labels i and IE(S)1 - i by fs must be a beau pair. C HAPTER 3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 76

Now for any pair of trees Tland T2 if they have locally bipartite graceful labellings fi

arid f2 then condition (2) for beau pair labellings is automatically satisfied; but to then meet condition (1) we need Tland T2 to be of the same order, that their bipartitions

split into sets of the same size, and that f and f2 assign the same labels to the high and

low bipartition sets of Tl and T2 respectively. Given any such pair of trees, we could use the Koh, Rogers, and Tan construction to obtain a graceful labelling of any tree formed

by sending an edge between vertices in each possessing the same label (we just let the

skeleton tree S be the edge P2). The question is, how often do Ive encounter trees that can be broken down into two subtrees of equal order (let aIone into subtrees which have the same sized bipartition sets)? Later in this section we dlbc loolcing at some data

that suggests that the constraints placed upon the size and structure of the tnro trees

here are in fact much more restrictive than those placed on their labellings; the following construction shows that for pairs of locally bipartite graceful trees a simple variation of

the labelling requircment is al1 that is necessary.

Definition 3.4 Let Tl and Tsbe trees with bipartitions (A1,BI) and (.A2, B2),and let the labcllings fl of Tiand f.L of T2 be bijections from their respective vertex sets lo consecutive sets of integers. We say that the labellings fl and f:! are bipartition matched if there

exzsts a set of integers {si,s2, . . . , ss) such that

and

.f2(.'2) = {O, 17.--7cr2) \ Skia? - Sk-1i---,112 - SI}

Ive further Say that the labellings fi and f2 of trees Tland T2 are bipartition sym- mctric ij fl and f2, or (equivalently) fi and fZ, are bipartition rnatched, wheze f ts the complementary labelling off as gzven in definition 1.12. Essentially, bipartition matched labellings are pairs of labellings that have the same gaps in the ranges where the labels of high and low partitions cross over; assuming, for example, that for the trees of the definition lAll > lAll:we have (abusing notation a bit)

where a = 1.44 - IAl 1 = a2- aiand (fl(Al) + a) denotes the set obtained by adding the constant a to al1 integers in fl(ril). Simply put, the high ]-A2( - a values in the image

of f2 are the those in the image of fl shifted up by a. Since the definition is restricted

to tree labellings, ive know that the missing values in fi (-41) and f2(A2)are supplied by f1(BI) and f2(B3) respectively; that is

When I.A1 1 = I.&l and IBi 1 = IB21 then any pair of bipartition matched labellings fl and f.r which are as well locally bipartite and graceful will be beau pair labeilings, but of course the nice thing about this property is that we do not require the sizes of the partitions or even the order of the trees to be the same; for example, every pair of bi partitely labelled trees are bipartition matched.

Theorem 3.5 Let Tl and T2 be disjoint trees with nl and n2 vertices, bipartitions (-41, Bi) and (-A2,B2), and locally bipartite graceful labellings fi and f2 respectively, and distinguish the vertices ulE V(Tl)and us E V(T2).If the labellings fi and f2 are bipartition matched &th fi(q) = f2(u2)+ lAl 1 - 1.421 and either ui E Al and uz E A2 or ul E Bi and u2 E B2,the tree T formeci by sending an edge from u1 to 212 has a graceful labelling. CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 78

Proof: The labelling f of T given by

is graceful (and, in iact, fully bipartite). The construction consists essentially of stretching

fi, flipping fst shifting it appropriately and then inserting it into fl like a piece of a jigsaw puzzle; figure 3.6 gives an illustration.

Figure 3.6: Example of construction for theorem 3.5

Verification of the construction: It is easy to confirm that the edges in T are a bijection to {l,2,. . . , IE(T)():f applied to Tl shifts the edge labels up by 722, covering

{n2+ 1,722 + 2, . . . , n2 + nl - 1 = IE(T)I};f applied to T2is the LB-reverse labelling of f2 with a constant added al1 vertex labels, so by lemma 3.1 (b) it covers (1, 2, . . ., nz - 1); C HAPTER 3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 79

and since fi (ul)= f2(74 + lA11 - IA21, the missing edge label n2 is supplied by the edge

Ul'U2:

3. Iful E B1,ZL~ E B:,

That the vertex labels are a bijection to (O, 1,.. . , IE(T)I)is guaranteed by the fact

that fl and f2 are bipartition matched- Let al and a2 be the highest labels in fi (Al)

and f2(&) respectively, let ,Ol and be the lowcst in fl(BI)and f2(B2),and let a = 1.111 - 1-42] = al - a?. By the definition of bipartition matched labellings, we have

d) {i: i = fl(v),v E -41,h(v) > pl) = {j: j = f2(4 + a, u E 4,f&) > iB2)

C) {i: i = f1(21), v E Bl, fl(o.) < al) = {j: j = f2(v)ta, v E B2, f2(~)< Q}

From (a), we can conclude by lemma 3.1 that f applied to Tl is a legitimate (n2 + 1)- graceful labelling Le. f (Al)and /(BI)are disjoint. From (b) and (c) we have that

f (If (S2))does not send any vertex labels outside of the range {O,. . . ,nl + 722 - 1). From (d) and (e) we know that the missing values of f (Al) = fl(h) in the critical range

pl,. . . , al are supplied by f (&), and likewise the critical range of f (BI)from ,& + nz to al+ 722 is filled out by f (A2). O C HAPTER 3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 80

Corollary 3.6 Let Tl and T2 be disjoint trees with nl and n2 vertices, bipartitions

(-41,BI) and (Az, B*),and locally bipartite graceful labellings fl and f2 respectivefy, and distinguish the vertices u1 E V(Tl)and uz E V(T2). If the labellings fi and f2 are bipartition sgmmetnc with f (ul)= 1.41 [ + [A2[- 1- f2 (24)and either ul E .+Il and u2 E B2 or ul E BI and u2 E A2, the tree T formed by sending an edge from u1 to us has a graceful labelling.

Proof: We use the construction in theorem 3.5 with either fi and f2 or fl and f2. i Since the above is a variant of the Koh, Rogers, and Tan tree product construction wherc the base tree S is P2, one may wonder about extensions to to a general tree product construction. Such an extension is indeed possible, but it involves considerable complications, so for space and aesthetic reasons it has been omitted. In the remainder of this section we will consider the prob1em of specifjing ahead of time the values that ni11 (and will not) be assigned by a Iocally bipartite graceful labelling of a tree to respective sets of the tree's bipartition. Offhand, for a tree T on m edges, bipartition into sets -4 and B, and graceful labelling f,we know that f cannot assign O and m to the same set; and if f is as well locally bipartite with O in f (-4) then we cannot have that 1 E f (B) and rn - 1 E f (A) at the same time. -4 brief look at the locally bipartite graceful labellings of small trees seems to indicate that these are the only rigid constraints; wbile for most LB graceful labellings we will see the low labels belonging predominantly to the low partition, there are some where the labels seem to be scattered almost at random. Our concern, however, is with the rnajority, where the Iow partition does take almost al1 of the low labels.

It may be useful to have some notation here. If S is a set of integers O < SI,s2, . - ., sk7 we will say that the locally bipartite graceful labelling f of the tree T with bipartition (-4, B) is LB(S) if CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 81

Note that any two labellings that are LB(S) for some set S are bipartition rnatched graceful labellings. For convenience, in the case of singleton sets we will Say LB(i) for LB((}). The class of trees that possess an LB(S) labelling will be denoted by LB(S); furthermore, we will define LB0(S),the closure of LB(S),by

LB*(S)= (J LB(R) RCS We see that within this scheme the class of bipartitely graceful trees is LB(0),while the class of ail locally bipartite graceful trees is LB*(N). Our interest in this section is in labellings with easily specified assignrnent of labels to partitions; the simplest of these after bipartite graceful labellings are LB(1) labellings: if a! is the highest label in the low partition, and p the lowest in the high partition, ive have cu = P + 1, and the other labels distributed to the low and high partitions according to whether they are < a or not. The tests which have been run to date have looked at the classes L B(1) and LB8(I)=

LLS(1) U tB(0). Since the nurnber of trees of order n that do possess LB(1) labellings makes up large majority of the total, the prograrns only kept track of counter-examples, the numbers of which are more manageable. Table 3.4 gives the numbers for trees of order n 5 l'i which have no bipartite graceful labelling, no LB(1) labelling, and no labelling of cither type: WC can sce that the number of trees nithout LB(1) labellings is growing much more slowly than the number of trces without bipartite graceful labellings, with the former down to about 15% of the latter wlien n = 17 (admittedly, considering positive cases this represents only about a 1% difference for LB(1) trees over bipartitely graceful trees). As

NT can see, a significant number of the LB(1) counter-examples do have bipartite graceful labellings (some were to be expected, from section 2.2); trees in CB'(1) constitute les than 10% of al1 trees in m(0). This gives us some reason to be confident that any two trees picked at random will in the vat majority of cases have LB(I) labellings, and hence be fodder for the construction given in theorem 3.5; for orders n < 17 alone there is more than a 99% CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 82

Trees =(a) D(1) LB'(1) 1 O 1 O

TabIe 3.4: Trees of order n without bipartite graceful, LB(l), or either labelling chance that ariy pair of trees dlboth be in LB(1) . For each pair of trees Tl and

T2 having bipartition matched graceful labellings fl and f2y where the bipartitions are respectivcly (-41, BL)and B2),we can in this manner obtain graceful labellings for min(!rli1, 1-421) + min(lBl 1, IB21) larger trees (up to automorphisrn): for every k E f2(A2) wliich is 2 (-44 - IA1I ive will always have k + +A1l- 1&1 E fi(Al), and similarly for consider ail trecs in CB'(1) we have even more options, since the canonical amalgamation Iabelling given in theorem 1.14 (or one of its several variants) can be applied when one of the subtrees involved has a bipartite graceful labelling and the other has any graceful labelling.

Unfortunately, when we consider the growth rates for the number of counter-examples, - - those for tB(1) and LB'(l),while slower than that for LB(0),seem to be exponential as well. This makes the possibility somewhat remote that a finite set of configurations will be able to cover al1 trees i.e. that there is a finite set of integers S' such that al1 trees are in Li3*(Sr). Other possiblities, it should be mentioned, are still wide open; for esample, the possibility that there is a finite k such that every tree belongs to CB(S") for some S" with [Srri5 k. More mrk needs to be done in this vein.

3.4 Some explicit constructions for trees which are

not bipartite graceful

In this section we dlreturn to the robustness theme of section 2, from a theoretical standpoint. Constructions for locally bipartite graceful Iabellings for two infinite classes

\dl be given; for neither of these classes is a fully bipartite graceful labelling possible. These classes would not be considered "large" with respect to the set of trees in general (which lias as much to do with the problem of finding trees with no bipartite graceful labelling as anything), but together they cover some cases of trees with arbitrary maximum degree and trees with arbitrary diameter.

3.4.1 A construction for k-cornets

Recall from chapter 1 that the k-cornet is defined as the tree obtained by subdividing every edge of a k-star (a generic k-cornet is shown in figure 3.7). From tlieorern 1.16 (restated at the head of this chapter) al1 trees of diameter 4 are either caterpillars or have no bipartite graceful labelling; hence we know that for k > 2 no k-cornet has a bipartite graceful labelling. The 3-cornet is in fact the smallest tree which does not have a bipartite labelling.

The empirical evidence seems to suggest that for k 2 5 no k-cornet possesses an LB(1) labelling either.

Theorem 3.7 Evenj k-cornet T ha a locally-bipartite graceful labelling. Figure 3.7: A k-cornet

Proof: For k f 2 (mod 3) ive provide the foltonring construction. Let T have the

vertes set

where w is the centre vertes, each vi is adjacent to w and vi, and each u, has degree one; hcnce the vertices of T have the bipartition into {w) U {ui}l

If k is odd,

f(w) = 0 f (q) = 2k - L4i-331 f(%) = 2k - 2i, i is odd, i < k = [y],i is even, i < k f(uk) = r5k-1 (= f(4- 1) If k is even,

/(ut) = as above

/(ci) = as above

f(ui) = asabove, i

- L~$J, i is odd, r < i < k = 2k-22, iiseven,r

/(ur) = "2 k f 2 (mod 8)

= *+l:2 k E 2 (mod 8) I(fJ (= L+] ) in either case f (uk) = as above

The idea is that in the high partition, labels decrease by one going frorn z.1 to vk, skipping a label at vs and every third vi after that; this progression wraps around at vk, going from ut to u2 in the same manner, with values we have skipped interleaved going in the opposite direction. When k is even we need to jog the progression a bit at around the f k point to get things to fa11 into place. Figure 3.8 gives a pair of esamples.

Figure 3.8: Construction from theorem 3.7 applied to the 9-cornet and 10-cornet If k = 2 (mod 3), there is no locally bipartite labelling of the k-cornet with f (w) = 0, sincc any such labelling must satisfy

2 C f (q) - C f (ui) = C edge labels = C vertex labels = C f (vi) + C f (ui)

which implies that sum of al1 vertex labels is divisible by 3. However, when k = 2 (mod q,?"(?'l) F 1 (mod 3).

In this case, we can obtain the desired labelling f of T as follows: Let f' be a locally bipartite graceiul labelling of the (k- 1)-cornet Tt = T - {vl) - {ul)such that ft(w) = 0. Define f by: f(ud = 2k

f(W> = 0 f(u) = f'(u)+l ifu E V(T1)

It is easiiy seen that f is locally bipartite and graceiul if f' is; and if we can confirrn that the construction given earlier works for k O or 1 (mod 3) then we always have an available f' for this case.

Verification of the construction for k O or 1 (mod 3) For conwnience, u7ewill define the following subsets of vertices in T:

1. f is locally bipartite: We have by definition f (uk) < f (uk) and (where applicable) CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 87

f (u,) < f (u,). Where ui E B, we have

Where .rti E C, we have

2. The mapping f : V + N is a bijection: We start by noting that:

1. f maps -4 to {2k, 2k - 1,2k - 3,2k - 4%- 5,3k - 7,. . . , r$k]}Le. counts down from 2k, missing 2k - 2 and every 4th value thereafter.

2. f maps Bodd to {2k - 2,2k - 6,2k - 10,. . . , *} terminating at f (ur-2) or f (u,-2) depcnding on the parity of k.

3. f maps Ce,, to {1,2,3,5,6,7,9,. . ., *} terminating at /(u~-~)or f (w-,) depending on the parity of k.

If k is odd, then 2k - 2 G O (mod 4), so the sets ,4, Bdd, and Cm,, partition

{ 1: 2' . . . 2k} and f is onto. Othenvise, Our labelling has to incorporate a "correction" at u, in order to get the -4and C sets of labels to meet up properly, since C starts off by skipping al1 O (mod 4) values on the way up and A in the b even case misses al1 2 (mod

4) values on the way down. The value of r itself is chosen so that for vertices in both

B and C we have f (ui)approaching f (u,) as i approaches r; the principle then is that jogging the indices at r keeps u, out of both B and C and f (u,) from being assigned to any of t heir vertices.

The actual point at which the jump cornes will not be the same for al1 values of k; CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 88 considering the cases k (rnod 8) we have:

f (G~1 = {l,2,..-, $+T-2]\.{j: j=O (rnod4)) f()= {$+r,$+r+l,--.72k}\{j: jr2 (rnod?)}

f (Bmm) = {4,8, ...,$+6-6) = {j: j 3 O (mod 4) and O < j < 5)

f (Bodd) = ($+6,5+6+4, ...,2k-2}

= {j : j 2 (mod 4) and 5 < j < 2k} where our values for y and 6 are:

k (mod 8) 7 6

O 12

2 O 5

4 14

6 13

We note that in eveqr case but one Soth the B and C labelling sequences jump over f (u,) entirely as they transfer from even to odd indices. The exception occurs for C when k 2 (rnod 8); here ive have f (u,) = $ + 1, while y is given as O and + -, + 1 is the secorid term that appears in f (Codd).This will not, however, cause us any problems: if k G 2 (mod 8) then + 1 = 2 (mod 4), so this value is not assigned to any vertex in Caddanyhow. f (A U B U C) hence covers al1 of {1,2,. . . ,2k) except for 5 or + 1 depending on whether k = 2 (mod 8); by the construction, this missing value is supplied b~ f (4

3. The edge labelling g induced by f is onto: For each edge wvi, we have g(wui) = f (vi),so it is sufficient to show that the sequence { f (vi) - f (u~))~=~,~,~.~+is a permutation of {f('k)}i=l,2, ...,k- CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 89

If ui E C:

- f(vi) - f(ui) - 2k - 191- i?J = 2k - 2i+ 1 - - l*J where b = i (mod 3)

= 3k-2i+2 checking cases b = 0,1,2

= f(ui-i) ~i-1E B

If E B, 2 5 i 5 k - Li, and (when applicable) i < r - 4 or i 3 r + 2:

wfiere b = i (mod 3)

Thcse two cases leave the vertices u2, ur, u,+2, and uk, and the edges u,-2ur-2, v,u,, vk-2uk-2: and 't'kui; unâccounted for.

U7c have:

f (4- f (~k) = 1 = fb2) 4(k-2)-3 (u) (2) - (2k 2(k - f = 2k 1 3 J - - - 2)) = k-1-LyJ

= rfkl since k f 2 (mod 3)

= fbk) C HAPTER 3. BIPARTITEAND LOCALLY BIPARTITEGRACEFUL LABELLINGS 90

When k is even, by checking cases k (mod 8) we can confirm that

ik+1 k~O(mod8) f(~,-~)- f(~,-~) = ryi k r 2,4,6 (mod 8)

Iicnce, if k O or 2 (mod 8)

and if k r 4 or 6 (mod 8)

3.4.2 A construction for certain lobsters

In the previous subsection. we dealt with a subclass of trees having arbitrary maximum degree. Hcrc we n-il1 give a construction for a subclass having arbitrary diameter:

Definition 3.8 The tree L(k), the "schematized lobster u;ith k legs", ts obtained from the path of length k + 1 by adding a pendant edge to each of the k vertices of degree 2, and then szlbdiuiding every edge in the r~sultinggraph. CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 91

Figure 3.9: L(5), the schematized lobster with 5 legs

Since the intermediate tree obtained by adding pendant edges has (k+Z)+k = î(kf1)

vertices, atl of odd degree, by theorem 1.27 L(k) has no bipartite graceful labelling when k is odd. It should be noted that, while the diameter of L(k) increases with k, the ratio of the diameter to the number of edges never goes above f, and approaches $ as k -+ oo; so our previous speculations about the relation between bipartite graceful labellings and diameter are not invalidated here. The thcoreni says nothing about the case when k is even, since then the number of

vertices in the base graph does riot violate the n $ O (rnod 4) parity condition. In fact,

the cven members of this class do have a (fairly simple) bipartite labelling.

Theorem 3.9 The tree L(k) has a fully bzpurtite labelling when k is even.

Proof: Wc will start by dividing the vertex set of L(k) into 3 sets, based on degree:

V = verticcs of degree 3

U = vertices of degree 1

1V = verticcs of degree 2

Thus L(k) has a bipartition into sets IV and (U u V). In each set the individual vertices will be given the following ordering:

{duo, vi, VÎ,. . . , vk, UL+I) wi11 be the vertices on the original path from which Our L(k)

is derived, with uo and uk+~being endpoints adjacent to vl and vk respectively, and

the v,, I 5 i < k, being adjacent to ui-1 and vi+l otherwise. CHAPTER3- BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 92

u,, 1 5 i < k, will be the degree 1 vertex created by attaching the pendant edge to

L'i.

~2~ will be the vertes created by subdivision of the edge between ui and vi,

rn wzi+~will be the vertex created by subdivision of the edge betwee vi and ui+lr

r wl lays bctween uo and vl in L(k), and

Figure 3.10: Layout of ui, ui, and wi in L(k)

The labelling f is thcn simply a function of the ordering n-e have imposed upon the verticcs within each set:

f (v,) = i - I 214

= i+k+1 i>zk

f(ui) = 23k-i i is even = :k+2-i- iisodd f(wi)= 4k+3-z

AS figure 3.11 makes clear, starting at one end uTelabel the degree 3 vertices on the spine with values starting at O until ive get half way, and then jump to the other end and work our way back through the degree 1 vertices in pairs; when done, we finish the degree three vertices (going in the original direction), then work our Ray back through the degree 2 vertices in order. CHAPTER3. BIPARTITEAND LOCALLY BIPARTITEGRACEFUL LABELLINGS 93

Figure 3.1 1: Construction from theorem 3.9 applied to L(4) and L(6)

Let Li, = {vi: i 5 g}, let Vhighbe V \ Km, let Li,, = {ui:i is even) and let Uodd= U \ Ueuen-Tt i~ easy to confirm that:

Which covers {O, 1, . . . , Ili + 2 = IE(L(k))1); hence the vertex labelling is a bijection. iI:e also sec that every label in /(VU CI) is 5 2k+ 1, rvhile every label in f (W) is > 2k + 1, so the labelling is bipartite. In the same vein, we can see that the edge labels are a bijection to {1,2, . . .,4k + 2).

The following caçes are disjoint and e-uhaustive:

1. Edges between IV and 6,: CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 94

Since 4k - 3i + 4 - (1) = [?k - 3(i + 1) + 1 - (-1)] + 1, each triple of edges pendant on vi is labelled distinctly from those on z;i+~; al1 triples taken together

cover {$k ;3, :I; i 1,.. . ,4ki 2}.

2. Edges between W and Vhigh:

These similarly cover {l,2,. .. , qk).

3. Edges between IV and U:

These respectively account for the sets {$k + 3, $k + 5, $k + 7, . . . , $k + 1} and

{$ + 2, $k + 4 , qk + 6,.. . ,2k)-5

4. Finally, the last two edges:

fil1 in the remaining gaps.

When k is odd, there are certainly enough locally bipartite labellings of L(k) which bcar some rcsemblance to the one above, but nothing which has led to any ground up construction of this sort. Instead, we will provide a recursive construction for a slightly stronger labelling, an extension of the LB(1) labelling from the last section: CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 95

Definition 3.10 Let G be a bipartite graph with bipartition (A,B). The labelhg f of

G is LB(1) with the tail {x,y, z} if

1. / zs LB(l),with A the low partition and B the high partition.

2. There are vertices x, y, and z such that

a) z, y, and z fonn a path protruding from G, cg. degsee(z) = 1, z is adjacent

to y, degree(y) = 2, and y is adjacent to x and z,

b) y E B, hence x and z E A

c) f (y) = 1-41 - 1 (2.e. the lowest labelled vertex in the high ~artztion),f (x)=

f (Y) - 1, f (4= f (Y)- 3-

Lemma 3.11 If L(k) has an LB(1) labefling with the tazi {x,y, r}, where z is an endpoint of a longest path in L(k), then L(k + 1) has an LB(1) labelling with the tad

{xf,y', 21, where 2 is an endpoint in a Iongest path of L(k + 1).

Proof: Let L be any tree tree isomorphic to L(4). We will start by giving a particular range-relaxed graceful labelling for L, which will be denoted by IL:

Figure 3.12: L(4) with labelling fL

Ive note that the range of the vertex labels for fL is O to 19, with 3 being the onIy value missing. The edges take the values {1,2,. . .,18} with no gaps. Note also that the labeliing is locally bipartite, with Il being the highest label in the low partition, and 10 being the lowest iabel in the high partition. In the following we wïll denote the vertices labelled with 2, 19, and 11 as x", y", and z" respectively, and the vertices labelled with

9, 10, and 8 as x', y' and 2'.

Now let T be a tree isomorphic to L(k) with bipartition (A,B), and with vertices x, y, and r such that z is an endpoint of a longest path in T, y E B is the degree 2 vertex adjacent to z, and x is the degree 3 vertes adjacent to y. Let T* be the tree forrned by identifying the path x - y - t in T with the path x" - y" - z" in L, as shown in figure 3.13; T*is isomorphic to L(k + 4):

Figure 3.13: Formation of T* from T and L

If fr is any labelling of T that is LB(1) with the tail {x,y, z), the labelling f of T' is dcfined by:

The labelling f is LB(1) with the tail {x', y', z'}. We note that the labels fT give y and z are thrown away (they are supplied via fL),while f (x)= fT(x) = fL(zf')+ 2k - 1.

Verification. Since by assumption fT is LB(1) with the tail, T has a bipartition into sets (A, B) with B çontaining the degree 2 vertices and A containing the odd degree CHAPTER3. BIPARTITEAND LOCALLYBIPARTITE GRACEFUL LABELLINGS 97 vertices. This gives us 1-41 = 3k + 2 and [BI = 2k + 1, with the tail labels then being:

The highest labei given to the low partition A is [Al = 2k + 2, which is given to some vertes u somewhere else in T;hence fT(.4 \ (u,x, z}) covers al1 values from O to 2k - 2. Li kewisc, fr(y) is the lowest label in the high partition B, and only u7slabel is higher in

-4. so fT(B \ {Y}) COV~Wal1 values from 2k + 3 to 4k + 2 = IE(T)I. Thus we have

Since T* has 4(k + 4) + 2 = 4k -+- 18 edges, the vertes labelling is a bijection.

Considering the edge labelling g~ induced by fT, we note that the edges xy and yz provided the values 1 and 2, so that the remaining edges would have taken al1 values between 3 and Ik + 2 inclusive. The action of f upon T - {y} - {z) is simply to add the constant 16 to each of these labels, covering (19,20, .., 2(k + 4) + 2}. Since f merely shifts al1 vertes labels given to L by IL by a constant, the range of the edge labels is urichanged; this range, as mentioned above, is {l, 2, . . . ,181. Hence the edge labelling g induced by f is also a bijection. So we have that f is graceful.

Finally, let (A', B') denote the bipartition of T* such that the B' contains the degree

2 vcrtices (hence y' E B*). Recall that in the labelling fL of L, fL(zM)= 11 was the highest label in the low partition of L, and fL(yl) = 10 \vas the lowest label in the high partition. Since f(zn) = 2k + 10 > 2k + 2 = the highest label in f(.4), f(9)is the highest label in f (A*); and since f (y') = 2k + 9 < 2k + 19 = the lowest label in f (B), f (y') is the lowest label in f (B*).So f is LB(1). Now 1-4'1 = 2(k + 4) + 2 = 2k + 10, so we have f(y') = 2k+9 = 1-47 - 1 f (x') = 9 + 2k - 1 = f (yr)- 1 f(2) = 8+2k-1 = f(yr)-2

Hence f is LB(1) with the tail {XI, y', z'}.

Theorem 3.12 For al1 k 3 O L(k) has a locally bipartite labelling.

Proof: By theorem 3.9 above, ive have that for k e~~enL(k) has a bipartite and hence a Iocally bipartite labelling. L(1) is the 3-cornet, and so that it is locally bipartite graccfui follows from theorern 3.7 earlier in this section. For the remaining L(k), by lemrna 3.11 it is sufficient to provide labellings for L(3) and L(5) which are LB(1) with the appropriate tails; these are given in figure 3.14.

Figure 3.14: LB(1) labellings (with tails) for L(3) and L(5) Chapter 4

O-Centred and m-Edge-Centred

Gracefulness

In the last chapter ive esamined some larger aspects of the structure of graceful labellings,

with respect to how vertes and edge labels in aggregate interact with the bipartite struc-

ture of a trcc. Here we will deal with some questions relating to the assignment of

specified Iabels to specified vertices; our particular focus will be on where the label O can bc assigned.

4.1 O-centred and m-edge-cent red graceful labellings

4.1.1 Gracefully assigning the O label

Whcn one sets about to actually label a graph gracefully, the first question is "Which label goes where?" Here we wilt look at some possible ways of answering the question. For the most part we will be concerned with where we can assign the O label; this is a natural starting point for the more gencral investigation since every gracefully labelled C HAPTER 4- O-CENTREDAND ~EDGE-CENTREDGRACEFULNESS 1O0

graph has a O-labelled vertex-

There are some things we can say immediately, which follow either from the definition of gracefulness or some well knmn folklore constructions. If G is a graph on m edges, v is a vertes of G, and f is a labelling of G:

- If f is graceful and f (v) = O, then there exists a vertex u in the neighbourhood of 2: suc11 that f (a)= rn; this is the only way the edge label rn can be generated.

0 If j is graceful and f (v)= O, there is a graceful labelling f' of G such that f'(u) = m; the converse is also true. This follows from the existence of the complementary

labelling (definition 1.12).

0 If G is a tree, f is a bipartite graceful labelling with f (v) = 0, and a is the size of the bipartition set containing v; then there are bipartite graceful labellings /' and f" of G such that ff(v) = a - 1 and f"(u) = m - a + 1. We can also infer bipartite graceful labellings of G with v labelled O if ive have any with v labelled a or m - a + 1. This follows from the existence of the reverse labelling (definition 1.13).

0 If the degree 1 vertex u is adjacent to v, and the graph Gf = G - {u} has a graceful

labelling f' such that f'(v) = O, then G has a graceful labelling such that u is

labelled O and u is labelled rn (which is merely f' augmented by the assignment of

the label m to u).

That thcre are trees which have vertices which cannot be gracefully labelled O is also ml1 known; the first counter-esample, of order 6, is smail enough to stumble upon by accident (see figure 4.1). This might Iead one to believe that for larger n such counter- esamples will be plentiful, but as we shall see that is not necessarily the case.

Some work (though not a great deal) haç been done on trees for which every vertex is assigned O in some graceful labelling. Most notable are the following two results: CHAPTER4- O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS

Figure 4.1: The smallest countcr-example

0 Rosa [30] showed t hat for every vertex u in the path Pk there was a graceful labelling

f of Pk such that f (u) = 0, and that with one exception (the centre vertex of Ps) there was in fact a bipartite graceful labelling f' of Pk such that ff(v) = 0.

-4s mmentioned in chapter 1, Chung and Hwang [12] showed that if two trees have this same property then the product of the two trees had this property. In their

terminology such trees were called rotatable graceful.

-4s wcll, Abrham and Iiotzig have Iooked at the assignment of non-0 labels to the end vertices of paths [l-L].With respect to connected cyclic graphs there are some theorems and conjectures concerning labels that are neuer used in any graçeful labelling of a given graph. For esample, Grace [18]proved a conjecture of Frucht that in al1 graceful labellings of CnO KI (a "polygon with pendant points") the ommitted label is even.

4.1.2 Some empirical results

Motivated in part by the smallest counter-example, and partly by the fact that every trce has either a unique centre vertex or unique centre edge, we will begin by looking at graceful labellings of trees where the centre vertex is labelled O; or, more precisely, by looking for trees which have no such labelling. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFWLNESS 102

Definition 4.1 Let T be a tree on m edges. If T has euen diameter with unique centre

vertex û., we Say the labelling f of the vertices of T is O-centred if f (v) = O. IfT has odd

diameter with unique centre edge UV, we say f is O-centred if f (u)= O or f (v) = 0, and

that f is m-edge-centred if I/(u) - f (v)1 = m.

Our intcrcst then is in O-centred and m-edge-centred graceful labellings. A convention ive will adopt is that if a tree T on m edges has odd diameter we will only cal1 it O-centred graceful if there esist O-centred graceful Iabellings for both of the vertices on its centrai cdge. Note that a tree of odd diameter can be O-centred graceful in this sense yet not rn-cdgc-ccntred graceful. -4 second con\*ention will be that we dlnot consider trees of cven diameter "non-m-edge-centred graceful" in the sense of being counter-examples, since the definition has xio relevance in this case; in questions conccrning m-edge-centred graceful labellings only trees of odd diameter will be considered.

The first bank of tests checked for the esistence of O-centred and m-edgc-centred graceful labellings for trees of order n < 18. The prograrns also checked whether there werc 0-centred and m-edge-centred graccful labellings which were either locally or fully bipartite; for fully bipartite labellings the tests only go up to order 16, due to the many ncgative instances encountered (these are more time-consuming for the program). The rcsults are given in table 4.1.

The first thing that catches the eye is how astoundingly small some of the numbers are. In particular, we sec that the nurnber of non O-centred graceful trecs does not even monotonicallÿ increase with the order, n; it just secms to jump about, and never makes it above 7. But if we take a look at the actual specimens, there is a very pronounced pattern, as the reader can see from figure 4.2. In fact, every one of the 26 negative instances of order 5 18 has diameter 4 and centre degree 2. In the nest section we will Iook more closely into this. C HAPTER 4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS

No m-ECGL Trees Odd D B Any LB B

Table 4.1: Non-O-centred and non-rn-edge-centred graceful trees of order n

-4lniost as eyebrom-raising is the number of odd order trees without an m-edge-centred graceful labelling; it is of roughly the samc magnitude as the number of non-0-centred g-raccful trees, but ive see an increase of precisely one at each step, starting at order

9. 1 personally thought this so intriguing that 1 ran the test for m-edge-centrcd gencral graceful trees of order 19 as well to see if the pattern held; it did. Looking at particular negativc instances here ive see a strong resemblance to the counter-exarnplcs for O-centrcd

Figure 4.2: First 3 non-0-centred graceful trees CHAPTEK4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 104

graceful labellings: they al1 are of diameter 5, nith the vertices of the centre edge having degrce 2; figure 4.3 gives the first several encountered.

Figure 4.3: Smallest 3 non-m-edge-centred gt-acehl trees of odd order

In the even orders the results were more along the espected lines. While the numbers are still vcry small relative to the number of trees in general, there does seem to be an csponential growth rate. Particular instances here are more heterogeneous; trces likc

thosc above, and some trees with diameter 3 stand out. Later in this section 1 will give two theorenis about m-edge-ccntred graceful labellings (4.9 and 4.3) which account fully for both classes of trees. The remaining counter-instances in the cven orders (which

consti tiite the majority of cases) did not provide such easy handles for theoreticai ivork, but they do seem to sharc a common property: every longest path terminates in a "porn-

pom" with an even number of leaves Le. the second and second last vertices in every lorigcst path have odd degree 2 3.

\['hile the numbers of non-0-centred and non-m-edge-centred Iocally bipartite gracefiil trees were also very small, the particular instances did not show as much similarity; for esample, in the vertex case most have diameter 4, but there is a scattering of higher diameter trees, and a sizable contingent with centre degree > 3. In both the edge and vertes case the numbers of offenders seem to be shadowing the numbers for general graceful labellings, rising and falling at the same time, but the differences show no discernable pattern other than to (not at al1 consistently) become more pronounced; the increase is not rnonotonic even when we split into even and odd cases. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 105

\Yi t h non-0-centred and non-m-edge-centred bipartite graceful labellings we see more substantiai growth; for this reason, as mentioned above, tests urere tenninated after order

16. In both cases, however, the number is a rninority of dl trees, and the proportion secrns to be decreasing fairly straight-fomardly after order 9. Now trees without any bipartite graceful labelling obviously cannot have any O-centred or m-edge-centred bipartite graceful labelling, so one might ask how their numbers relate. As we can see from table 4.2, trees without any bipartite graceful labelling make up about 32% of those without a O-centred bipartite graceful labelling, with the proportion seeming to be somewhat stable. The number of non-rn-edge-centred bipartite graceful trees, on the other Iiand: seems to be puliing away from the number of non bipartite graceful trees of odd diameter, albeit slowly.

No O-C O O 1 2 6 11 24 43 83 147 273 450 798 1342

Tabic 4.2: Non-bipartite graceful vs. non-O-CBG and non-m-ECBG trees

4.1.3 Two theorems about m-edge-centred labellings

Before we turn Our attention to the main results of this chapter concerning non-0-centred graceful trees? we will confirm some of the observations prompted by the empirical data CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS

ive have on non-rn-edge-cen t red graceful t rees.

Theorem 4.2 Let T be a tree of diameter 3 with m edges. Then T has an rn-edge-

centred graceful labelling if and only if the vertices on the central edge do not bath have

even degree.

Proof: Let f be an m-edge-centred graceful labelling of T, with u being the vertex Iahelled O by f, and v being the the vertex Iabelled m. Let -4 be the set of degree 1

\-ertices adjacent to u,and let B be the set of vertices adjacent to u; IAl = degree(u)-1 and 1 B 1 =degree(v)-1.

Figure 4.4: m-edge-centred labelling of diameter 3 tree

Every other vertes in T takes a Label s with I 5 s 5 m - 1, with each pair of labels

(s,m - s) necessarily assigned to pairs of vertices adjacent to just one of u or u. If 1.41 and IBI are both even this can always be done by assigning 2 of the pairs arbitrarily to vertices in A, and the rest to vertices in B. If 1-41 + [BI is odd, then rn = 1-41 + 1B1 + 1 is even and 7 = m - "-2' hence the label ? can be assigned to a vertex in ivhichever of -4 or B have an odd number of vertices, and the rest divided as before. In each of these cases we always obtain a graceful labelling.

On the other hand, if (Al and [BI are both odd, we are Ieft with one pair of distinct labels (s,m - s) which cannot both be assigned to vertices in one of the sets; hence any m-edge-centred labelling of T must contain at least one repeated edge label, and cannot be g-raceful. rn CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 107

We note that the ungraceful case can only happen when the order of the tree is even, giving rise to [yJ distinct trees without m-edge-centred graceful labellings. The next theoren deals with a more interesting class of trees, those rvhich coastitute our only counter-esamples for odd n 5 19.

Theorem 4.3 Let T be a tree of diameter 5 on m edges, un'th the degree of both vertices

on the centre edge equal to 2. Then T hm an rn-edge-centred graceful labelling if and only

if at least one other vertex of T has degree 2 i-e. at least one of the branches terminates

in a single leaf.

Proof: Let f be an rn-edge-centred labelling of T, with u being the non-central vertex adjacent to the central vertes labelled 0, and v being the non-central vertex adjacent to

the central vertes labelled m. Let -4 be the set of degree 1 vertices adjacent to u, and B be the set of degree 1 vertices adjacent to u.

Figure 4.5: m-edge-centred Labelling of diameter 5 tree

We can assume that f (u) = m - 1, since othenvise f (u) = 1 and we can use the complementary labelling f instead. If IBI = 1, f can be finished gracefully as in figure

4.6:

Otherwise: Il31 > 1. Let f (v) = r for some r between 1 and m - 1; the following cases show that no value of r allows a graceful labelling. CHAPTER4. O-CENTREDAND m-EDCE-CENTREDGRACEFULNESS

Figure 4.6: Labelling for case 1 BI = 1

CASE: r = 1. This is impossible, since then we have tnro distinct edges labelled m - 1.

CASE: 1 < r < rn - 3. Since (m - 1) - (r - 1) = m - r: the vertex labelled T - 1 is necessarily adjacent to v. This gives us Our edge nith label 1, forcing r + 1 to be

assigned to a vertex adjacent to u and m - 2 to be assigned to a vertex adjacent to W. But (m - 1) - (r+ 1) = (m- 2) - r, so we again have duplicate edge labels.

CASE: r = m - 3. Xs above, r - 1 = m - 4 is assigned to a vertex adjacent to u; but wc tiien have (rn - 1) - (rn- 2) = (m- 2) - (m- 3) = 1 as well, so the label m - 2 cannot be assigned to any remaining vertes without causing a conflict.

CASE: T = m - 2. If the vertex labelled s is adjacent to u then the vertex labelled s - 1 is aiso adjacent to u, since otherwise ive have distinct edges labelled m - 1 - s and rn - 2 - (s - 1). Now m - (m- 2) = (m- 2) - (771. - 4)? so m - 4 and al1 lower labels must be assigned to vertices adjacent to u;this leaves only one value, m - 3, as a label for vertices in the set B,but by our assumption IBI > 1.

The number k can be written as the unordered sum of 2 numbers greater than 1 in exactly ways; hence 191trees of order n > 6 are not m-edge-centred graceful by theorem 4.3. This accounts for al1 the known non-m-edge-centred graceful trees of odd order n. CHAPTER4. O-CENTRED AND m-EDGE-CENTREDGRACEFULNESS

4.2 O-centred graceful trees of diameter 4

4.2.1 Trees of diameter 4 with centre degree 2

As we can see from figure 4.2: while al1 trees we know of that lack a O-centred graceful labelling have diameter 4 and centre degree 2, it is not the case that al1 trees with

diameter 4 and centre degree 2 lack a O-centred graceful labelling. In fact, from Stanton

and Zarnke's [37] construction for radially symmetric trees (an example was given in

chapter 1, figure 1.11), we know that any tree of diameter 4 for which al1 vertices adjacent to the centre have the same degree has a O-centred graceful labelling. The following theorem will provide a complete characterization of trees of diameter

4 and centre degrec 2 in terms of û-centred graceful labellings. Here and throughout the

rest of the chapter a branch -4 will refer esclusively to a vertex u adjacent to the centre

plus any degree 1 vertices adjacent to v, and 1-41, the size of -4,will refer to the number of degree 1 vertices (or equivalently, the number of edges) in A; a trivial branch will be one consisting solely of a degree 1 vertes adjacent to the centre: while euen and odd branches will be branches with even and odd sizes respectively.

Theorem 4.4 Let T be a t~eeof diameter 4 having 2 branches A and B, with 1-41 2 \Bi.

T hus a O-centred graceful labelling if and only zj there exist zntegers x and r such that

where 2

O 5 x 5 min(r- 1,IBI)

x is euen if T is odd; its parity is not constrained otherwzse.

Proof: As we shall sec, every O-centred graceful labelling of T can be constructed using acceptable values of x and r, with some room for arbitrary choice if x > 1 and CHAPTER3. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 110

r > 2. Let T be a diameter 4 tree Rith centre degree 2 and m edges, let u and u be the

vertices adjacent to the centre, and let A be the branch containing u and B the branch

containing u. Let f be any O-centred graceful labelling of T. For the time being ive mil1

relav the assumption that 1-41 2 IBI, and instead assume that f (u)= m; as will becorne clear later, the m label can only go to the larger of the two branches if they differ in size. Let f (u) = m - r, where r is any number between 1 and m - 1.

Figure 4.7: O-centred labelling of diameter 4 tree with 2 branches

Our first step is to determine the acceptable values of r. As it turns out, as wll as

satisfying the bound given in the theorem, r must divide rn; the particular value chosen for r dlalso force at least half of the remaining label assignrnents:

1. r 7. Since rn and n - r have already been used as labels for non-adjacent vertices, the orily way we could obtain an edge labelled r is to label a degree 1 vertes in

B with rn - 2r; however, if r > 7 we have rn - 2r < O, so that is not a possibilit- 2. r divides m. The vertex labels between m - r + 1 and m - 1 cannot generate any edgc labels with m or m - r that are > r - 1, so if the edge label m - s > r cannot be gencrated by assigning s - r to a vertex in B,the label s must go to a vertex in

-4. Since by necessity the labels 1,2, . . . ,T - 1 are assigned to vertices in A in order to generate edge labels m - 1, m - 2, . . . , m - r + 1, we are forced to assign al1 vertex labels cr + 1, cr + 2,. . . , cr + r - 1 for c 5 - 2 to A in order to generate the edges m - cr - 1:. . . ,m - cr - r + 1; that is, al1 labels less than rn - r and f O (mod r). On C HAPTER 4. O-CENTREDAND m-EDGECENTREDGRACEFULNESS 111

the other hand, the label r must go to B; and if the label s < m - r is assigned to B, the label s + r cannot be assigned to -4, so al1 miiltiples of r which are greater than O

and less than m - T aïe as well given to vertices in B.

Now in order to generate the edge label r itseIf some vertes adjacent to u has to be

labelled rn - 2r. However, if m - 27- $ O (mod r) this label has already been assigned to

a vertes in -4 in order to generate the edge label 2r. If f is graceful, therefore, it must

be that m - 27- is a multiple of r (note that this includes the case where the 27- = m).

3. 1 < r. By the above we have that al1 s in the range 1,. . .,rn- r - 1 that are

multiples of r must be assigned to the branch B. However, if r = 1 this is simply al1 the

intcgers between 1 and m - 2 inclusive, leaving no iabels for the branch A.

The assignmcnt of al1 labels < rn - r has been forced by Our choice of r; at this point, thoiigh, we still need to assign the r - 1 labels between m - r and m. We start by noting

t tiat if rn = kr, there are k - 2 distinct multiples of r greater than O and less than (k - l)r,

and (k - l)(r - 1) non-multiples. This accounts for our size constraint, since even if ive can in fact assign al1 the remaining labels to B we still haïe k - 2 + r - 1 5 (k - 1) (r - 1) for a11 values of k, r 2 2. Assuming that in fact (k - l)(r - 1) 5 -4 and k - 2 5 B, ive

define x: O

i.e. x is the number of high labels which are assigned to branch B. Solving the second of these for k and substituting into the first gives us the espression in the theorem.

The high labels can be given to either branch arbitrarily with one proviso: for each

s, 1 5 s 5 r - 1, the labels m - s and m - r + s must go to the same branch. This accounts for x's parity condition. If r is even, we have

1. Esactly one of x and r - 1 - x even, and one odd. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 112

2. .-;=mmT+f, so the label m - 5 can be assigned to whichever branch has the odd leftover without worry.

If, on the other band, r is odd, the values m - r + 1,. . . , m - 1 make up 9 full pairs, so x and T - 1 - x had better both be even if we uTantto assign these labels gracefully r

From this condition for O-centred gracefulness we can, for any diameter 4 tree T with nz edges and branches .A and B, conclude such things as the following:

1. T has no O-centrcd graceful labelling if rn is prime. 2. T always has a O-centred graceful Iabelling if m - O (mod 4) 3. T always has a O-centred graceful labelling if both both its branches have even size.

To obtain (2) and (3), we set r = 7 and x = IBI; the condition from theorem 1.4 is satisfied since either r or x will in that case be even. Given the empirical evidence, it is tempting to speculate that theorem 4.4 in fact charactcrizcs al1 non-0-centred graceful trees. \Ve are, of course, in no positition to go so far at this time, but a natural generalization to other trecs of diameter 4 is immediately at Land. Dcnote the class of trees of diarneter 4 having no O-centred gracefui labeliing as

D. It will be shown in the nest subsection that V consists precisely of al1 trees having centre degree 2 which fail the conditions imposed by theorern 4.4.

4-22 Other trees of diarneter 4

Theorem 4.5 Al1 trees of diameter 4 and centre degree > 3 have a O-centred graceful la belling.

Proof: We will begin nith some convenient notation.

Definition 4.6 Let H be o tree on m edges, and let f be a partial labelLing of H such that the vertices labelled by f Jorn a eonnected subtree I? on rn - r edges, with f being a valid CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 113 range-relazed graceful labelling of H in the range {O,. . . ,m). Let X = {zl,Q, . . . , x,) C

O,nt be the set of unused vertex labels, and let Y = {ylly,, . . . ,y,} c (1,. . . ,m} be the set of unused edge labels. We cal1 the pair of sets {X,Y) the configuration of f; zf 60th X and Y are sets of consecutive integers, we vil2 say it has a closed configuration

(or sirnply that j is closed). If f has a closed conjiguration, the parameters of f are the pair (x;y), where

While an ordinary configuration of a partial graceful labelling is merely a list of what still necds to be done, a closed configuration has sorne useful features. Let H, j, H, X, and Y bc as above, let {-Y, Y) be a closed configuration, let (x;y) be the parameters of and let v be a vertes in H adjacent to a vertex u in H - H. The condition that X and 1' be sets of consecutive integers insure the following two properties:

1. If j(u) = Z, labelling IL with xi generates the edge label gi-

2. If j(z) = y, labelling u with zi generates the edge label yr-i+l.

The following lemma tells us that if a partial graceful labelling has a closed configuration: its parameters contain al1 the information ive need to obtain a new, more extensive partial graceful labelling with a closed configuration.

Lemma 4.7 Let f be a closed partial graceful labelling of H with parameters (x;y), let v be a vertex labelled by f, let u be an unlabelled vertex adjacent to u, and let xl be the Zowest vertex label not used by f; it follows from definition 4.6 thot 3: + y - xl 2s the highest unused vertez label, x1 - x is the lowest unwed edge label, and y - xl is the hzghest unused edge label. CHAPTER4. O-CENTREDAND m-EDGECENTREDGRACEFULNESS

1. 1-1j(u) =

a) assigning XI to IL generates the edge label XI - x, and the new partial labelling

is closed with parameters jx;y + l),

6) assigning x + y - XI to u generates the edge label y - x1, and the new partial

labelling is closed with parameters (x;y - 1).

a) assigning xi to u generates the edge label y - XI,and the new partial labelling

is closed with parameters (x + 1; y),

6) assigning x + y - xl to u generates the edge label x - x1, and the new partial

labelling is closed with parameters (x - 1; y).

3. If j(v) = either x or y and u has at least two unla6elled neighbours ui and uî,

assigning xl to one of them and x + y - zlto the other give us both edge labels

XI- x and y - XI,and the new partial labelling is closed with the same parameters

as the old, (x;y).

4. (Increment lemma) if u is adjacent to k unlabelled vertices UI,74, . . . ,ul;, where

k is odd, and f (v) = z or y, we can label al1 ul,. . . , uk such that the new labelLing

is as well a valid range-relazed gracejul labelling having a closed configuration. If

j(v) = 2, the effect will 6e to shijl the value O/ the second parameter y up or down

bg any odd arnount < k we choose; sirnilarly, if f(v) = y, the e&t will be to shift

the first parameter x up or dom by any odd amount 5 k we choose. In particular, CHAPTER4. O-CENTREDAND m-EDGE-CENTREDG RACEFULNESS 115

we can label al1 k of the vertices such that the new labellzng is closed and the new

parumeters are (z; y 3~ 1) or (z f 1; y), depending on whether j(u) = x or y.

5. (Maintenance lemma) If v is adjacent to k unlabelied vertices ut : uz, . . . ,ui, - where k is even, and f (v) = x or y, we can label al1 ul,.. . , uk such that the new

labellin9 is as: well a valid range-relaxed graceful labelling having a closed configura-

tion. If f(v) = z we can in the process shij? the value of the second parameter y up

or dom ly any even number $ k we choose; similady, if f(v) = y, we can shift the

first parameter z up or down by any even number <_ k we choose. In particular, we

can label al1 k of the vertices such that the new labelling is closed and the parameters

maintain their previous value, (x;y).

6. (Finishing lemma) If al1 remaining unlabelled vertices of H are adjacent to v,

and j(u) = z or y, j can be cornpleted to a full gracefvl labelling H.

Proof: (1) and (2) follow dircctly from the definition and the 2 properties stated bclorv it; for (3) ive of course apply both parts (a) and (b) to one of them. (1) and (5) then follow by repeated application of (3) with (1) or (2) as ueeded; (6) is merely the case mhere the resulting configuration is empty and the new values of the parameters are irnmaterial. The important things to note is that in the coristructions that follour, ire do not need to keep track of specific vertes labels assigned or edge labels generated; these are determincd (to the extent that they need to be) by the various parts of the Iemma invoked.

14'e are now ready to proceed with the proof of our theorem. Let T be a tree of diameter 4, with rn edges and k 2 3 branches. The reader may notice a pattern familiar from the last chapter, with the construction dividing into even and odd cases. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 116

CASE: k is odd. Starting (where else?) with the centre labelled O, we assign to the k vertices adjacent to the centre the labels k-1 k-l k-l 172,..., ,m-- ,m-- +l,...,m-1,n. 2 2 2 Since these labels are fixed, we wdl refer to the vertex labelled i as ui (rather than the hulkier f-'(i)), and the branch rooted at ui as Ai- The basic plan is label degree 1 vertices a branch at a time, in an order based on the labels gi~7ento the root vertices; ive start with branches at the estremities of our range and work our way in, alternating betwcen high and low branches in (generally) the following manner:

Of course, these starting labels cannot be assigned randomly to the vertices adjacent to tlic centre. The prior constraints upon the labelling are as follows:

1. Even branches: ttiese will be labelled in pairs, so WC would like something like (adil is cvcn iff l=lm-iI is men; whether this will work, however, is affected by whether

the number of even branches is even or odd, and hon- many of them are trivial.

\Vtiile there are a number of ways of handling these situations, one simple method that works consistently is to assign the labels

to the roots of the even branches (i.e. the middle values of the starting labelling

givcn above). s2 = m-SI if the number of even branches is itself even, and m-sl + 1

otherwise; if we have only one even branch it becomes -4, k-1.

3. Trivial branches: note that since these have O degree 1 vertices they are even

branches as well. Since Our intention is to be finished labelling by the time ive encounter the first of thcse, they are given values in the middle of the range assigned to the even branches i.e. k-1 k-1 k-l Tl,Tl + 1, -.- , ,m- ->m- - + 1, ..., r2 2 3 2 C HAPTER 4. O-CENTREDAND m-EDGE-CEKTREDGRACEFULNESS 117

where sl 5 rl and r2 5 s2; similarly to above, r2 = rn - r1 if we have an even number of trivial branches, and m - rl + 1 otherwise.

-49 --lm-61

Even Branches

Figure 4.8: Initial Iabelling for diameter 4 tree with k branche

Con~pletingthe starting fabeifinggracefully. We will let fi dcnote the partial labelling of T when the lowest branch with unlabelled degree 1 vertices is Aiand thc highest branch wi th unIabelled degree 1 vertices is A,; our starting Iabelling is then f l,,. In cases where it is of technical iniportancc fj will denote the partial labelling where only the branch -qj rernains to be labclkd; howevcr, WC dladopt a slight abuse of notation from the onset, and use f,, in cases where tliere may or may not be more than one non-trivial branch rcmaining to be labelled, if this does not materially affect Our strateg?; at that point (the idea bcing that generally we can carry on until we find out we are finished; there are only two situations which rcquire spccial handling at the end, and these are both dealt with bcfore ive are down to a single untabelled branch anyhow). f will of course denote the complete 1abeIling of T.

Initially ive have that since (1,. . . , ,m- 2,.k- 1 . . ,m} have been assigned to the vertices adjacent to the centre, the set of unused vertex labels is exactly the set of unused edge labels. {y+ 1, . . . , ,-k-i- l};this gives us a closed configuration, and parameters (0; m). As we shall see, from this starting point we can always finish gracefully given the way the even and trivial branches are arranged. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 118

Branches are labelled either singly, or in pairs, according to one of the 4 following procedures: 1. Fizcl sizgle. If Al is the last unlabelled branch, with the parameters bebg eit.her

(1; j) or (i;l), then we nced not trouble ourselves with what happens to the parameters from here; by the finishing Iemma assigning al1 remaining unused labels to ill will result in a graceful labelling. In al1 cases where we implicitly arrive at the partial labelling fi t his is the procedure that wilI be invoked.

2. Odd sangle. If the parameters of the current partial labelling are (i;j) and -4 is an unlabelled odd branch, by the increment lemma we can label al1 degree 1 vertices in a-li so as to advance the parameters to (i;j- 1). SimiIarly, if the parameters are (i;j) and dj is an unlabelled odd branch, IW can label .-?, so as to advance the parameters to (i + 1;j). 3. Ezien pair. If the parameters of the current partial labelling are (i:j + 1) with diand .-lj unlabelled even branches, ive can label both Ai and Aj so as to advance the paramcters to (i;j - 1) as follottls:

* Label one degree 1 vertes in .4,, advancing the parameters to (i;j) (by lemma 4.7 part l(b)).

O Label al1 degree 1 vertices in .$, holding the parameters at (i;j) (by the maintenance lernma).

O Label the remaining unlabelled vertices in Ai;advancing the pararneters again to

(2; j - 1) (by the incremcnt lemma).

Similarly. if the parameters are (i - 1; j) with Ai and Aj unlabelled even branches, are can label both with the parameters being advanced to (i + 1;j); the procedure starts at -Aj in this case.

4. Euen triple. If the parameters of the current partial labelling are (2; j + 1) with -Ai: and -Aj al1 unlabelled even branches, we can label the three as follou~s: CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 119

Label 1 degree 1 vertex in Ai to advance the parameters to (2; j) (by lemma 4.7

W)).

Label 1 degree 1 vertex in Aj to advance the parameters to (i + 1; j) (by lemma 4.7 3(a)).

Label al1 degree 1 vertices in -li+l, holding the parameters at (i + 1; j) (by the maintenance lemma).

Label the remaining unlabeiled vertices in .Li, this time so as to bring the param-

eters back to (2; j) (advancing in the opposite direction).

Finish off the remaining unlabelled vertices in -.ii (where the parameters go from here is irnmaterial, since this procedure will only be used when Ai, Ai+i and Aj are the iast three unlabelled branches).

If the parameters are (i - 1; j) with Ai, and .Aj unlabelled even branches, rve can labcl a11 tliree in a similar manner, starting at .Aj-

It only remains nom to confirm that these 4 procedures are sufficient to finish the labclling f. We start by defining an acceptability criterion for our partial labellings.

Lct fi ,j (alternately, fj, f) be a partial labelling of T such that

9 The branch .IIwith root vertex labelled l exists for 1 = i, . . . ,c, m - cl.. . , j.

Ko branches in the range . . . ,-4 ,-,-1 require labelling.

0 -411 trivial branches, and no non-trivial branches, are in the range A,, , - . . , A,,

where c + 1 5 rl and m - c - 1 2 r2 = m - TI or m - TI + 1.

Al1 even branches, and no odd branches, are in the range A,, , . . . :AS=, where sl 5 rl

and rz 5 s2 = m -sl or m - s1 + 1. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 120

fi,, (ah. fj, f)is then an acceptable partial labelling if it is in one of the five following states:

State Labeliing Paramet ers Extra conditions

(a) fi,m-i+l or fm-i+l (i - 1;m - i + 1) -

(b) fip-i of fi (i;m- Z+1) -

(4 fi.m-i or fm-i (i-mi) [A,-iIeven

(b') f*,m-i+t Of fi (i,m-i+2) lAileven (* f - T is gracefully labelled

Thcse statcs will be denoted therefore as acceptable states.

The nest step will be to show that if a partial labelling of strîctly less than V(T) verticcs is acceptable we can always apply one of the procedures to obtain a larger acceptable labelling. Note that our initial labelling fi,, is in state (a), and is therefore acceptable: by design it satisfies the global conditions at the beginning of the definition. In fact, thcse conditions are only made esplicit here so that the following lemma and its corollarics can be used again mhen Ive attack the k even case:

Lemma 4.8 If the partial labelling fij of T is in an acceptable state then either

* f*,j = f , a finished graceful labelling of T,

0 fij can be irnmediately finished to a graceful labelling of T,

There exists a partial labellzng filal with i' > i or j' < j which is in an acceptable

state.

Proof: Since each of the labelling procedures provided takes care of at least one branch if not more, it is sufficient to show that for each of the acceptable states except (*) some one of the procedures puts the new labeiiing into an acceptable state. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 121

State (a), l=lm-i+l 1 odd. Use the odd single procedure to label Am-i+l; the resulting

labelling is fi,,-i with parameters (2; m - i + l),so it is in state (b); or (*) if Am-i+i is the last unlabelled branch in T.

State (b), l-+( odd. Use the odd single procedure to label Ai; the new partial

labelling is fi+,l,-i Mth parameters (i; m - i). Letting if = i + 1 and j' = j, we

have fi+ilm-i = fil,,-iI+l and (2; m - i) = (if - 1; m - if + l),so the labelling is now in state (a), or (*) if Ai happens to be the last unlabelled branch in T.

Now since there are no odd branches between A,, and A,, inclusive, with s2 = rn - sl or m - si + 1, we know that if we are ever in state (a) with ~A,-i+ll even or state (b) n-ith J=ii[even al1 remaining unlabelled branches are even branches as well; hence while therc arc unlabelled odd branches we alternate between states (a) and (b).

State (a), l.-lm-i+l 1 even, Ai and .4i+l non-trivial even branches. Use the even pair procedure starting at -rlm-i+l to label both -'Im-i+l and -Ai- The resulting labelling

is fi+l,,-i mitii parameters (i+1; rn-i+l); setting if = i+1 gives these as fit,m-il+l

with (if;m - i' ;2) with = -4: an even non-trivial branch, so we are in state

(b') -

State (a), 1 even, Aitl trivial. If is also trivial, the evcn single or pair scheme \sri11 finish the labelling (depcnding on whether is also triviaI) and the new state is (*). Othernise, -4, and Am-i+L are the last unlabelled branches and al1 even, so the even triple procedure starting at tvill likewise put us in

State (b), lAil even, A,n-i and -4m-1-l non-trivial. Use the even pair procedure

starting at Ai; the new labelling is fi+llm-i- 1 with parameters (i;m - i - 1);setting

if = i + 1 we have with parameters (if- 1; m - if) and non-trival, putting us in state (af). CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 122

State (b), lAi1 even, Am-i-l trivial or non-existent. If is also trivial, the even single or pair procedure puts us iu (*). Othenvise, Ai, -Aiil, and are the last

unlabelled branches, so the even triple procedure starting zt -di as we!! gets us to

(*)-

m State (a'). If is non-trivial so is -4,and if there is no non-trivial there is no non-trivial Hence we use the even double procedure every time we find

oursclvcs in this state; this takes us eittier to (*) or state (b), since the resulting

labelling is either f or fi+l,m-i-i with parameters (i + 1; m - 2).

r S tatc (b'). If ili is non-trivial so is rlm-i+l,arid if Ai+* is trivial so is Am-i; so again Ive use the even double procedure every time. The resulting labelling is either f or

ji+ mith parameters (2, m - i), which puts us into either (*) or (a).

While we are labelling even branches, the states either alternate between (a) and (b'), or

(a') and (b).

Corollary 4.9 If the tree T has a partial labelling j svch that jis in one of the acceptable stutes (a): (b), (a') or (b'), f can be compieted to o graceful labelling ojT.

Corollary 4.10 For k odd every diameter 4 tree T with k branches has a O-centred gracejul labelling.

Two esamples of trees labelled açcording to the construction are shown in figure 1.9. The reader might note how strikingly regular the 5-cornet's labelling appears in corn parison wi t h the larger tree's.

CASE:k is euen. We cannot apply the above construction directly when the number of branches is even, since if ure label the vertices adjacent to the centre with the same nurnber of consecutive high and low labels we have no way to generate the edge m - 5.k

Instead ive will provide a series of reductions to the odd case. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS

Figure 4.9: Two O-centred graceful labellings of 5 branch trees

The major rcductions will al1 need to make assumptions about the size and parity of sorrie number of the branches, so as a matter of convenience we will restrict our attention to trees witli no trivial branches. This is not a problem, since if T has an even number of branches with one of them trivial, it foi1ows "almost trivially" that T is O-centred graceful: letting u be a degree one vertex adjacent to the centre of T, we have by lemma

4.8 corollary 4.10 that T - {u} has a O-centred graceful labelling /', from which we directly can obtain a O-centred graceful labelling f of T by merely sctting f (u) = m (the range of f' is {O, . . . , m - l}).

For each of the five reductions we start with a generai base labelling similar to the initiai labelling fi,, of the odd case; for the first three we will use the following, fa: the centre vertex is assigned 0, and the k vertices adjacent to the centre are assigned the values C HAPTER 4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 124

Note that n - 5 is always > $ - 1, since T has diameter 4. Again, we ad1 refer to the vertex labelied i as ui, the branch rooted at vi as Ai,and the partial labelling when

the lotvest linlabelled branch is A, and the highest unlabelled branch is A1j as fij (or f,, or f,as applicable). The reductions for the most part work by labeiling several branches in way that does not follom the pattern establisbed in the odd k construction, but which puts the reinainder of the tree into one of the acceptable states. To achieve this some branches of spccified parity or minimum size will need be set aside to take particu root labels (usually tliose with with the highest or lowest values). The assignment the rernaining root labels will then largely follow the odd k construction with respect parity: the branch -4 ni11 be even if and only if sl 5 1 5 s*, where s2 = rn - L - SI or nz - sl (this slight modification of the relation we saw between sl and s;, in the odd k constructiori is meant to distribute the even branches as equally as possible below 6 - 1 and above m - ,).k- Sirice no assignments are made to degree 1 vcrtices yet at tliis point, for f, the set of unuscd vertes labels is the same as the set of unused edge labels, !j+ 1, . . . m--- 1}; (5;- - - 3 it is closed with parameters (O; rn - 1).

1. T has at least one even branch, and at least two odd branches, one with size 2 3.

UTestart with the labelling fa, with these extra specifications: IA,/ is even, IAm-l 1 is odd, ].-LI 1 is odd and 2 3. To get to the rcduction, we label .li, and -4, as follows:

r Label al1 degree 1 vertices in so as to advance the configuration to (1; m - 1) (by the increment lemma).

m Label esactly 1 degree 1 vertex in ;li,nith a low label; by lemma 4.7 (la) this moïes our parameters to (1; n).

a We are now in a position to deal with A,; it is labelled completely so as to keep the parameters unchanged (by the maintenance lemma).

a The nurnber of unlabelled vertices in Al is even and > 2; ive label it so as to advance CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 125

the parameters by two, to (1; rn - 2) (by lemma 4.7 (5), generaf case). The three

branches are fully labelled, so the partial labelling is now f2,2,m-2.

Let c = $- - 1, let mf = m - 1, and let if = 2. Wë have that the branches remaining to be labelled are A:, . . . , A,; =I,c,, . . . , ..lml-if+l, ewn branches range frorn AS1to A,,

where $2 = mf- s1 or m' - SI + 1, and the labelling is currently fil,m-*f+l mith parameters (if - 1; rn - if + 1); hencc we are in state (a) as defined in the odd k construction, and by lcmma 4.8 corollary 4.9 can complete the partial labelling to a full graceful labelling.

Hence T is O-centred graceful.

Figure 4.10: -4 O-centred graceful labelling of a 6 branch tree

This reduction takes care of a large majority of diametcr 4 trees with an even number of branctics. Thc cases we are left with are:

i. T has al1 odd branches.

ii. T has < 2 odd branches.

iii. T has at least one even branch, and al1 odd branches have size 1.

(i) involws extra complications, so we will leave it for last. The folIowing variation on the previous reduction ni11 deal with practically every tree in case (ii).

2. 7' has k > 6 branches, at least 5 of which are euen. \Ve start with fa, letting -;lr, .A2, and -4, al1 be even branches. They are then labelled in the foilowing order. tu obtain partial labelling f3,*-3 with parameters (2; rn - 3): CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS

Parameters: (O; n - 1) (1m - 1) (1; m) (1; m)

Branch: Am- 1 -4 1 Am AI No. labelled: 1 1 All(Even) Rem(0dd) Advance Param: LOW,+l HIGH,+l - HIGH,-1

Parameters: (1;rn - 1 (2; m - 1) (2;m - 2) (2;rn - 2)

Branch: A,- A2 Am-2 -42 No. labelled: Rem (Odd) 1 -411(Even) Rem (Odd) Advance Param: LOW,+l HIGH,-1 - HIGH,-1 If k = 6, there is only one unlabelled branch left, so with the parameters standing at (2; m - 3) we are done by the finishing lemma. Othemise, letting c = 5 - 1,

rn' = rn - 1: and i' = 3 this time ive see the labelling is effectively fi~,m~-i~+lwith

paraineters (if- 1; m' - i'+ 1) and hence in state (a), so T has O-centred graceful labelling.

We note that in this reduction, al1 odd branches can have size 1; so this gives us case (iii) above as well if the number of even branches > 5. The following reduction is designcd to fil1 the gap.

3. T has at least 1 even branch, and at lemt 3 branches with size = 1. Starting again

wit h fa, ive let Al, .4,-1 and -4, be size 1 branches, and be an even branch. The

labelling f: is obtained from fa by assigning to one arbitrary vertex in A,-2 and to the

one vertes in the other stated branches the following values:

Note that we can always count on n - 5 - 2 being greater than 5 + 1, since T has diameter 4. These generate edges m - 5 - 3, m - 2k - 2; 5, and m - 5 - 1 respectively, so the unused vertex labels are {$ + 2,. - . , rn - - 3) and the unused edge labels are CHAPTER4. O-CENTREDAND rn-EDGE-CENTREDGRACEFULNESS 127

{&+ 1, . . . , m - - 4); f; has a closed configuration with parameters (1;m - 2). Let -2 k c = - - 1; mf = m- 1, and i' = 2; also, iet A&-* be the branch minus the one degree ? x.vcrtesthat is !abelled by fi. A&-, is effectively an odd branch in the partial labelling

= /i~,m-i~il;the other even branches (if there are any) are distributed according

to the odd k construction about c; and the parameters are again (if - 1; m - i' + 1). Hcnce we are in state (a), f: can be completed gracefully, and T has a O-centred graceful labclling.

For cases (ii) and (iii) reduction (3) takes care of al1 but a couple of tedious Ieftovers, trees having too few branches to work with. Constructions for these will be presented

briefly; thcy al1 start with the labelling fa as well-

3a. k = 4 or 6: two branches vith size = 1, the rest even. One vertex ui in each of

the outer 4 branches is assigned a label in order to clear two branches, change the parity

of the tivo others, and nudge the parameters to (1; m).

Parity/Size Even Size=l Even Size= 1

Effective P/S 1 Odd 1 Nil 1 Odd 1 Nil

The labelling then proceeds as folloms; for k = 4 we of course stop aftcr -4,:

Parameters: (1; m) 1 m - 1 (2; rn - 1) (2; m - 3)

Branch labelled: -4L -LI-1 -42 Am-3 -4dvance Parameter: HG- LOIV,+l HIGH,-2 NA.

3b. k = 1: 5 1 branch odd, the rest even, at least one even branch with size 3 4. Assign an even branch with size 2 4 to A,; if one branch odd, assign that to Arnd1,and label al1 the vertices in A,-l in the first step. C HAPTER 4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS

Parameters: (O;m - 1) (1; m - 1 (1; m) (1; m)

Branch labelled: .Arn- 1 -41 Am -4 No. labelled: Odd 1 A11 (Even j 2 Advance Parameter: LOW,+i HfGH,+l - HIGH,-2

Parameters: (1; rn - 3) (1; m - 2) (1; m - 1)

Branch labelled: Am-* *4 .Am- 1 No. labelled: -411 (Even) Rem (Odd) Rem Advance Parameter: - HIGH,+l N.X.

3c. k = 4: al1 branches have szze > 2, ut least 3 have szze = 2. Let -A1, and .A,,, hc the sizc 3 branches: and let AmVI be the variable size branch. CVe then label two vertices in cach branch according to the following scheme:

Branch A Am-1 A, Vertes Label 4 rn - 4 3 rn-6 5 m-3 3 m-5 Gcneratcd Edge 3 m - 5 m - 4 4 m-6 3 m - 3 5

This prcprocessing brings us back to essentialIy what we started with (minus the 8 vcrciccs): a closed partial labelling with parameters (O; rn- 1). If there are any unlabelled vertices rernaining in we can assign them any unused vertex labels.

3d. -At this point we have al1 diameter 4 trees with at least one even branch covered escept for esactly one tree. Figure 4.11 provides an esplicit labelling for this annoyance.

Finally, we come to the case where al1 the branches are odd. At this point we have to abandon our base labelling fa, since it can be shown by parity arguments that for k r 2 (mod 4) there is no graceful labelling of a diameter 4 tree T with k branches, al1 odd, such that the vertices adjacent to the centre are labelled in such a manner. For k O (mod 1) there is, but these labellings are more complicated then the following reductions, which work in both instances. We will cal1 the new base labelling used for these fB: the CHAPTER4. O-CENTREDAND m-EDGECENTREDGRACEFULNESS

Figure 4.1 1: -4 tree not covered by any of the cases centre vertex is assigned 0; the k vertices adjacent to the centre are assigned the values

Tliat is, we are using the base labelling from the odd k construction with the "slight" iriodifiçatiori that ive skip the labei 2. vi, Ai,and fij will retain their meanings from bcfore. Since the trees we will be working with have al1 branches odd, no assignment of brariches according to parity is required. Note that this base labelling does not have a ciosed configuration at the onset; the first thing that each of these two 1st reductions do is label a small selection of degree 1 vertices first in order to obtain one.

4. T Ims al1 branches odd, at least 3 with szze 2 3. As in the previous case. this breaks down into subcases which require chasing down progressively smaller trees (though thankfuily riot so piddly this tirne).

4a. T 2 8. Starting with f8 we label 6 vcrtices in 5 of the branches so as to obtain a closcd partial labelling with parameters (1; m):

Branch Constraint Vertex label Edge label Effective P/S Even 1 Odd CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 130

We thcn label branches Al, Am-2, and -4, in the following order; if A, and .A,-i are left with no unlabeUed vertices by the initiai labelling they can be ignored:

Parameters: (1;rn) (1; m) (1; m - 1) (1; m - 1) (1;m - 2)

Brancb labelled: -4, A -~X-I A Xo. Iabelled: Rem 1 Rem Rem(0dd) Rem(Even,> 2) .+IdvanceParameter: - HIGH,-1 - HIGH,-1 LOW:+Z?

This leaves the paranieters at (3; rn - 2), while A,- ++, has an odd number of vertices reniaining to be labelled, and the other branches between .A3 to Am-3 have not yet been touched; so our effective partial labelling is fi,,-, with parameters (3; m - 2). Letting c = -2 Y m' = m, and if = 3 we see that the labelling and parameters are fi~,m~-z~and (2'; m' - i' + 1), which puts us into the acceptable state (b); hence by lemma 4.8 corollary 4.9 we can finish gracefully, and T has a O-centred graceful labelling.

4b. k = 6. In this case .-lm-;,, - so the above procedure needs this branch

to have size 2 5 to work, but it is othenvise unaffected. If none of the branches have size > 3, the following construction will work:

Branch AL -4,- i -& Size Constraint 2 ].4m-11 Size=3 5 Irll 1 Sizef3 - lTertteslabel 6 3 m-4 4 m - a 5 Edge label 5 m-4 2 m-6 4 rn-5 Effective P/S Even/Nil Nil Even/Nil Even

No degree 1 vertices in .A3 or .4,-3 are touched, but A3 mut be a size 3 branch. This lcaves us, as before, with the parameters (1;rn). The idea this time is that if there is anything lefi to label in .41 or A,-, we can take care of it without affecting this situation; ive then use the turo vertices left in A, and the 3 left in ,A3 to "jump the parameters" to where the nest active branch will be. If the first stage completes Al (and hence .4,-1) we can start imrnediately with -4,: CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 131

Parameters: (1;m) 1-1(1;-) (1;m) (3;m) (3;m - 3)

Branch labelled: -41 .Arn- l -4 1 -4, 4 &-3 No. labelled: 1 Al1 1 Al1 ,411 -411 -Adv. Param.: HIGH,-1 - HIGH,+l LOW,+2 HIGH,-3 NA.

4c. k = 4. This time we have that A,-j,, = A,-1. The original prelabelling given for k 2 S will work, since the constraints on .+lm-i+l and Am-' are not incompatible; in

the initiai stage we just label threc vertices in A,-'. Since the second stage will then

Iabcl T completely, taking us to state (*): we can relax the size constraint on Am-2 as well.

5. T fias ail branches odd, at least one with size 3 3, ut least two with szze = 1. In the first stage ive augment fa by labelling 4 degree 1 vertices so as to obtain a closed partial labelling with parameters (1; m):

Branch AI A,-$ Am-$+l Am Constraint None SiLc=l Size=l Sizel 3

Vertex label 4 + 2 2 m-k-2 1 $+l Edge label 2 + 1 m - -2 - 2 2 m---2 Z Effective P/S Even/Nil Nil Nil Even

In the second stage we merely label whatever unlabelled vertices are left in -Al, main- taining the parameters, then label the remaining vertices in A, so as to advance the pararncters by 2, to (3; m). If k = 4 we are now done; otherwise, since the prela- bclling has taken care of -4,- and -4,- $+, , the unlabelled branches are now -A3, . . . ,4 2

4 + . . . 4 Setting c = 2, m' = mi-2, and i' = 3 we see that we have unlabelled branches -ili!:. . . ,A, and Am-,, . . . .4,t,it, al1 odd, so the partial labelling is effectively fi1 ,,nt -,r with parameters (2'; m' - i' + 1). This of course is again the acceptable state (b), so by lemma 4.8 coroIlary 4.9 we can finish gracefully, and T has a O-centred graceful labelling. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 132

These two reductions cover al1 T tvith an even number of branches, al1 of which have

odd size, except for the one case where al1 the branches have size 1. These are of course

fi-cornets with k even. For k-cornets there is at least one known constrwtion (based on Stanton and Zarnke [37]) whicli always results in a O-centred graceful labelling: Let T

be a k-cornet with k even, where vc is the centre vertex, wl,w2, ..., WA:are the vertices adjacent to vc,and ui is the degree 1 vertex adjacent to wi. The labelling f of T defined

b~ f(.c) = 0 j(wi) = 2i

f(ui) = 2(k-t)+l is a O-centred graceful labelling. .4n esample is given in figure 4.12.

Figure 4.12: O-ccntred graceful labclling of the ô-cornet

Now that we have confirmed that al1 diameter 4 trees with 2 3 branches have a O- ceritred graceful labelling, we can see that D is a somewhat special subclass of 2 branch diamctcr 4 trees. Whether 2) is al1 the trees that have no O-centred graceful labelling is anotlier question. One way to take things from here is to look at higher diameter trees; another is to espand the range of our inquiry to cover vertices other than the centre, and labels othcr than O. While the former is a definite future possibility, for the tirne being we will follow the latter course. CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS

4.3 Trees t hat are L4ubiquitously"graceful

4 -3.1 O-ubiquitously graceful trees

The most direct line of attack from here is to consider al1 vertices of the tree at once; the nature of the counter-instances will tell us if any particular thing (such as e.g. degree) is a dctcrmining factor on Our ability to gracefully label a vertes with some specified value.

Definition 4.11 Let G be any graph rn edges. G is k-ubiquitously graccful if for every vertex .(: in V(G)there is a graceful labelling f of G such that j(u) = k. G is simply ubiquitously graceful if it is k-ubiquitowly graceful for al1 k = 0,1,. . . , m.

Furthemore, ijfor every edge uv E E(G) there exists a graceful labelling f such that

1 f (u)- f (v)l = k, we will say G zs k-edge-ubiquitously gracefiil, and if G is k-edge- u6iquitously graceful for al1 k = 1,.. . , m we vil1 say G is edge-ubiquitously graceful.

\\.%will begin with the cornputer results concerning O-ubiquitously and ubiquitousIy graccful trees, as this estends Our previous espIorations of O-centred gracefulness. Since cstablishing whether a given trce lias either of these properties is considerably more time consurnirig than merely confirming whether the tree is O-centred graceful, the tests only ncnt up to order 11. -4s before, along rvith checking the general case the program checked how often the condition holds in conjunction tvith locally and fully bipartite gracefulness. The rcsults are given in table 4.3. The numbers for O-ubiquity in the general case are, if anything, more startling thaa before ...particularly when one realizes that the number of non-0-ubiquitously graceful trecs of order n is simply the cumulative sum of the number of non-0-centred gracefu1 trecs up to order n. What does this mean? Weil, looking at some of the actual trees (given in figure 4-13), ure see that the first, fourth, and fifth encountered are members of the class D,which is not too surprising; CHAPTER4. O-CENTREDAND m-EDGECENTREDGR~CEFULNESS

n Trees Any LB B ALB B 3 1 1 1 100 O 4 2 5 3 6 6 7 Zt 8 23 9 47 10 106 11 233 12 551 13 1301 14 3159

Table 4.3: O-ubiquitously graceful trees the other two are trees obtained from the first one by attaching to its centre copies of P2 and P3 respectively. This as well is not too surprising. What is surprising is that every ncgative instance of order n 5 14 fits this description; they are al1 trees obtained from a trec TD in the ciass D and the path Pk by idcntifying an end-vertes of Pk with the centre of Ta;the only vertes not taking O as a label is the other end vertex of the path, which will bc the centre of the tree if k = 1.

Figure 4.13: The first few members of D'

That any such tree cannot be labelled with O at this vertex of course follows directly from the fact that no tree in V can be labelled with O in the centre: if any such labelling

\vas possible we could, bÿ repeatedly complementing the labelling and then pruning the CHAPTER4- O-CENTREDAND m-EDGE~ENTREDGRACEFULNESS 135 degree 1 vertex consequently holding the high label, obtain a O-centred graceful labelling of the base tree from the class 2). Ive will dcfine Dyas the chss of graphs obtained by identifq-ing an end vertes 3f an arbitrary path with the centre of a tree in 2). There is good reason to believe that these are the only trees which are not O-ubiquitously graceful. The following theorem, the last of this thesis, establishes that this is the case for diameter 4.

Theorem 4.12 Cet T be a tree with diameter 4.

1. If T 2s not in a', it zs O-ubiquitowly graceful.

2. If T zs in D', it can be gracefully labelled with O assigned to any vertex but one (up

to autornorphism).

Proof: The proof will make hcavy use of the following two facts; :

1. Let G bc a graph with distinguished vertex v, let S be a Il--star, and let G< be the graph formed by identifying the centre of S with v. If there is a graceful labelling

f of G such that f (u) = O, then there is a graceful labelling f < of G< such that one of the degree 1 vcrtices of the attached S is given the label O. This follows by the canonical amalgamation construction of theorem 1.14, with an application of complementary labelling to the result (definition 1.12). Since switching the labels

of any two of these degree 1 vertices results in an automorphism, al1 of them can be O-labelled.

2. Let G be a graph with distinguished vertes v, let S be a &starl and let G' be the

graph formed by identifying a degree 1 vertex of S with v. If there is a graceful

labelling f of G such that /(u) = 0; then there are graceful labellings fi and f; of G' such that fi assigns O to the centre of the attached S, and f,' assigns O to any

of the degree 1 vertices of S except the one identified with W. This follows from iterating (1) twice. CHAPTER4. O-CENTREDAND m-EDGECENTREDGRACEFULNESS 136

Let T be a diameter -1 tree, and let v be a vertex of T. We will look at the following cases:

1. T hm 2 4 brmches. By theorem 4.5, if z is the centre vertex, ive have a graceful labelling f of T such that f (u) = O. Otherwise, let A be the branch of T containing u. Sirice T' = T - -4 still has 2 3 branches, theorem 4.5 still applies and Tr has a O-centred graccful labellirig; hence by (2) above there is a graceful labelling f' of T such that v in -4 has the Iabel O (WCuse an (1-41 + 1)-star). 2. The centre of T hm degree 2. If T is in D then by definition it is in Dr, and the centre of T is the unique vertex which cannot be gracefully labelled O; othenvise by the cliaracterization theorem (4.4) T does have a O-centred graceful labelling. In either case if .ü is any other vertes in T there is a graccful labelling f of T such that f (v) = O. Since

T is a caterpillar, this is in fact the canonical caterpillar labelling given figure 1.6 of cliapter 1; al1 vertices other than the centre are either the endpoint of a longest path in

T: or adjacent to such an endpoint, so we can start the canonical labelling (by assigning

O to u) anywhere but the centre. 3. The centre of T has degree 3. By theorem -1.5 T always has a O-centred labelling; so our conccrn is nith non-centre vertices. The members of 2)' which have diameter 4 fa11 into 3 groups:

a) Trecs in D (handled in the previous case).

b) Trees obtained from those in V by attaching a pendant edge to the centre.

c) Trees obtained from those in V by growing a path of length 2 from the centre.

Let m be the number of edges in T, and let A be the branch containing the vertex u ive tvant to label O. There are 2 subcases:

1. u is adjacent to the centre vertex:

a If -4 is an odd branch, by the construction for the odd case of theorem 4.5 there is CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 137

a O-centred graceful labelling f of T such that u, the root of -4, is labelled m; the cornplementary labelling then has f(v) = 0.

a If 1-41 is even and >: 2, let u be a degree I vertex adjacent to u; by exactly the same

reasoning as above T - {u) has a û-centred labelling f with f (v) = m - 1. Taking the complement and reattaching u with label m results in a graceful labeiling f' with I1(v)= O.

r If .L' has dcgce 1: then cithcr T - {u) is not in 27, so it has a O-centred graceful labelling and by (1) thcre is a graceful lahelling of T with u assigned 0; othemise, T is in V', and v is the unique vertes in T which cannot be gracefully labelled O.

2. u is not adjacent to the centre:

If 1-41 is even (hence necessarily 2 2) we can use the construction for even branches

irnrnediately above; taking the cornplement again gives us f'(u) = O, so reversing

the roles of u and u does the trick.

rn If IAI is odd and 2 3 the same approach works, this time removing two degree 1 vertices from -4 instead of one to obtain the intermediate labelling.

If 1-41 = 1, then if T - {v} is not in V' we know by the degrce 1 case for vertices

adjacent to the centre that there is a graceful lahelling of T - {v} such that u's

neighbour is labelled O; hence by (1) T has a graceful labelling where u is labelled O. Otherwise, T is in 27' and either v is the unique vertex nhicli cannot be gracefully

labelled O? or T has exactly two branches with size 1, and the degree 1 vertices of ncither can be labelled O gracefully (if al1 3 of T's branches have size 3? T is not in

-0').

a Trccs of diameter 4 are often the "spoilers" for assumptions about gracefulness (as in, for esample, bipartite graceful labellings); generally low diameters place many restrictions CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 138

on possible graceful labellings, while higher diameters offer greater freedorn. Hence one rnight have expected that if there were more negative instances of O-ubiquity to be found,

some would likely be lurking in diameter 4 somewhere; bct by theorem 4.-5 thk canmt be the case.

Conjecture: The set of trees which are not O-ubiquitously graceful is esactly the

class D'. Every member TDI of 2)' has esactly one vertes v or two automorphic vertices

v and û such that there is no graceful labclling f of TD with f (v) = 0; for ewry other

vertes w # v in & there is a graceful labelling of TD where w is assigneci the label O.

4.3.2 Furt her empirical results

This ctiapter will wind down with the last of the empirical results. \Ve wiil start with the riurnbers for trees of order n that are not m-edge-ubiquitous, which are given in table

m-edge-u biqui tous n Trees -4ny LB B 3 1 1 1 1 4 3- 1 1 1 5 3 3 2 1

6 6 4 3d 1 7 11 9 5 -3 8 23 15 5 1 9 4'7 41 29 6 10 106 84 66 12 11 235 336 189 32 12 551 489 436 93 13 1301 1285 1218 316 14 3159 2990 2893 1007

Table 21.4: m-edge-ubiquitously graceful trecs

rn-edge-ubiquity displays very similar behaviour to m-edge-centred gracefulness; in the general and locally bipartite case the nurnbers of counter-examples are understand- ably higher (about 5 or 6 times) but they exhibit the same oscillatory pattern with even C HAPTER 4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS 139 and odd orders. It should be noted that nothing directly linking every negative instance here to a negative instance of rn-edge-centred gracefulness has been found.

Table 4.5 displays the number of ubiquitously graceful and edgcubiquously graceful trces of ordcr n.

ubiquitious edge-ubiqui tous n Trees A LB B

Table 4.5: ubiquitously graceful trees

Alost notable is that for botli ubiquitous gracefulness and edge-ubiquitous graceful- ncss, if no furtlier restrictions are placed on the labclling ire still have the vast majority of trees as positive instances: at order 11 about 80% of al1 trees are ubiquitously graceful and about 60% are edge-ubiquitously graceful. The other thing that catches the eye is the total collapse in the number of edge-ubiquitously bipartite graceful trees; for al1 orders n only the (n - 1)-star has the property. This bears some looking into when time perrnits. IVith respect to locally bipartite graceful labellings there is a similar, if not so total, collapsc; at order 11 irehave 70% of trees being m-edge-ubiquitously LB graceful while only a srnattering are edge-ubiquitously LB graceful (around 5% or so). Lastly, it should be noted that vertex-ubiquitous gracefulness is not very compatible tvith fully bipartite gracefulness, since a tree can only have both properties if the sizes of its two bipartition sets are the same; u-hich explains why the number of positive instances of CHAPTER4. O-CENTREDAND m-EDGE-CENTREDGRACEFULNESS ubiquitously bipartite graceful trees drops to O for every odd order. Chapter 5

Conclusions and Future Work

\Ve have covered a fair amount of territory; it is perhaps a good idea to review the main results at this point:

1. Proof that evcry tree T of order n and size rn = n - 1 has a range- relaxed graceful

labclling jRsuch that fR(V(T)) C {O;. . . ,2m} (theorem 2.2) and a vertes-rela~ed graceful labelling fv such that 1fc-(VV(T))j > 5- (theorem 2.4).

2. Evidence that probably al1 trecs are locally bipartite graceful, nith machine verification for al1 trees of order < 19 (chapter 3 section 2.1) and proofs that some infinite classes of trees diich are known to not have bipartite graceiui labellings do have locally bipartite graceful labellings (theorcms 3.7 and 3-12).

3. A general recursive construction making use of locally bipartite graceful trees (theorem 3.5): with some data indicating that the extra assumptions necessary for the construction to work are not very restrictive.

4. Isolation of the classes V and V' (theorem 4.4 and section 3.1 of chapter 4), and empirical evidence suggesting that thcy may well be the only non-0-centred and nori-0-ubiquitously graceful trees respectively; along with the proof that al1 trees of CHAPTER5. CONCLUSIONSAND FUTUREWORK 142

diameter 4 are O-centred graceful if and on1y if t hey are not in 27, and O-ubiquitously graceful if and only if they are not in D' (theorems 4.5 and 1.12).

Thcse results have in the course of the thesis prompted us to make the following

gcneral conjectures: Conjecture 5.1 All trees are locally bipartite graceful.

Conjecture 5.2 For euerg tree T,T zs O-centred graceful if and only if T 4 V

Conjecture 5.3 For evenj tree T, T is O-ubiquitously graceful if and only if T 4 V'

Thc results and conjectures may seern sornewhat diverse, but as we shall sec shortly there is a sense in which certain elernents that have appeared in the various parts of the

thcsis can be put together to useful effect; what WC have been doing is in essence building

a -'graceful toolbos" for more direct attacks on the graceful tree conjecture. How the

ideas brought up in this thesis can be appiied to further work towards the GTC will be the subject of section 2 of this chapter; ive will first look at some interesting questions of

a more independent nature which are brought up in several places.

5.1 Some open problems

In chapter 3, the obvious place to start is with the bounds obtained in theorems 2.2 and 2.4: while decent they are definitely not the best that can be done. ,As stated in the chaptcr there are some constructions for both RRG and VRG iabellings of trees which seem to do consistently better in practice than the ones used in the proofs of the theorems, but did not lend themselves to rigorous analysis as readily. So one first step nre can take from here is to take a harder look at these constructions, and either corne up with cases wbere they do no better than the bounds, or show why they are superior and in this nray extract tighter bounds. X related problem is nitb reductions between the various relaxation schemes we have. In the one reduction we have, from a VRG labelling of a tree to an RRG labelling of a smaller tree, the size of the smaller tree is directly dependent upon the particular labelling used; so we still cannot Say very much about the range of RRG labellings of

trecs of order nl giver, some bourd fcr all VRG labetlings cf trees of crder 22. Ideally

we would like to have the fuli set of reductions between any two of edge-, range-, and vertes-relased graceful labellings; if those from edge-relaxed graceful labellings did not involve much loss in terms of the quantities we are concerned with then a better bound for RRG and VRG labellings could be obtained, from Rosa and Siran's bound of Fm for ERG labellings.

In chapters 3 and 4 something that is not an open problcm as much as a future target of investigation is the mass of data collected from the cornputer tests and verifications. The bulk of this data has not becn really sifted through with any thoroughness; only those patterns which "Ieapt off of the priutouts" went on to be dcveloped as theorerns. Subtler but more extensively applicable patterns may be lurking beIow the surface. It would be very nice to estract some readily recognizable subclasses for any or al1 of the following:

6 Non-caterpillar trees which have bipartite graceful labellings.

a Trees not covered by theorems 1.16 and 1.17 which do not have bipartite graceful Iabellings.

a Ubiquitously graceful trecs.

O-centred bipartite or locally-bipartite graceful trees (subclasses of either positive or negative instances). AS well, any of the ubiquitously graceful trees under bipartite or locally-bipartite constraints in either positive or negative terms.

Aside from these, the follon-ing problems seem to be worthy of attention:

1. Invcstigating the relation between the size and diameter non-bipartite graceful t.rees. This \dl likely have a pronounced number theoretical aspect; from data we have certain prime orders seem to bring out worst cases (in these orders there is a single tree of diamcter exactly Zrn wit hout any bipartite graceful labelling).

Conjecture 5.4 Al1 trees of size m with diameter > $n have bipartite graceful la bellings

2. -4 closer look needs to be taken at the handful of trees not covered by theorem 3.3 for which al1 locally bipartite graceful labellings are fully bipartite; hopefully this will be able to tell us whether we can expect more esceptions in higher orders like that of figure

3.4.

3- The finding reportcd in section 2.3, concerning trecs without locally bipartite label1irigs t hat induce path-decreasing edge labellings: t his really requires more scrutiny.

In particular, the last two trees @-en in figure 3.5 do seem to imply that some pattern is evol\.ing, but the shape of the first two leads one to wonder what possible condition the ~vholcgroup sharcs other than having diameter 6. Path-decreasing edge labeilings in gencral should be given a thorough treatmcnt (Le. for the trees not in the figure 3.5 List we still have no idca whether many or rnost locally bipartite graceful labellings induce path-decreasing cdge labellings, or if this particular combination of labellings is rare and hard to corne by).

4. One other thing that would be nice to do is finish the characterization of known trees withou t any m-edge-centred graceful labellings (chapter 4 section 1.2), by figuring out esactly which of the "trees with pom-poms" are in this set and which are not.

Question 5.5 1s there a single jormula characterizing al1 non-rn-edge-centred gmceful trees not covered by theorems 4.2 and 4.3? 5.2 Working towards the graceful tree conjecture

Our main concern, of course, is with the graceful tree conjecture. In my opinion, the most promising ways to develop the ideas presented in this thesis are the following:

1. Taking advantage of the stronger assumptions about graceful tree labellings that the first three conjectures stated at in this chapter give us.

3. Exploring the relation between vertex-relaued gracefulness and proper gracefulness.

3. Espanding and tightening the construction technique of theorem 4.5, partial labellings with closed configurations.

1. Our conjectures concerning loca! bipartiteness, O-cent redness, and O-u biquity each in\-olved imposing estra conditions on top of gracefulness; al1 imply the graceful tree conjecture (since al1 trees in the class D are caterpillars). Their extra strength niay give us the lcverage we need to make a substantial dent in the GTC. IfTith respect to locally bipartite gracefulness, theorem 3.5 represents a first step towards any generai application of the condition, but much rieeds to be done. Two things which should be priorities hcre are:

1. X simple construction for a broad, casily recognized class of trees, along the lines of the canonical caterpillar Iabclling given in figure 1.6.

2- The reader wiil rccall from chapter 3 section 3 the definition of an LB(S) labelling,

and of the sets of trees LB(S) and LB'(Sf). There WC made some speculations about the possibility of al1 trees belonging to the class CB'(Sf)for some finite set

S', but concluded from the empirical evidence that this was not Iikely to be the case. It would be worthwhile to take another look at the 66 trees of order 5 17 which did not have LB(0) (Le. bipartite graceful) or LB(1) labellings, in order to sec what the worst case(s) turns out to be. The nature of the "bad" sets S might CHAPTER5. CONCLUSIONSAND FUTUREWORK 146

give us some idea whether the folloning two questions are true or not; or whether

ive should look at some other way to manage our assumptions about how labels are disiributed in locdly bipartite grâceful labzllings.

Question 5.6 1s there a finite set S' such that eveq tree belongs to LB*(St)? Question 5.7 Does there exist a finite k such that euery tree belongs to LD(St') for sorne S" satzsfying lS"[ 5 k.

O-centred graceful and O-ubiquitously graceful trecs have already given us something to really work with, since we have al1 of diarrieter 4 except for the sets D and 2)' respectively, and cverythirig in the lowcr diameters. Hence we now know such things as:

The tree T obtained 63 identifying any vertex v of the path Pi,i # 5, and any vertex

IL in any diameter 4 tree H not belonging to the class V' is graceful.

This follows froni theorem 1.15 (which tells us al1 paths escept P5 are O-ubiquitously bipartite graceful) and the canonical amalgamation construction of theorem 1.14.

For O-centrcd labellirigs, a very straighforward way of proceeding is to tackle even diameters greater than 4, one at a time; it may be that a pattern develops by diameter 10 or so. Considcring the compIications presentcd by the second part of the proof for thcorem 4.5, a viable option may bc to limit ourselves to those trees of even diameters that are casier to work with; if ive can obtain a construction for a large proportion of al1 even diarneter trees it may bc possible to extract general gracefulness proofs for the rernaindcr.

2. As reported in chapter 2 section 3, no ungraceful connected graph with a vertes- relascd graceful labelling has been found; however, a thorough search through orders > 6 still needs to be done. One thing on the agenda will then of course be to adapt somc of the programs used in this thesis to such a purpose; this will involve some work, since these programs for efficiency reasons are very tree specific (a bnel description of CHAPTER5. CONCLUSIONSAND FUTURE~ORK 147 the programs is given in the appendk). Consideration of what kind of structure a graph must have if it is VR graceful but not simply graceful will as well be a priority, but it is not sû immediately clear how to proceed ~lththis.

Question 5.8 Does euery connecled graph with a vertex-relaxed graceful labelling have a proper graceful labelling ? It should be noted that if the above question is restricted to bipartite graphs and locally bipartite labellings, a positive answer still implies that the graceful tree conjecture is truc.

3. Wliile the proof given for thcorem 4.5 leaned somewhat heavily on the structure shared by al1 trecs of diametcr 4, the underlying machinery given in definition 4.6 and lcrnma 4.7 was independent of such structural conccrns. Hence 1 think that this technique bascd around partial labellings with "closed configurations" could potentially be used on other sirbclasscs of trees to good effect (1 am not so sure about general graphs: cycles will probably have a disruptive effect on the conditions we want to maintain here). There are t hrec nori-exclusive ways to go wit h t his:

a) Develop the notztion further so that the relation betwcen vertes degree and the graceful labelling givcn is more direct and automatic. Degree sequences of trees

have a good body of theorems already, and things such as probabilistic arguments may be brought to bear (graceful tree labellings have not yet given a very good foothold for methods based in probability, geometry? or matroids).

b) Try to confirrn whether for a11 graceful labellings of trees there is some order of assignments such that each partial labelling has a closed configuration. Certainly the "half' versions of this hypothesis (for consecutive unuscd vertex labels and consecutive unused edge labels individually) is extremely easy to confirm for al1 graceful labellings of graphs in general; so it is likely that the full version will not bc hard to establish for trees. This is of both theoretical and algorithmic interest: CHAPTER5. CONCLUSIONSAND FUTUREWORK 148

closed configuration labelling promises to be a much more efficient way of computing graceful labellings for specific trees than the "juiced-up" brute force method of the programs used for this thesis.

c) The closed configuration technique as it stands is best suited to trees which are

structurally "shallow", in the sense that removal of al1 degree 1 vertices gives us

a base tree which can be given an initial partial labelhg with very predictable features. -4s such there is a chance that the class of lobsters will respond favourably

to its use (you will remember from chapter 1 section 2.3 that Berrnond stated a "graceful lobster conjecture" in the late 1970's as a trial run / first step towards the GTC). Trees that are "deep", on the other hand, will probably have to be attacked recursively. One possibility is a construction for range-reked graceful labellings with certain closed configurations and some mandated label locations. Thesc can be used in a version of the canonical amalgamation construction given

in theorem 1.14; the bipartite graceful labelling of the first tree is replaced with a "pre-strctched" labelling with the appropriate gap and one vertex labelled so as to facilitate attaching the second tree to it. It is likely that the recursion will have

to bottom out at an ordcr considerably larger than 1, which dlrequire a certain

amount of case work as well. Appendix

Programs used for the empirical results

Since many of the results stated in this thesis u7ereobtained by cornputer programs, some

Ilricf description of these is marranted. Space does not permit reprinting the source code or data files verbatirn, but anyone wishing to see these for themselves can contact me at

and I dlbe happy to fonvard copies of the requested items.

INPUT/OUTPUT: h4ost of the programs were dcsigned to check al1 trees of a given ordcr, with optional low and high diameter specifications. Output for these programs n-ould consist of the total number 01 instances (usually negative) found plus the total riuniber of trees checked; in almost al1 programs the option was present to print out the found instances as ASCII drawings designed to look ok with a monospace font such as courier; an example is given below: UNDERLYING MODULES: The algorithms used were al1 very tree specific, since t-arious time and space optimizations were available under the assumption that the graph ir connected and acyclic. The basic data format used throughout for a tree of order n is an integer array of size n containing the "parent Iist" of the vertices (such as would be obtained from dept h-first or bread-first search algorit hms) ; vertices are given a provisional ordering from O to n - 1, vertes O is considered the root and is given parent "-ln,and the i-th place in the array contains the nurnber of the one vertex closer to the root that vertex i is adjacent to.

Every program made use of several or al1 the following modules. -411 code was written in C.

treegri : The graceful tree labelling cngine. Input consists of an integer giving the size of the trce, an integer flagging whether the found labellings should restricted to one of the bipartite variations or not, an integer array which holds the parent list of the trce, optional arrays for automorphism information and a fked initial labelling of some verticcs respectively, and a pointer to a function mhich will be activated on every successful labelling.

The lahclling technique used is essentially brute force; to establish counter-examples ive have as of yet no better way. Partial labellings prog-rcss through the parent list in such a rnanner that at every stage a connected subtree is labeHed; vertices are assigned only unused labels which gencrate needed edge labels; and the algorithm backtracks if it rcaches a dead end or we request more labeilings. This brute force is rnitigated somewhat bj- two optimizat ions:

1. Automorphism information is utilized so as to not repeat labellings or go down the samc dead end more than once.

2. Extensive fonvard checking is implemented. Various sums arc taken to determine if assignment of a particular label to a particular vertex will violate one of the APPENDIX

following conditions, making completion to a graceful labelling impossible:

Every edge difference we need can still be generated from the vertex labels are have left.

Every unused vertes label contributes to a needed edge difference.

For every labelled vertex there are vertex labels available for their unlabelled neighbours which generate needed edge differenccs.

The total number of distinct edge differences which can generated between labelled and unlabelled vertices is sufficient to label al1 edges out of the labelled subtree.

The number of distinct edge differences generated among unused vertes labels alone is sufficicnt to label al1 edges in the unlabelled part of the tree.

\.STLiilc the brute force base of the algorithm makes applying it to, Say, al1 trees of

order 20 or greater infeasible, for individual trees of somewhat larger size a labelling can

usually bc obtaincd in a livable amount of time.

treegen: The trce generator. Gencratcd trees in a standard ordering based on the depth and sizc of branches from the centre wrtes (or central edge for odd diameter trccs). Output is a tree in its parent list reprcsentation with an additional array holding automorphism information for the tree labeller.

treeut il : Some utilities for e.g. detecting autornorphisms, rotating trees, or changing the root of arbitras, trees in the parent list representation. treeprn: Prints a given tree in the .WC11 format shown above. PROGRAMS: These are the programs that were used for the various tests. Al1 escept the first were designed to check al1 trees of a given order.

grt: The basic one tree at a time graceful labeller. Required input is the order and the parent Iist, with elements of the list separateci by commas only. Various other options are available to specify whether the program should find locally bipartite or fully biparti te graceful labellings (the default is any graceful labelling) ; to give some 6xed initial labelling to some of the vertices; and to speciS how the output is to be given. By default al1 graceful !abellings in the ASCII format mentioned above are output, with the

count; the option to stop after the first positive instance was found, or to page through theni is also present.

Bipartite and locally bipartite graceful trees and labellings:

lbpg (chapter 3 section 2.1): Used to confirm that al1 trees of order 18 and less had locally bipartite graceful labelling. Process involved first testing if tree had fully

bipartite graceful labelling, so option was included to Save and count al1 non-bipartite graceful trees encountered. A modified version which checked directly for the existence

of a locally bipartite graceful labelling was used for order 19.

labcount (chapter 3 section 2.2): Used to count atl graceful labellings by order/diameter and by type (G, LB, or B).

glabprn (chapter 3 section 3.2): Did the same as labcount but on a tree by tree basis;

printed cvery tree with their particular statistics. This prograrn rvas run up to order 8;

its use \vas not really feasible past that point.

bornlb (chapter 3 section 2.2): \Vas used to find both (1) trees for which al1 graceful

labellings were locally bipartite, and (2) trees for which al1 locally bipartite graceful labellings were fully bipartite.

lbl, lblstar (chapter 3 section 3): Counted trees not in the classes LB(1) and LB(1)' respectively.

glball (chapter 3 section 2.3): Used a slightly modified fonn of the treegrl module that only accepted Iabellings which induced path-decreasing edge labelliogs as defined prior to theorem 2.3.

O- centred, m-edge-centred, and ubiquitously graceful trees: ctrO and ctre (chapter 4 section 1.2): Gave counts of trees without O-centred and m-edge-centred graceful labellings respectively. Specification of all, localiy. bipartite, or fully bipartite labellings %-asan option. gut and gute (chapter 4 section 3): gut Counted O-ubiquitously and ubiquitously graceful trecs while gute counted m-edge-ubiquitously and edge-ubiquitously graceful

trecs. For output we included the option to print either negative or positive instances of O-ubiquity and m-edge-ubiquity; only positive instances of ubiquity and edge-ubiquity

werc kcpt track of- Bibliography

1. R.E.L. Aldred and Brendan D. McKay, Graceful and harmonious labellings of trees,

Bulletin of the ICA 23 (1998) 69-72

2. 1. ,Anderson, Cornbinatorial Designs and Tournurnents, Oxford University Press,

(1997)

3. R. Balakrishnan, R. Sarnpathkumar, Decompositions of regular graphs into KG V

2K2, Discrete Malh. 156 (1996) 19-28

4. J-C. Bermond, GracefuI graphs, radio antennae, and French windmilis, in Gruph Theory and Combznaton'cs, R.J. Wilson (ed.), Pitman Publishing Ltd. (1979) 18- 37

5. G.S. Bloom and S.W. Golomb, Numbered complete graphs, unusual rulers, and as- sortcd applications, in Theory and Applications of Graphs, Lecture Notes in Math., 642, Springer-Verlag (1978) 53-65

6. B. Bollob&, Random Graphs, Academic Press (1983)

7. B. Bollob&, Modem Graph Theory, Springer-Verlag (1998)

S. C.P. Bonnington and J. siran, Bipartite labelings of trees with maximum degree three, preprint. 9. H.J. Broersma, C. Hoede, Another equivalent of the graceful tree conjecture, Ars Combinaton'a 51 (1999) 183-192

10. M.Burzio, G. Ferrarese, The subdivision graph of a graceful tree is a graceful tree, Discrete Math. 181 (1998) 275-281

11- 1- Cahit, Status of the graceful tree conjecture in 1989, in Topics in Cornbinatorïcs and Graph Theonj, R. Bodendiek and R. Henn (eds.), Physics-Verlag (1990) 175-

184

12. F.R.K. Chung and F.K. Hwang, Rotatable graceful graphs, Ars Combilratorica 11 (1981) 239-250.

13. F.R.K. Chung, Labellings of graphs, in Selected Topics in Graph Theory, L. Beineke and R. Wilson (eds.), Academic Press (1988) 151-168

14. K. Eshghi, The Existence and Constmction of a-vafuations of 2-regular Graphs with Three Components. Ph.D. Thesis, Industrial Engineering Dept., University of Toronto (1997)

15. S. Finch, 0tter7s Tree Enumeration Constants, http://wuu.mathsoft.com/asolve/constant/otter/otter-html

16- J.-4. Gallian, A dynamic survey of graph labelling, Electronic Journal of Cornbina- ton'cs 5 (1998) 1-43

17. S.W. Golomb, How to number a graph, in Craph Theory and Computing, R.C. Read (ed.), Acadernic Press (1972) 23-37

18. T. Grace, New proof techniques for gracefully labelhg graphs, Congressus Numer- anlium 36 (1982) 403-408

19. T. Grace, More graceful trees, Congressus Numerantium 39 (1983) 433-439 20. R.L. Graham and N.J.A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Disc. Meth. 1 (1980) 382-404

21. F. Harary and D.F. Hsu, Node-graceful graphs, Comput. MaW. Applzc. 15 (1988) 39 1-398

22. C. Huang, A. Kotzig, and A. Rosa, Further results on tree labellings, Utilitas Mathematica 21C (1982) 31-48

23. D. Jin, S. Lee, H. Liu, S. Liu, ,Y. Lu, and D. Zhang, The joint sum of graceful trees, Cornputers Math. Applic. 26 (1993) 83-87

24. D. Jin, S. Lee, 14. Liu, S. Liu: X Lu, and D. Zhang, The radical product of graceful trees, Cornputers Math. Applic. 26 (1993) 89-93

25. K.hl. Uoh, D.G. Rogers, and T.Tan, Products of Graceful Trees, Discrete Math.

31 (1980) 219-292

26. -4. Kotzig, On certain valuations of finite graphs, Utilitas Mathematica 4 (1973)

161-390

27. -4. Kotzig, Recent results and open problems in graceful graphs, Congressus Nu-

merantium 44 (1984) 197-219

28- J. Leech, On the represcntation of 1,2,. . . ,n by differences, J. London Math. Soc. 31 (1956) 160-169

29. A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Inter- national Symposium, Rome, July 1966), Gordon and Breach, NY (1967) 349-355

30. A. Rosa, Labeling snakes, Ars Combinatoria 3 (1977) 67-74

31. A. Rosa and J. siriin, Bipartite labelings of trees and the gracesize, Journal of Graph Theory 19 (1995) 201-215 32. P.D.Schumer, Introduction to Nurnber Theory, PWS Publishing Company, (1996)

33. D. A. S heppard, The factorial representation of balanced labelled graphs, Discrete .&fath. 15 (1976) 3'79-388

34. J. Singer, -1 theorem in finite projective geometry and some applications to number theory, Tram. Amencan Math. Soc. 43 (1938)377-385

35. N.J..-\. Sloane, The On-Line Encyclopedia of Integer Sequences (Look-Up), http://uw.research.att.com/-njas/sequences/eisonline.html,

lookup sequences A033472 (gracefully labelled trees) and -A000055 (unlabelled trees).

36. D.E. Speyer and Z. Szaniszl6, Every tree is 3-equitable, Discrete Math. 220 (2000) 383-259.

37- R.G. Stanton and C.R. Zarnke, Labellings of balanced Trees, Congressus fimer- antiurn 8 (1973) 479-495

35- D. West, Introduclion to Graph Theory, Prentice Hall (1996)