The Path Partition Number of a Graph

The path partition number of a graph

ELIZABETH JONCK

THESIS

submitted in the fulfilment

of the requirements for the degree

PHILOSOPHIAE DOCTOR

MATHEMATICS

in the

FACULTY OF SCIENCE

at the

RAND AFRIKAANS UNIVERSITY

PROMOTER: PROF. I. BROERE

October 1997 Acknowledgements

I wish to thank:

• RAU for their financial assistance,

• Professor Izak Broere for his motivation and instructive guidance,

• Cobus and Lindy for their professionalism and friendly help,

• Andries for his encouragement and help with the proofreading.

I wish to dedicate this thesis to my family, in particular to "Pa Frik" and "Ma Mart"; Your love and support are precious to me!

Contents

1 Introduction and overview 1

1.1 Introduction ...... 1

1.2 Overview ...... 2

2 The induced path number of some classes of graphs 4

2.1 The induced path number of bipartite graphs ...... 4

2.1.1 Introduction ...... 4

2.1.2 The induced path number of some classes of bipartite graphs . 6

2.2 The induced path number of a complete multipartite graph ...... 14

2.2.1 Introduction ...... 14

2.2.2 Proof of Theorem 2.17 ...... 16 I

3 The induced path number of some products of some graphs 31

3.1 Introduction ...... 31

3.2 Products of complete graphs ...... 32

3.2.1 The induced path number of Km X K ...... 32

3.2.2 The induced path number of the complement of Km X K ...... 42

3.2.3 The induced path number of Cm X Cn ...... 49

4 Uniquely partitionable and saturated graphs with respect to linear ar-

boricity 56

4.1 Introduction ...... 56

4.2 Uniquely partitionable and saturated graphs ...... 57

4.3 The order of uniquely partitionable graphs ...... 62

5 Vertex and edge addition and deletion 65

5.1 The path partition number and the addition or deletion of a vertex or an edge. 65

5.2 p-criticality ...... 68

Bibliography 75

Index 78

List of Figures

2.1 An induced path partition 6

2.2 The binary trees B2 and 133 ...... 7

2.3 The cartesian product C1 x C2 ...... 9

2.4 The 2-dimensional mesh Md,,d2 ...... 9

2.5 The butterflies B1 , B2 and B3 12

2.6 Partition classes of cardinality one, two and three ...... 17

2.7 Partition classes containing a vertex of V€ ...... 18

2.8 Partition classes containing one or two vertices of V1 ...... 20

2.9 The remaining subcases of case (iii) ...... 21

2.10 The remaining subcases of case (viii) ...... 22

2.11 The situation after y iterations ...... 23

2.12 The remaining subcase of case (ix) ...... 24

111 iv

2.13 The remaining subsubcase of case (ix) ...... 24

2.14 The situation after z iterations ...... 25

2.15 The seventh case ...... 27

3.1 Rows and columns of C ...... 32

3.2 Km x K, m odd (the first case) ...... 34

3.3 If, x K, in even (the first case) ...... 35

3.4 Km x K,, in a multiple of 4 (the second case) ...... 37

3.5 Kn x K,, in not a multiple of 4 (the second case) ...... 38

3.6 Km x K, m odd (the third case) ...... 40

3.7 K, x If,,, m even (the third case) ...... 41

3.8 E3 x ...... 42

3.9 K3 XIc, n odd ...... 45

3.10 K3 X K, n even ...... 46

3.11 ......

3.12 Cm X C,-, fl = m ...... 51

3.13 Cm X C, fl > m ...... 52

3.14 Cm X C, n 4a and m = 2a(2k - 1) + 1 ...... 53 V

3.l5CmXCn ,m even and n>in 55

4.1 Possible partitions ...... 61

4.2 Possible partitions ...... 63

4.3 (a) Graph F (b) ...... 64

5.1 C3.K5 ...... 70 Opsomming

Die gemnduseerde padgetal p(G) van 'n grafiek C word gedefinieer as die minimum aantal

deelversamelings waarin die puntversameling V(G) van G verdeel kan word so dat elke

deelversameling 'n pad induseer.

Ons bepaal die geinduseerde padgetal van 'n volledige £-ledige grafiek.

Verder ondersoek ons die geInduseerde padgetal van produkte van volledige grafieke, van die

komplement van sulke produkte en van produkte van sikiusse.

Vir 'n grafiek C word die lineêre puntbosagtigheid Iva(G) gedefinieer as die minimum aantal

deelversamelings waarin die puntversameling van C verdeel kan word so dat elke deelversameling 'n lineêre bos induseer. Omdat elke pad 'n lineêre bos is, is Iva(G) p(G) vir elke grafiek G. Ons noem 'n grafiek C uniek rn-lineér-bos-verdeelbaar as Iva(G) = in en daar slegs een verdeling van V(C) in in deelversamelings is so dat elke deelversameling 'n lineêre bos induseer. As Iva(G) in en lva(G+e) > in vir elke e E E(), sê ons die grafiek C is rn-lva- versadig. Ons konstrueer grafieke wat uniek m-lineêr-bos-verdeelbaar en m-lva-versadig is.

Ons karakteriseer die ordes van uniek m- pad- verdeel bare onsamehangende, samehangende en in-p-versadigde grafleke.

vi vii

Ons bekyk die invloed van byvoeging en weglating van 'n punt of 'n lyn op die geInduseerde padgetal. As C 'n grafiek is so dat p(G) = k en p(G - v) = k - 1 vir elke v E V(G), dan sê ons dat C k-minus-kritiek is. Ons bewys dat as C 'n samehangende grafiek is, bestaande uitsikliese blokke Bi met p(B1 ) = b1 yin = 1,2, ... ,n waar n >2 en k= b1 —n+ 1, dan is C k-minus-kritiek as en slegs as elke blok B 'n b1-minus-kritieke grafiek is. Summary

The induced path number p(G) of a graph G is defined as the minimum number of subsets

into which the vertex set V(G) of G can be partitioned such that each subset induces a path.

In this thesis we determine the induced path number of a complete £-partite graph.

We investigate the induced path number of products of complete graphs, of the complement

of such products and of products of cycles.

For a graph G, the linear vertex arboricity lva(G) is defined as the minimum number of subsets into which the vertex set of C can be partitioned so that each subset induces a linear forest. Since each path is a linear forest, Iva(G) p(G) for each graph C. A graph G is said to be uniquely rn-li near- forest- partition able if lva(C) = in and there is only one partition of V(G) into m subsets so that each subset induces a linear forest. Furthermore, a graph

C is defined to be nz- Iva- saturated if Iva(G) < in and lva(C + e) > iii for each e E

We construct graphs that are uniquely n2-linear-forest-partitionable and in-lva-saturated.

We characterize those graphs that are uniquely m-linear-forest-partitionable and rn-lva- saturated. We also characterize the orders of uniquely in- path- partitionable disconnected, connected and rn-p-saturated graphs.

viii ix

We look at the influence of the addition or deletion of a vertex or an edge on the path partition number. If C is a graph such that p(G) = k and p(G - v) = k - 1 for every v E V(G), then we say that C is k-minus-critical. We prove that if C is a connected graph consisting of cyclic blocks Bi with p(B1 ) = b, for i = 1,2, ... ,n where ii > 2 and k bi - n+ 1, then C is k- minus- critical if and only if each of the blocks B1 is a bj- minus- critical graph. Gliapter 1

Introduction and overview

1.1 Introduction

We generally use the notation and terminology of [8].

For a graph C, the induced path number p(G) is defined by Chartrand e.a. in [9] as the

minimum number of subsets into which the vertex set V(C) of G can be partitioned such

that each subset induces a path. They investigated the induced path number for bipartite

graphs and presented formulas for the induced path number of complete bipartite graphs

and complete binary trees. They also determined the induced path number of all trees and considered the induced path numbers of meshes, hypercubes and butterflies.

1 1.2 Overview

In Chapter 2 we summarise the results of [9] and determine the induced path number of a

complete £-partite graph.

We investigate the induced path number of products of complete graphs, of the complement

of such products and of products of cycles in Chapter 3.

For a graph C, the linear vertex arboricity Iva(G) is defined as the minimum number of

subsets into which the vertex set of C can be partitioned so that each subset induces a linear

forest. A graph G is said to be uniquely rn-linear-forest-partitionable if Iva(G) = in and

there is only one partition of V(G) into m subsets so that each subset induces a linear forest.

Furthermore, a graph C is defined to be rn-lva-saturated if Iva(G) in and lva(G + e) > in for each e E(). In Chapter 4 we construct graphs that are uniquely rn-linear-forest- partitionable and m-lva-saturated. We characterize those graphs that are uniquely rn-linear- forest-partitionable and rn- Iva- saturated. We also characterize the orders of uniquely in- path-partitionable disconnected, connected and rn-p-saturated graphs.

In Chapter 5 we look at the influence of the addition or deletion of a vertex or an edge on the path partition number. If C is a graph such that p(C) = k and p(G - v) = k - 1 for every v E V(C), then we say that C is k-minus-critical. We prove that if C is a connected graph consisting of cyclic blocks Bi with p(B1 ) = bi for i = 1,2,... , n where n > 2 and k = b1 - n + 1, then G is k-minus-critical if and only if each of the blocks B1 is a b-minus-critical graph.

In [2] an encompassing theory of partitions of the vertex set V(G) of a graph G is discussed. 3

The contents of this thesis does not fit into the framework given in [2] since the property

"to be an induced path" is not hereditary. Nevertheless, the topic studied in this thesis has given rise to interesting results on notions that are typical in [2], viz, uniquely partionable graphs and critical graphs. Chapter 2 =

The induced path number of some

classes of graphs

2.1 The induced path number of bipartite graphs

2.1.1 Introduction

The arboricity of a nonempty graph C is the minimum number of subsets into which E(G)

can be partitioned so that each subset induces a forest. In one of the major results in graph

theory Nash-Williams [17] proved that the arboricity of a nonempty graph C is given by

IE(H)i 1 max IV(H)I - 1 where the maximum is taken over all nontrivial induced subgraphs H of G. Harary [13] later defined the linear arboricity of a nonempty graph C as the minimum number of subsets into

4 5

which E(G) can be partitioned so that each subset induces a linear forest, that is a forest

in which every component is a path. He went further and defined the path number of a

nonempty graph C as the minimum number of subsets into which E(G) can be partitioned

so that each subset induces a path. A number of results on the path number of a graph were

obtained by Stanton, Cowan and James [20] =

Partitioning the vertex set V(G) of a graph C into subsets according to certain rules is a

fundamental process in Graph Theory. The study of partitioning V(G) into the minimum

number of subsets such that each subset induces an independent subgraph of G, that is, the

study of the chromatic number of a graph, played a vital role in Graph Theory.

Generalizations of this idea are described amongst others in [6], [8], [15], [16] and [19]. One

of these generalizations was by Chartrand e.a. in 1994. They defined in [9] the induced

path number p(G) of a graph G as the minimum number of subsets in a partition of V(G)

such that each subset induces a path. For the graph G of Figure 2.1(i) (p 6), p(G) = 2. A

partition of V(G) into two subsets, each of which induces a path, is shown in Figure 2.1(u)

(p 6). In this chapter a partitioning using at most r subsets will be called an r-pat h-part ition

of G and the elements of the partition will be called partition classes.

In [7], Chartrand, Kronk and Wall defined the vertex- arboricity a(G) of a graph G as the

minimum number of subsets in a partition of V(C) such that each subset induces a forest,

while Harary introduced in [12] the linear vertex arboricity lva(G) of a graph C as the minimum number of subsets in a partition of V(G) such that each subset induces a linear forest. Consequently a(G) lva(G) p(C) for every graph C.

The minimum number of components in a spanning linear forest in a graph C was introduced 3

Y /

(i) (ii)

Figure 2.1: An induced path partition

by Boesch, Chen and Mc Hugh [1] and called the island number is(G) of G by Slater [18].

Thus is(G) p(G) for every graph C.

In [9] Chartrand e.a. investigate the induced path number of bipartite graphs and in partic-

ular complete bipartite graphs, complete binary trees, meshes, hypercubes and butterflies.

They also determine the induced path number of trees. These results are summarized in the

following subsection.

2.1.2 The induced path number of some classes of bipartite

graphs

All the results in this subsection are from [9].

First we present a formula for the induced path number of the complete bipartite graph

Km,n where rn n. 7

V0,1 level 0 - level 0 edges V7 2 level 1 - level 1 edges level 2 V2 ,1 V2,2 V2,3 V2,4 - level 2 edges level 3 -+-

Figure 2.2: The binary trees B2 and B3

Theorem 2.1 Let rn and n be positive integers with in n. Then

un + ni if 77-1 2in.

We now consider the induced path number of a complete binary tree which can be described as follows:

We denote the complete binary tree of height h by Bh. The vertices of B,, are labelled according to the level to which they belong. In particular, the root (at level 0) is labelled v0,1 . The two vertices at level 1 are v1,1 and V1 ,2. In general the vertices at level 1(0 l h) are labeled v1,1 , v1,2 , . . . , v1,1.

The edges between the vertices of level i and level i + 1 (0 i h - 1) are referred to as level i edges. We depict B2 and B3 in Figure 2.2 (p 7).

A formula for the induced path number of a complete binary tree is given in the following theorem. 8

Theorem 2.2 For the complete binary tree B h of height h, we have

(2 h+1 + 1) if h is even p(Bh ) = J 1 1(2h+1 —1) if h is odd.

The induced path number of meshes is investigated in the following paragraph.

Following [8], the cartesian product C = C1 x C2 has V(G) = V(G1 ) x V(G2 ) and vertex

(ui , u2 ) of C is adjacent to vertex (v1 , v2 ) of G if and only if either

ul = v1 and u2v2 E E(C2)

U2 = v2 and u1v1 E E(G1).

For example, if C1 = K2 and G2 = K3 the cartesian product C1 x C2 is the graph C shown

in Figure 2.3 (p 9).

For positive integers d1 and d2 , the 2-dimensional mesh Md,,d2 is defined as the cartesian

product Pd, x Pd2 . In what follows we assume that the mesh Md1 ,d2 is drawn and labelled

as shown in Figure 2.4 (p 9).

Clearly, p(Md1,d2 ) = 1 if and only if either d1 = 1 or d2 = 1. The following theorem gives the

induced path number of Md,,d2 when neither d1 = 1 nor d2 = 1.

Theorem 2.3 If dl , d2 ^! 2, then p(Md1,d2 ) = 2. 0

1111I111

Figure 2.3: The cartesian product C1 x C2

Vd12 Vd1 ,1 40 . . . --40 Vd1,d2

-H - —H -

—H - —H —H

• S

• •

• S

—H - - - V2d2

V1,1 —H------H— • • S —H------V1,2

Figure 2.4: The 2-dimensional mesh Md1,d2 10

We now extend this result to 3-dimensional meshes. In general, the k-dimensional mesh

Md,,d2... . dk is defined as the Cartesian product Pd, x P 2 x ... X Pd, Given a 3-dimensional

mesh Md1 ,d2 ,d3 , we form 2-dimensional meshes as follows: For k = 1,2,.. . , d, let

Vk = { v1,,kl1

Each set Vk , 1 k d3, induces a 2-dimensional mesh. Using Theorem 2.3, we see that each

1/1k set Vk can be partitioned into subsets and V2k which induce paths PjIc and P respectively.

Note that each vertex in P (respectively P) is adjacent to a corresponding vertex in p1k+1

(respectively P 1 ) for 1 k < d3 . Therefore, the vertex sets V1' = U. 1 V and 1/' = UIV2k

induce two 2-dimensional meshes. Using Theorem 2.3 again, we conclude that we need at

most four subsets in a partition of the vertex set of Md, ,d2 ,d3 such that each subset induces

a path. Thus p(Md1,d2,d3 ) < 4.

Using the idea of decomposing a k-dimensional mesh into two (k - 1)-dimensional meshes, we obtain the following result:

Theorem 2.4 If dl , d k 2, . . ,d are any positive integers, then p(Md1d2dk) 21. 0

For a discussion of the induced path number of hypercubes, recall that the d-dimensional hypbercube Qd is defined recursively by Qi = K2 and Qd = Qd—i x K2 for d > 2. The problem

of determining P(Qd) seems to be quite complex, but the following is known for small values

of d.

Theorem 2.5 P(Q2) = p(Q) = 2, p(Q) 3 and p( Q) <4. 0 11

From the definition of hypercuhes, it follows that p(Qd+1) 2p(Q4. Since we also know

that p(Q) 4, we can conclude that p(Qd) <2d_3 for all d> 5. The actual value of p(Qd) is believed to be much smaller.

Conjecture 2.6 P(Qd) 2.

We consider now another class of bipartite graphs, namely the butterflies. Like the hyper-

cubes, the butterflies are defined recursively:

2r-1 Consider Br-1 with its bottom vertices 1,2,. .. , 2'. Br is constructed from two copies of Br-1 with bottom vertices 1,2, ... , 2r-1 and 1',2',. .. , (2)', by adding 2r new bottom vertices v1 , v2 ,. .. V2--1, Wi, W2,.. . W2r-1 with each vi and each wi adjacent to i and

1,2,... ,2r_i. The construction of the first three butterflies are shown in Figure 2.5

(p 12) with B3 partially labelled according to the definition.

r Clearly the butterfly B,. contains 2 induced paths of length r and thus p(B,.) < 2 for all r.

Since B1 is a cycle, p(B1 ) = 2. For r = 2k, the butterfly Br = B2k is a k . 22k x (k + 1) . 22k bipartite graph, that is, its bipartition can be chosen with k . 22k vertices in one set and

(k + 1) . 22k vertices in the other set. For r = 2k - 1, the butterfly B,. = B2k_1 is a k 2211 x k . 22k-1 bipartite graph. Every induced path in a bipartite graph contains at most one more vertex in one of its partite sets than the other. Since B2k has 2" more vertices in one partite set than the other, it follows that p(B2k) 2k Hence we have the following result:

Theorem 2.7 p(Br) 2 for every even positive integer r. 0

BI: N B2:

B3:

it 4 2f 41

V2 V3 V4 W1 W2 W3 W4

Figure 2.5: The butterflies B1 , B2 and B3

Chartrand e.a. (in [9]) conjecture the following:

Conjecture 2.8 p(Br) = 2' for every odd positive integer r.

Finally we investigate the induced path number of trees. We begin by giving the character-

ization of the n-path-partitionable trees of [9].

Theorem 2.9 A tree T is n-path partitionable if and only if there exists a set of fewer than

n edges whose removal from T results in a linear forest. 0

As an immediate consequence of Theorem 2.9, we obtain

Corollary 2.10 Let T be a tree of order p with p(T) = ii. Then the minimum number of

edges whose removal from T results in a linear forest of size p - p(T) is n - 1. 0 13

Let T be a tree. For a vertex v of T with degv 3, we define the excess degree e(v) of v by

(v) =degv-2.

Theorem 2.11 Let T be a tree and let H be the forestinduced by the vertices of T having degree 3 or more. Let H' be a spanning subforest of H of maximum size such that degl V

6(v) for every vertex v of H. Define £ = IE(H')I + [e(v) - deglv]. vEV(H) Then p(T)=t+1. UI

From Theorem 2.11, we obtain

Corollary 2.12 Let T be a tree and H the forest induced by the vertices of T having degree three or more. Let H' be a spanning subforest of H of maximum size such that deg, v e(v) for every vertex of H. Then

p(T) = 1 + JE(H')I + [c(v) - deg, v]. VEV(H)

The next two theorems give an upper and a lower bound for the induced path number of a tree. We call a tree T an in x n tree if its bipartition can be chosen with sets having cardinalities m and n.

Theorem 2.13 Let T be an m x n tree of order p(= m + n) with 3 < in < n. Then p(T)p-4. 0 14

Theorem 2.14 Let T be an in x ii tree of order p with in ii. Then p(T) > p - 2in. U

Combining Theorems 2.13 and 2.14, we obtain

Corollary 2.15 If T is an rn x n tree with 3 m < n, then max {1,n - in) p(T) :5 rñ+ñ-4. - - 0

The bounds given in Corollary 2.15 are indeed sharp as can be seen from the following interpolation result.

Theorem 2.16 Let k, m and n be integers such that 3

Then there exists an m x n tree T such that p(T) = k.

In [10] the induced tree number r(G) of C is defined as the minimum number of subsets into which V(G) can be partitioned so that each subset induces a tree. The induced Li-tree number r(G) requires that each induced tree has maximum degree at most L. Consequently, the induced 2-tree number is the induced path number.

2.2 The induced path number of a complete multi-

partite graph

2.2.1 Introduction

A complete €-partite graph C is an £-partite graph with partite sets 1 /1, V2 ,.. . , V having the additional property that uv E E(G) for each u E V, and v E V with 1 0 j. If 1V1 = 15 then this graph is denoted by I(, ,2.....T1 In general, the order of the numbers n1 , n2 ,. .. , is not important. For every complete £-partite graph considered in this chapter however, we order and rename the numbers n1 , n2 ,.. . , nt so that n1 <2 < ... <

Chartrand e.a. present a formula for the induced path number of a complete bipartite graph (p 7). In this section we present a formula for the induced path number of the - complete £-partite graph It',21,, ...,. In the remainder of this chapter we assume that n > 2 for at least one i and £ > 2, since otherwise Kni,n2 ,-.-,nt is Ift or N 1 with p(Ke) = 1/21 and p(N,) = ni respectively. Throughout the remainder of this chapter, we denote K1,2 by C.

Since the largest order of an induced path in is three, we have p(C) > ( ni +n2 + . and since p(C) is an integer,

Denote the number of odd n3 's by d and let

a=d — n+ (n1-1) ni even ni odd

We can now formulate

Theorem 2.17 (i) If n < 2(n i + n2 + + n1_1 ) then

Iii + [a] (n - 1)1 if a E {3,5,6,. . even p(G)=J n, odd ] 1 I nh/3 otherwise. I I 1=1 1

t-1 (ii) If nj > 2(n 1 + n2 + + n_ 1 ) then p(G) = n - 16

£ Note that in the first part of case (i) as well as in case (ii) we have p(G) >

Theorem 2.1 (p 7) is a special case of Theorem 2.17 (since a if £ = 2).

2.2.2 Proof of Theorem 2.17

Suppose throughout that p(G) = r.

To prove Theorem 2.17, we start off by showing in three lemmas that there exists an r-path- partition of C in which a particular type of partition class is forced to occur.

Lemma 2.1 For each r-path-partition of C which has two partition classes of cardinality

one contained in V and each i j, there is a partition class of cardinality three with one vertex in V, and two vertices in V.

Proof Consider any r-path-partition of C in which two vertices of Vj belong to partition classes of cardinality one. A vertex of V, (for any i j) can appear in partition classes of cardinality one, two and three in the seven ways indicated in Figure 2.6 (p 17).

In cases (i) - (vi) of Figure 2.6 (p 17) however, an (r - 1)-path-partition of C is easy to find. 0

Lemma 2.2 There is an r-path-partition of C with a partition class of cardinality three which contains two vertices of V. 17

vi Vi Vk

L___ J

I • I

L___ J L ------

V3 1/;

(i) (ii) (iii)

[I+II] [I+I]

I • • I I • • L J L J [1.1 V3 113 Vi Vi

(iv) (v) (vi) (vii)

Figure 2.6: Partition classes of cardinality one, two and three

Vt Vt Vt 1/. I I I I I I I I I I L___ J

(i) (ii)

Vt I I I I r - - - Vi - Vt - - I I I I I• I I I I I I I L7I L__JL/_J I •I I L__J L Vi (iv) (v)i (vi) -Y L I

- VI Vt r -- 1 I ------r Vi Vt '------'I II I r - -I r-- -I I I I I I I I I - J I 4HI I I I I I I -j j L_ j I L __ J (vii) (viii) (ix) L___J

Figure 2.7: Partition classes containing a vertex of V€ 19

Proof Observe the nine possible ways depicted Figure 2.7 (p 18) in which two vertices of

V can appear in partition classes without the occurence of the desired partition class.

In case (i), Lemma 2.1 applies (with j = 1) to give the desired partition class. In case

(ii), a repartitioning in r - 1 partition classes is possible. In cases (iii) to (vi) and (viii), a repartitioning of C in r partition classes containing the desired partition class is easy to find.

In cases (vii) and (ix) we proceed as follows: At least one vertex of V does not belong to a partition class of cardinality two or three with the other vertices of these partition classes in

V, otherwise n3 > nt. Such a vertex of V, however, can occur in another partition class of cardinality one, two or three. In each instance, a repartitioning of C either in r - 1 partition classes or having the desired partition class is easy to find. 0

Lemma 2.3 Suppose C '(1,2,2. Then there is an r-path-partition of C containing a partition class of cardinality three which consists of a vertex of V1 and two vertices of V.

Proof We prove this lemma by combining the result obtained in Lemma 2.2 with all the possible ways in which one or two vertices of V1 can appear in partition classes in an r- path-partition of C without the occurrence of the desired partition class. Suppose that the vertices which form together with a vertex in V1 an induced path, are in 1'. Suppose further that the vertex which form together with 2 vertices of V an induced path of order three, is in V. Note that V = Vj or V1 = V€ is possible. In total we need to consider the nine cases depicted in Figure 2.8 (p 20).

Except for cases (iii), (vi), (viii) and (ix) a repartitioning of G in r partition classes containing the desired partition class is immediate.

V1 VI V1 VI V1 Vt

Lii L i Li LJ L\L!,r ]

1<, Vi Vi v=Vj (i) (ii) (iii)

i L -1\j, L -V- J, L J, VII

Vi Vi Vi Vj (iv) (v) (vi)

V1 ye V1 V

I EE4j/ V I I -V -

I Vi Vi (vii) (viii) (ix)

Figure 2.8: Partition classes containing one or two vertices of V1 21

Vf V1 V V1 Vt [I\ L F Vi=Vj Vk Vi Vj vç= (i) (ii) (iii)

Figure 2.9: The remaining subcases of case (iii)

Case (iii): The graph G must have at least one more vertex than shown, otherwise G =

K1,2,2 . By considering a few subcases, it is easy to show that such a vertex cannot form a

partition class on its own, otherwise p(G) < r. There are 18 ways in which two or three

vertices can be added as a partition class in case (iii); 15 of them can easily be handled with

a repartitioning giving the desired partition class. The remaining three subcases are as in

Figure 2.9 (p 21).

In each of these subcases we proceed as follows: At least one vertex of V€ does not belong to a partition class of cardinality two or three with the other vertices of these partition classes in V = l/3 otherwise n3 > nt. Now we consider the possible ways in which such a vertex may occur in partition classes of cardinality two and three in an r-path-partition of C. In each of the six subsubcases in (i), the ten in (ii) and the six in (iii) of Figure 2.9 (p 21), a repartitioning of G in r partition classes containing the desired partition class is immediate.

Case (vi): Since nt ^! ri1 = n, the graph C must have at least one more vertex in V than shown. At least one of these extra vertices in V does not belong to a partition class of cardinality two or three with the other vertices of these partition classes in 17, V, otherwise 22

r--,

[I I

V.1 1<7(2) 1<7(1) 1<7

(i) (ii)

Figure 2.10: The remaining subcases of case (viii)

n3 > nt. Such a vertex of V however, can appear in another partition class of cardinality one, two or three in eight possible ways in an v- path- partition of G. For each of them, a repartitioning of G having the desired partition class is immediate.

Case (viii): Since n3 > n 1 , the graph C must have at least one more vertex in V than shown. Such a vertex can appear in partition classes of cardinality one, two or three in ten possible ways in an v-path-partition of C. Nine of them can easily be handled with a repartitioning giving the desired partition class. The remaining subcase is the one of

Figure 2.10(i) (p 22).

In this subcase we proceed as follows: Since 12,(1) ^! n 1 , there must be at least one more vertex in V, (1) than shown. Such a vertex can appear in partition classes of cardinality one, two or three in an v-path-partition of C; there are thirteen such subsubcases to consider.

rTiE -I •.• ...I••

- - - -I •.• (y) V.j(y-1)

Figure 2.11: The situation after y iterations

Twelve of them can easily be handled with a repartitioning giving the desired partition

class. The remaining subsubcase is the one of Figure 2.10(u) (p 22). In this subsubcase we iterate the construction described in the preceding paragraph. Note that this process

must end, say after y iterations, since f is finite. Hence we obtain the situation depicted in

Figure 2.11 (p 23).

This is impossible, since fl() > 1.

Case (ix): In this case we may assume that both V1 and V are of even cardinality (since

otherwise we are in one of cases (i) to (viii)). Since n, ^! n1 , there must be at least one more

vertex in V than shown. Such a vertex can appear in partition classes of cardinality one, two

or three in an r-path-partition of C in nine possible ways. Eight of them can again easily

be handled with a repartitioning giving the desired partition class. The remaining subcase

is depicted in Figure 2.12 (p 24).

In this subcase we proceed as follows: Since flj(i) ^! ni, there must be at least one more vertex

V1 Ve I I I I I I I I

IV :v;

V1 Vi

Figure 2.12: The remaining subcase of case (ix)

V1 Ve I I I I I I

L I_...__i L_...J V(2) V1 V1(i)

Figure 2.13: The remaining subsubcase of case (ix)

V1 vi

L__J L_ - _J

I I I I I I 11 V t.:::::.. •.t..:::::.. L I I ö I L L L____J L_ -- J L - - - J

Vi z) Vi

Figure 2.14: The situation after z iterations

in Vi( 1 ) than shown. Such a vertex can appear in partition classes of cardinality one, two or

three in an r- path- partition of G in 12 possible ways; 11 of them can easily be handled with

a repartitioning giving the desired partition class. The remaining subsubcase is depicted in

Figure 2.13 (p 24). In this subsubcase, we iterate the construction just described and note

again that this process must end, say this time after z iterations. We therefore obtain the

situation depicted in Figure 2.14 (p 25).

Again this is impossible since i(z) > L

Let C '(1,2,2 and consider an r-path-partition of C with x E V1 and {y, z} c V the vertices

of the partition class whose existence is guaranteed by Lemma 2.3. Let G' = C - {x, y, z}.

It is easy to see that C 1 is again a complete £ 1 -partite graph with p(GW) = r - I. If

'(i,, or a complete graph or an edgeless graph, this construction can be repeated

to form In the five results to follow we assume that ne < 2(72 i + n2 + + n 1 ) and

we consider how an iteration of this construction then ends.

Lemma 2.4 The process C - G 1 -' C 2 -' C(m) ends with G(m) = or

G(m) = ' 1,2,2 or with G as the empty graph. 26

Proof If G(m) is the empty graph, the process clearly has ended. Suppose therefore that

the graph G(m ) is not empty. Then = '1,m2 .....m3 s > 2 and rn, > 2, then the

process is not complete, unless G(m ) = K122 Hence we may assume that s = 1 or in, = 1.

If m, = 1, then G( m ) = K1,1

Suppose now that .s = 1 but rn, > 2. This means that there is only one partition class left after the process C - --4G (2 -p C() has ended; denote this partition class by - V. In the process C(m ) .._, G( m i) G(m 2) _+ ... -* C of reconstructing G we see with every step taken that IVI increases by two (to form V) and one of the other partition classes increases by one. But then nt > 2(n i + n2 + + nt_ i ), which is a contradiction. U

Lemma 2.5 Every r-path-partition of C has at most one partition class of cardinalily one.

Proof Suppose there are two partition classes of cardinality one in some r-path-partition of C. Then these two vertices must be in the same partition class of C, say they are in

V. By Lemma 2.1 there is, for each i 34 j, a partition class of cardinality three with one vertex in 1/, and two vertices in V. Now consider the seven ways in which a vertex of any vç, i j, can occur in a partition class of cardinality one, two or three: in six of these an

(r - 1)-path-partition is immediate. The seventh case is depicted in Figure 2.15 (p 27): in this case every vertex of every V, i j, is in a partition class of cardinality three with two vertices of I/. But then n3 = nt > 2(ii i + n2 + + n t_i), a contradiction. 0

Remember that a = d - n1 + E (n1 - 1) and note that, if nt > 2(n + n2 + 1-j even ni odd + nt_ i ), then a = d - [n1 + n2 +.•. + nt - dJ

Figure 2.15: The seventh case

- (n 1 + n 2 + + nt-1)] which is negative in most cases. In the next result we present an interpretation for a if a > 0.

Lemma 2.6 If a > 0, then a is the total number of vertices which are in partition classes

of cardinality one and two in an r-path-partition of C which has the maximum number of

vertices in partition classes of cardinality three.

I Proof Since a > 0, we have d > E n 2 + (n, - 1) . Hence at least n1+ n even odd n even

(n 1 - 1) partition classes of cardinality three can be formed in the process G -, CW nj odd G 2 - G(m) described above. The subgraph induced by the set of vertices that then

remain to be partitioned is a graph in which no partition class of cardinality three

can be formed. Therefore there are exactly I En + E (n - 1) partition classes of ni even n, odd 28 cardinality three and the result follows by elementary arithmetic. U

In order to consider the value of a for each graph in the process G —' — G 2 —*

we shall write aN for a of i 1,. . . , in.

Lemma 2.7 Suppose C '1,2,2• Then a(1) =- a ± 3 2if7i1 is eveñänd a' a if ni is odd. Furthermore, if a < 0, then a' <2 and if a > 0, then a 1 = a.

Proof Note that E n1 + E (ni ni — d and that the parities of IVeI and ni even ni odd I V — { y,zfl used in the process G — 0(1) are the same.

If ni is even, then a' = (d+ 1)— I [n —3— (d+ 1)] =a+3.

If nj is odd, then a(1) = (d— 1)— —3— (d —1)] = a.

It is clear from this that < 2 if a < 0.

Now suppose that a > 0. Then a' > a > 0 by the above so that a and aW both are the number of vertices in partition classes of cardinality one and two by Lemma 2.6. Since a partition class of cardinality three is formed during the process G —* 0(1), it follows that a = a

Applying this lemma to each step in the process C —' C 0(2) ... 0(m) , we obtain the following result:

Corollary 2.18 If a < 2, then a( m) < 2.

For the final stage of the proof of Theorem 2.17, we argue as follows:

To prove (i): Suppose n1 < 2(ni + n2 + + nt_i).

If p(C) ( n/3, then at most two partition classes are of cardinality one or two: if not,

then at most (1E ni/31 —3)3 + 3(2) = 3 - 3 < 1: n vertices are partitioned, which is impossible. Furthermore, if there are two partition classes of cardinailty one or two,

then both are of cardinality two: if not, then at most (1E Tl i / 31 - 2) 3+2+1 = 2nj/31 3 < E n1 vertices are partitioned, which again is impossible. Thus, if p(G) = 11 then a{3,5,6,...}.

Suppose now that a E {3,5,6, . . .}. Then, by the above and our remarks in Section 2.2.1,

p(G) > 1I/ 31 To find p(G), we argue as follows: If we perform the construction

C - - G 2 _ ... ) we note that a(m) e 13,5,6 ... .} can only occur if G(m) is a

complete graph. Thus, the number of partition classes of cardinality at most two is at most

Ia/21.

The number of partition classes of cardinality three is therefore n + E (n1 - 1)

71i even ni odd

Therefore p(C) n + (n - 1)] + Ia/21. The process C C .' C2 [n, even n, odd

G(m) and the results of Lemma 2.3 and Lemma 2.4 also imply that p(G) > [ n+ n even

> (n1 - 1) + Ia/21, therefore we have equality in this case. n, odd

To see that the equality p(G) = 11: nt/31 also holds if a E {O, 1, 2,4) and if a < 0, we argue as follows:

Suppose that p partition classes of cardinality three result from the process C —p G(1)

0(2) C(m) and let a E 10, 1, 2,4).

If a = 0, then = 3p so that p(G) = p = 1I/31 30

If a = 1, then i =3P+1 so that En p(G) = p + 1 = Yni/31. If a=2, then Enj =3p+2 so that p(G) p + 1 = > If a = 4, then ni = so that p(G) = E p+ 2 =

If a < 0, then we prove first that at most one partition class has cardinality one or two: If

two partition classes have cardinality one or two, then (by Lemma 2.5) at most one of them has cardinality one. Hence there are exactly three or four vertices in the graph G(m). But then G(m) is either or Is ,1,1,1 so that a(m) is either 3 or 4, contradicting the inequality a(m) 2 which holds in this case by Corollary 2.18.

We now conclude the argument in the following three cases:

If no partition class has cardinality one or two, then E n2 3p so that p(G) = p = If one partition class has cardinality one, then E n2 3p+1 so that p(G) p+l = 1I n/31.

If one partition class has cardinality two, then E ni = 3p + 2 so that p(G) = p + 1 =

To prove (ii): In this case the process G - -p -* ... -p G' produces n1 + n2 +

1_ 1 + n partition classes of cardinality three. The remaining ne - 2(ni + n2 + + ft-1) vertices are all in V€ and can therefore only be partitioned using partition classes of cardinality one. Thus we have p(C) = fj - n. 0

The main result of this section appears in [5] with a different proof. Chapter 3

The induced path number of some

products of some graphs

3.1 Introduction

The following result is known for the cartesian product of paths Pd, x P 2 for positive integers d1 and d2:

Theorem 3.1 (Chartrand e.a. in [9]) The induced path number of the cartesian product

Pd, X Pd, is two for d1 , d2 > 2.

This result leads us to investigate the induced path number of some products of some other graphs.

The results of this investigation are contained in the next two sections. 31 32

row 1

row 2 column 1 column 2 column 3

Figure 3.1: Rows and columns of C

3.2 Products of complete graphs

3.2.1 The induced path number of K7 x K,.

In Figure 3.1 (p 32) we indicate, with an example, what we mean by the rows and columns of a graph of the form C = C1 x G2 in the sequel.

Note that in this case there is a complete graph K3 in every row and a complete graph K2 in every column.

Theorem 3.2 Suppose n in. Then

n - if n even and n > 2

p(Km X I() = + if n is even and n = m

n—i 21m1 if nisodd. 2 +I- -I

Proof For the proof of the first case we reason as follows: 33

Each row and each column of Km x K,, induces a complete graph. Therefore, if we want

to partition the inn vertices of the graph Km x Kn in a number of induced paths, we can

choose at most two vertices per row and at most two vertices per column. This means that

we need at least = partition classes in any partition of Km x K into induced paths. 2m 2 Thus we have =

p(Km X I() >

A partition of the inn vertices of Km x K in induced paths, each of order 2m, is shown

in Figure 3.2 (p 34) for ii > in and m odd and in Figure 3.3 (p 35) for n > in and in even.

Note that the paths are named P1 , F2 ,... , Pi and that P1 ends in the vertex in row in and column m. Also, every P, contains two vertices from every row.

Therefore we also have that

p(Km X K) < m

For n even and n > in, we conclude that

p(Km X K) =

The proof of the second case:

Consider an induced path partition of the vertices of Km x K, into V1 ,V2 ,. . . , Vk . Suppose the set of vertices in a given row is denoted by R. We then consider the non-zero numbers among IV1 fl RI, 1V2 fl R I, ..., IVk fl RI and we arrange them in non-increasing order - the resulting sequence of positive numbers is called the form of R. Note that the form of every row will be2,2,...,2,1,1, ... 1 with an even number ofVs.

Among the forms of all the rows, let a be the minimum number of l's. 34

- -

Figure 3.2: Is x Ic, m odd (the first case) 35

-P1

• •

• • •

Figure 3.3: Km x K, in even (the first case)

71 n If a = 0, then there are at least paths as seen in the row with 0 l's. Suppose these paths

are longest paths (of order 2n - 1). Suppose further that there are k rows in which endpoints

of these paths occur. Let e1 (k e is the number of endpoints of the ) :re paths in say row i. In row z there are atleast 1 2 1_ (el +.+ej_1+ej+l++ek) = e j ] . n [eil n rn + e = other paths. Thus there are at 1ast ^ [ ] It 1 = i1 other paths in total. Thus,

n Fm] p(G) :5

ma na n If a > 2, then there are at least ma l's and thus at least -- = > paths of which

the vertices causing the l's are the endvertices. As in the proof of a = 0, we can procede to Iml proof that there are at least other paths. Thus,

n p(G)

[7,nl A partition of the inn vertices of Km x K,. in + induced paths is shown in Figure 3.4

(p 37) if in is a multiple of 4 and in Figure 3.5 (p 38) if in is not a multiple of 4. In both rn the paths P with £ > + 1 are indeed short, most having only two vertices.

Therefore we also have that n [,,nl. p(KmXKn )

So, for n even and n = m, we conclude that

[m] p(Km X K,.) =n +

The proof of the third case: 37

• 1

PM I S.. I I . I

F!!!2 P_1 I P+11 / • . .

S .

S S

. .

.P3

__ __ •\• P2

P11 I .S. __

Figure 3.4: Ii x K,, in a multiple of 4 (the second case)

S.. . •11

PM I / I 1 I . PM 2

P!!!_1 2

PM - 2 . ^-

S .

S S

S S 2I\. I P2 P1: I S.. _ P!1

Figure 3.5: If,, x K, m not a multiple of 4 (the second case)

The proof of this case is similar to the proof of the second case, except that the form of every

row will be 2,2,... ,2,1,1,... , 1 with an odd number of l's.

ma Hence there are at least ma l's in total and thus at least paths of which the vertices

causing the l's are the endvertices. But there are also 2 a other paths as seen in a row

containing a l's. Thus

n—a ma p(G)^ 2 = n_i_a+1+m+m(a_l) 2 n—i m ma—m— a+ - 2 +2+ 2 n—i in - 2

Since p(G) is a positive integer, we have that

n - 1 Iiml 2 + I--I . n —i I mi A partition of the mn vertices of I x K , into induced paths is shown in 2 + I I Figure 3.6 (p 40) if m is odd and Figure 3.7 (p 41) if m is even.

Note that in the above argument for the lower bound for p(G) we took a = 1. That it can

be realised is seen by Figures 3.6 (p 40) and 3.7 (p 41).

Therefore we also have that

< - 1 [m]. p(Km x K) - 2

Thus, for ii odd, we conclude that

n — i [m] .p(KmX 2 +2 0 40

P1 P2 P!iL P!!+1 SSS

P +11 SS• ___

Figure 3.6: Km x K, in odd (the third case) 41

P1 P2 Pfl-L P

P!2j-i +2 ...

S S

Figure 3.7: K x I(, rn even (the third case) 42

Figure 3.8: K3 X K2

3.2.2 The induced path number of the complement of K7 x K

An example of a graph of the form 'm x K,, is shown in Figure 3.8 (p 42).

This complement can also be described, as a product of graphs, by the following:

For the graphs 'm and If,.,the graph 'm x Ii has vertex set V(Km) x V(K) and the vertices (a, b) and (c, d) are adjacent if and only if

a is adjacent to c in Km

and

b is adjacent to d in K,,,

that is, a 34 c and b d.

Lemma 3.1 If C = Km x K,,, then 0 does not have a path of order six as induced subgraph.

Proof Suppose Z7 has an induced path of order six. Let this path be v1 v2v3 v4 v5 v6 . Since the three vertices v1 , v3 and v5 are mutually independent, they must all three be in the same 43 row or in the same column. Without loss of generality we may assume they are in the same row. Since v6 is nonadjacent to v1 as well as to v3 , it must then also be in the same row as v1 and v3 . But then v6 is in the same row as v5 , which is not possible, since v5 is adjacent to v6. 11

b Theorem 3.3 If and = 36 with a,b EN0 and C = Km x K,, then

r V.1 (0)

Proof

(i) Since by Lemma 3.1, no six or more vertices of 0 induce a path, we have that

p() > 1±1.

(ii) By using induction on a + b, we now prove that

(i)

Ifa=b=O, then Km xK=K1 x K1 =K1 and

p()=1< 1,11 . [mnl.

Now suppose that - [ mn] p(G) T

b 3b for all values a + b < k. Let in =3'.2 and n = with a + b = k + 1. Consider

M n mn in ii mn the graph of 0 and partition the vertices of 0 in x = -- (or -- x - =

blocks of 2 by 3 vertices (or 3 by 2 vertices) each.

In every block of six vertices, form an induced path of order five in such a manner that

the vertices not used form a graph where Ci = K q x I( or G, = Km x I).

This is always possible since m = 3' . b and n = 3b• Then we have

mn p(?7)

[- < Mn 5 l 6 - [inn + 5mn I 30 V

= inn]I We conclude that p() U

nnl1rn Conjecture 3.4 p(Km X K) = 5 except when

(i) m = n = 2

(ii) m=3 and nE {3,5,8,10,11,13,14,15,...}

(iii) m and n odd, rn,n > 5 and mn = 0 mod 5.

In cases (ii) and (iii) we conjecture that

[ mn] + 1. p(Km X I() =

This is clearly true in case (i) too.

It is easy to check that p(Km x I() = for the values of (3, n) excluded by (ii):

(3, 2), (3, 4), (3, 6), (3, 7), (3, 9) and (3, 12). For all other values of (3, n) we can support this conjecture by showing that p(Km X K) ITl + 1:

• 0 0 0 • • • • E1 • • EiIJ

• • • • • • • • • Elli 11111

• . . . . • . . .

Figure 3.9: K3 x K, n odd

Clearly p(K3 X K3) = 3 =+ 1.

3n n A partition of the 3n vertices of K3 x K, , n odd, in 1 induced paths where a = L] paths are of order five, one path is of order four, b = L j paths are of order three and at most one path is of order one or two (if 3n - 5a - 4(1) - 3b = 1 or 2 ) is shown in

Figure 3.9 (p 45).

3n n A partition of the 3n vertices of K3 x K, , n even, in + 1 induced paths where a = 3n 5a paths are of order five, b paths are of order three and at most one path is of = L1—I order one or two (if 3n - 5a - 3b = 1 or 2) is shown in Figure 3.10 (p 46).

Thus p(K3 X K) <+ 1.

We can further support this conjecture by showing that p(I x K,,) < [ an-] + 1 for the

pairs (m, n) of case (iii) too. For this we let in = 5a. 5an] A partition of the 5an vertices of K5a x K,, , a = 1,3,5,..., in+ 1 = an + 1 induced

paths can be found as follows:

. S.. • . E1 • • . . . . . • . . . • . . • . . . . .

• • . 001 . S • S • . .

Figure 3.10: 1(3 x K, n even

First partition the vertices in two classes V1 and V2 such that (V1 ) = K5a X 1(_ 3 and

(V2 ) = K5a x 1(3 . A partition of the 5a(n —3) vertices of (V1 ) into a(n —3) induced paths of

order five is shown in Figure 3.11 (p 47) - note that this partition is based on the fact that

n - 3 is even.

In case (ii) we showed that

] P(K5a X 1(3) 115a + 1 = 3a+ 1.

Thus we have

P(K5a x I() < a(ri - 3) + 3a + 1

= an+1

I5anl =

Theorem 3.5 p)(P = 1] if n > 4.

Proof The result is clear if n = 4. 47

1 2 n-4 n-3 1 •• ... . .

// zz

• • • • • •

5a-4 • . . .

AOID

Ea '! S..

Figure 3.11: < V > 48

Suppose there are five vertices in P, n > 5, that induce a path. Then there are at least six edges in the subgraph of P induced by those five vertices, which is impossible. Thus we have

1111 n5.

A partition of the n vertices of J in induced paths is now described:

Suppose the vertices of F,-, are {vi , v2 ,... , v,} with each v1 adjacent to i < n. Suppose n = 4p + r, 0 < r <4. If r = 0, the p sets of vertices of the form {v4+i, v412 , v423, v4+4}, i = 0, 1,. . . ,p - 1 suffice.

If r > 0, the p sets 1v4+2, v413 , v424 , v41+s} (i = 0 1 1 1 . . . , p - 1) and the set {vi , v4 , 2 , .

V4p+r} (of 1,2 or 3 vertices, depending on the value of r) each induces a path. [n] Thus p(P) > 5.

Combining the contents of Theorem 3.2 and Theorem 3.3 we have some evidence for the following conjecture which is of the type of a Nordhaus-Gaddum theorem.

Conjecture 3.6 For any graph C of order p

[3pl ^p(C)+p()^ i+11 .

Of course, this conjecture has been checked for other types of graphs and their complements.

The bounds given by this conjecture, if true, are sharp: For the complete graph iç we clearly have that [3p 1 p(I()+p(k)= 11 += -i-i 49 and for the path P, we have by Theorem 3.5 that

P(PP) + p(PP-) + [^] ,p>4.

3.2.3 The induced path number of Cm X C

The following two results by I. Broere and M.J. Dorfling [3] is based on Grinberg's ideas (in dual form). Consider 2 - partitions of the vertex set of a graph, that is, partitions into two subsets. Such subsets will be denoted by V1 and V2 , and the number of edges between these sets will be denoted by e(Vi , V2 ). For such a partition of a graph C, the number of vertices of Vj which have degree i in the graph C will be denoted by f(i,j).

Theorem 3.7 If V1 , V2 is a 2 - partition of C and there is a constant k such that e(V j ) =

IVI + k forj = 1,2, then

- 2)(f(1, 1) - f(i, 2)) = 0.

Proof We have that

- 2)f (1, 1) = (degGv - 2) i vEV1

= 2e(Vi)+e(Vi,V2)-2IV,

= 2 1 V11 + 2k + e(Vi , V2 ) - 21V11

= e(V1,V2)+2k

Similarly, (i - 2)f (1, 2) = e(Vi , V2 ) + 2k.

Fol

We now use this theorem to prove 50

Corollary 3.8 If the graph C is regular of even degree 2n > 4 and of odd order, then G does not have a 2 - partition in subsets inducing acyclic subgraphs with the same number of components.

Proof If G has such a partition with c components in each of the induced (acyclic) sub- - graphs, then the equality e(V,) = 1V1 - c holds for i = 1,2; therefore the theorem is applicable. But then (2n - 2)(f(2n,1) - f(2n,2)) = 0, implying that f(2n,1) f(2n,2) which is impossible since C is of odd order.

We are now ready to consider the induced path number of Cm x C. Clearly in > 3 and n>3.

Theorem 3.9 Suppose in and ii are odd natural numbers. Then

p(Cm X C) = 3.

Proof Note that the graph Cm x C is 4 - regular and has odd order. By Corollary 3.8 we have that p(Cm X C) > 3. .. (A).

A partition of the mn vertices of Cm x C, in three induced paths is shown in Figure 3.12

(p 51) if n = mn and in Figure 3.13 (p 52) if n > in. Thus p(Cm X C) :5 3.. . (B).

By (A) and (B) we have that p(Cm X C) = 3. 11 51

HE 4 S....

• __ • m-2 TTTI

1 2 3 4 5 ••• n-3 n-i

Figure 3.12: Cm x C, n = m 52

1 • • • .—. S S S S S S S 2 • • • • S . . . j

• I • __

n-rn n-m+2 n-m+4 n-4 n-2 n 1 2 3 4 5 . n-m+i n-m+3 n-m+5 • 3 n-i

Figure 3.13: Cm x C, n > m

Theorem 3.10 Suppose a,k E N, n = 4a and rn = 2a(2k —1) + 1. Then

p(Cm X C) = 2.

Proof Clearly p(Cm X C) > 2.

A partition of the rnn vertices of Cm X Cn, in this case in two induced paths, is shown in

Figure 3.14 (p 53). This pattern can be stopped after the completion of the 2a + 1 1h row or

11h the 6a + row etc. to form a 2 - partition of C2a+i X C4 , C6a+i X C4 etc. respectively.

Thus P(Cm XCn )<2. 11 53

1 2 3 4 5 2!T:t:fE!i1

6a-l-1r,---.--.-. • • • • • • : 6a +

Figure 3.14: Cm x C, ii = 4a and m = 2a(2k - 1) + 1 54

Theorem 3.11 Suppose in, n E N, ni is odd, n is even and n > in. Then

p(Cm X C) < 3.

Proof We can partition the mn vertices of Cm x Cn in three induced paths similar to the partition in Figure 3.13 (p 52).

Theorem 3.12 Suppose in,n Cz N, in is even and n m. Then

p(Cm X C) < 3.

Proof A partition of the mn vertices of Cm x C1r, in three induced paths is shown in

Figure 3.15 (p 55). o 55

n-m-4-1 n-m+3 n-m+5 n-4 n-2 n 1 2 3 4 5 S S S n-m+2 n-m-f4 n-m+6 • n-3 n-i

Figure 3.15: Cm x C, rn even and n m Chapter 4

Uniquely partitionable and saturated graphs with respect to linear arboricity

4.1 Introduction

In this chapter we consider partitions of the vertex set V(C) of a graph G for which each set in the partition induces a linear forest in C. Our main results are however applicable to path partitions as well. We define lva(G) as the minimum number of subsets in such a partition of

V(G) and, following [12], refer to it as the linear vertex arboricity of C. We say that a graph

C is m-linear-forest-partitionable if lva(C) in and uniquely m- lin ear-forest-pa rtit ion able if lva(G) = m and there is only one partition of V(G) into in subsets so that each subset induces a linear forest. Moreover, we say (as in [11]) that a graph G is m-lva-saturated if 56 57

lva(G) in for each e E E().

Note that the linear vertex arboricity of a graph is a generalization of the induced path

number of a graph defined in [9]. Also, the motivating concept behind the definition of

a uniquely linear-forest-partitionable graph is a uniquely colourable graph defined in [14].

Throughout this chapter we shall abbreviate the term linear forest by if and therefore refer

to m- If- part itionabie graphs and uniquely in-if-partitionabie graphs.

We construct the graph Km[Qi, Q2, . . , Qm] by replacing the in vertices of the complete

graph Km with the paths P 1 , P 2 ,. . 'atm and by adding all edges xy where x E V(Q1)

and y E V(Q3 ) with i j. Such a graph will be denoted throughout the remainder of this

chapter by cm. Since {V(Q 1 ), V(Q2 ),. .. , V(Q,)} partitions V(Gm) into subsets inducing

paths, it follows that Iva(Gm) in: we say that this partition is canonical. We remark that

every m-If-partitionable graph is a subgraph of some Cm. Denote the complement K,-,, of

the complete graph by Nm. We construct Nm[Qi, Q2, . . , Qm ] by replacing the in vertices

of N n with the paths 'ti I Pt2l...

Throughout this chapter we write [n] for the set 11, 2,. . . ,n} of natural numbers.

4.2 Uniquely partitionable and saturated graphs

For in > 1, a graph in the class cm is referred to as a graph of type 1 if £ I ^! 3 and

£ > 4, 2 i in and referred to as a graph of type 2 if £ 2 and e ^! 5, 2 i m.

Lemma 4.1 Each graph of type 1 and each graph of type 2 is uniquely rn-if-part itionable. 58

Proof We prove our claim for graphs of type 1 by induction on rn. For in = 1 the assertion

is clearly true. Let n E N and suppose the canonical partition is the only n-if-partition of each graph of type 1 in c.

Let G 1 be a graph of type 1 in cn+i. Suppose there is another (n + 1)-If-partition of G1

in partition classes V1 , V2 ,.. . , V. There is thus in this partition a partition class which

contains vertices of (at least) two Q t 's. Suppose (without loss of generality) that

ViflV(Qi)7^ q and ViflV(Q2)j4q5 ...... A

Then 1V1 1 3 (otherwise < V1 > contains a cycle or a vertex of degree at least three which

implies that < V1 > is not a linear forest) so that

I V2 U1 U . .. UVn+l I>4T1 ...... B

Also ( V1 fl V(Qi )( < 2 and ( V1 fl V(Q2 )1 < 2 with at least one of these inequalities strict.

Suppose that Ic is any index with 2 k n + 1. We show that there are indices i and j

such that

If C does not hold, then Vk c V(Qb) (say) for such a k. Suppose (without loss of generality)

that V2 c V(Q +1 ). By adding the other vertices of Q (if any) to V2 , we may assume

that V2 = V(Q1).

If we apply our induction hypothesis to the graph G i - V(Q +1 ), we conclude that each

Q (1 i < n) is contained in a partition class, contradicting A. Thus C holds. 59

We conclude that IVk I 3 for 1 k n + 1. But then I V2 U V3 U... U V+i I < 3n, which

contradicts B.

The proof for graphs of type 2 is similar. 0

Lemma 4.2 Each graph of type 1 and each graph of type 2 is rn -Iva -satitrated. --

Proof We prove our claim for graphs of type 1. Note that it is trivial for in = 1; hence we

may assume that in > 2.

Let Cm be a graph of type 1 in G,. Consider the graph Gm + e where e = UW E E(Gm).

Then {u,w} c V(Q1 ) for an i E [nn]. Suppose that lva(Gm + e) < in and consider an

in-If-partition V1 , V2 ,. .. , V of Gm + e.

No partition class V is contained in V(Q2 ) since otherwise the sets Vk - V(Q1 ) with k 54 j

would be an (m - 1)-if-partition of the graph Km_i[Qi,. . . , Q-i, Q+i,... , Qm] . But then

each of the sets V,, - V(Q 1 ), k 54 j, is equal to one of the sets V(Q3 ) with s i by Lemma 4.1.

Now, if some vertex of Qj is a member of Vk, k j, then the subgraph of Cm induced by Vk

contains a cycle which is impossible. Hence V(Q1 ) is contained in V. But then the subgraph

induced by Vj contains Pe, + e which contains a cycle and this is again impossible.

Note again that, as in the proof of Lemma 4.1, any partition class which has a nonempty intersection with at least two of the paths Q's, has at most three elements. Hence the (at least two) partition classes that contain vertices of V(Q1 ) have at most three elements each.

Each other partition class has at most £ (j 34 i) elements, where L > 3 if i = 1 and £ ^! 4 except possibly for one £, if i 1. KE

Therefore at most .s = 2 3 + >IIjEK 4 vertices occur in partition classes where K c [in] and

IKI = in - 2. But s

Thus lva(Gm + e) > m.

The proof for graphs of type 2 is similar. 0 =

Lemma 4.3 If C is an rn-lva-saturated graph and Iva(G) = in, then C E cm.

Proof Suppose lva(G) = in. Then there exists a partition Vi , V2 ,.. . , Vm of V(G) so that

< V >G is a path for each i E [m]. Let i E [in] and v E V. Since the graph G is rn-lva-saturated, v is adjacent to each vertex of each V,j E [n2] and j i. 0

Lemma 4.4 If C is a uniquely rn-If-partitionable graph that is also in -Iva -satuvated, then

C is a graph of type 1 or type 2.

Proof Since C is an rn-lva-saturated graph and Iva(G) = in, C E cm by Lemma 4.3.

Clearly {V1 , V2 ,. .. , Vm} with V = V(Q2 ) for each i e [in] is an rn-If-partition of C. This is the only partition of C, since C is uniquely rn-lf- part itionable.

If there exists an i E [in] with J1' = 1, then there is only one such i: suppose that I vil =

JVk I = 1. Then Vj U Vk can be one partition class in a repartitioning of C and hence lva(C)

Now suppose that i E [m] is the only index with JVI = 1. Then we can repartition C by adding an endvertex of any < V >, j 54 i to V, to obtain another rn-If-partition of G. Thus

4 > 2 for each i E [in]. 61

Vi Vi Vi Vi Vi Vi Vi Vi

(a) (b) (c) (d)

Figure 4.1: Possible partitions

Suppose that fVI = 2 for some i E [m]. If jVj I = 2,3 or 4 for some j i, then there is

another rn-if-partition of C as depicted with dotted lines in Figure 4.1(a) - (c) (p 61). Thus

IV, I > 5 for each j

Next we show that at most one partition class contains three vertices: suppose that 1V1 =

IVj I = 3, for some i,j E [m]. Again another rn-if-partition of G is possible as shown in

Figure 4.1(d) (p 61).

Thus C is a graph of type 1 or type 2. EM

Combining these results we obtain the following result:

Theorem 4.1 A graph C is uniquely rn-lf-partitionable and m-lva-saturated if and only if

C is a graph of type 1 or type 2.

Note that this result immediately implies

Theorem 4.2 A graph G is uniquely m-path-partitionable and rn-p-saturated if and only if

C is a graph of type 1 or type 2. 0 62

We are as yet not able to characterize uniquely m-If-partitionable graphs.

4.3 The order of uniquely partitionable graphs

Concerning the orders of uniquely m-path-partitionable graphs, we have the following:

Theorem 4.3 (a) For an integer in > 2 there exists a uniquely rn-path-partitionable

disconnected graph of order p if and only if p > in.

(b) For an integer in > 2 there exists a uniquely in-path-partitionable connected graph of

order p if and only if p > 2rn+ 1.

of order p if and only if p > 4m - 1.

Proof (a) The necessity of the condition follows from the observation that Nm is the

uniquely rn-path-partitionable graph of smallest order. The sufficiency of the condition

follows since each graph Nm[Pi, P1 ,. . . , F1 , P,.] is a uniquely rn-path-partitionable graph.

(b) To prove the necessity of the condition we argue as follows:

Suppose the graph C is a uniquely rn-path-partitionable connected graph. Suppose further

that V is a partition class with jV = 1, say V1 = {v}. Since C is connected, v is adjacent

to at least one other vertex of C, say w, and suppose w E l's. If w is an end-vertex of the path induced by l<, then another rn- path- partition of C can be obtained by moving w to V1.

Suppose now < 1' >G = v1 ,v2 ,. .. ,v and that Vk is the first vertex on this (v1 - v)-path adjacent to v. Then {v1 , v2 ,... , vk, v} and { Vk+I , Vk+21 . .. , v} both induce paths, so that again another rn- path- partition of C is possible. Thus fV ^! 2 for each 1 C [rn}.

v, -Vi VLX

(a) (b) (c)-

Figure 4.2: Possible partitions

Suppose now that Il' ( = 2 for an i E [m]. Say V = {v, w}. Since C is connected, at least one

of v or w is adjacent to at least one other vertex of C; suppose such a vertex is in V,. If only

one of v or w, say v, is adjacent to vertices of V, let < V >G = v1, v2,.. . , v, and suppose

that vk is the first vertex on the (vi —v)-path that is adjacent to v. Then {vi,v2,.. . , vk, v, w}

and {Vk+1,Vk+2,. . . , v,j both induce paths and again another rn-path-partition is possible.

Therefore v and w must both be adjacent to vertices of l',.

Now IV.I > 2, otherwise IV1 I = JVj J = 2. Applying the remarks of the previous paragraph to

V and V, there are three cases to consider. They are depicted in Figure 4.2(a) - (c) (p 63) and in each case another partition is possible as indicated with dotted lines.

It now follows that the order of C satisfies p > 2m + 1.

For the sufficiency of the condition we consider the graph F of order 2m + 1 of Figure 4.3(a)

(p64).

Clearly p(F) m and we proceed to prove that F is uniquely m- path- partitionable. Suppose

there is another rn- path- partition of F in partition classes W1 , W2 ,.. . , Wm. Then there is 64

V1. Vi J I IE= T=LJ

Vrn_i (a) (b)

Figure 4.3: (a) Graph F (b) < Vm U V1 > in this partition a partition class which contains vertices of (at least) two V,'s; suppose

(without loss of generality) that Wi fl Vm q and W1 fl V1 q for an i, 1 < rn — i. (The only possibilities are W1 = {u,v,w} and W1 = {v,w} - see Figure 4.3(b) (p 64).) Then p(F) ^! (m - 2) + 3 = rn + 1, a contradiction.

The proof is now completed by remarking that the same argument goes through if the edge uv is replaced by a path of arbitrary length.

The contents of this chapter will appear in [4]. Chapter 5 =

Vertex and edge addition and deletion

5.1 The path partition number and the addition or deletion of a vertex or an edge.

Theorem 5.1 For every graph C and every vertex v E V(G)

p(G) - 1 p(G - v) p(G) + 1.

Proof Suppose V1 ,V2 ,... ,V is an n-path-partition of C. Without loss of generality we may assume that v E V1 . Then the graph induced by V1 - {v} is either a path or a disjoint union of two paths. Thus we have p(G - v) p(G) + 1.

65 Me

To prove the other inequality, we begin by assuming that p(C - v) p(C) —2. Now suppose

that V1 , V2 ,. . . , V(c)-2 is a (p(G) - 2)-path-partition of C - v. Then V1 , V2,... , V(Q) 21 {v}

is a (p(G) - 1)-path-partition of C. Thus p(C) - 1 < p(C - v). D

All the statements in the following examples involving Bh, Km,n and K2(P,11 P 2 ) can be deduced from Theorem 2.1, Theorem 2.2, Theorem 2.17 and Lemma 4.1.

Examples of graphs C such that p(C - v) = p(G) + 1 are the following:

(i) C = P, and v is an internal vertex of P.

(ii) C = , n > 2m and v is one of the in vertices.

(iii) C = Bh , h odd and v the vertex at level 0.

Examples of graphs C such that p(C - v) = p(C) - 1 are the following:

(i) C=C.

(ii) C=K, n>3 and nodd.

(iii) C = , m < n 2m, v is one of the n vertices and in + n = 1 mod 3.

(iv) C = Ktn,n , n > 2m and v is one of the n vertices.

(v) C Bh , h even and v the vertex at level 0.

(vi) G = N n , 772 > 2.

Examples of graphs C such that p(C - v) = p(C): 67

(i) G=K, n>2 and neven.

(ii) G = Km,n , n = 2m and v is one of the n vertices.

(iii) G = B,, , h odd and v a vertex at level 1.

Theorem 5.2 For every graph G and every edge e E E(G)

p(G)-1

Proof Suppose V1 , V2 ,. . . , V, is an n- path- partition of G. If e E E(V2) for an i, then it is possible to partition V(G - e) into n + 1 subsets V1 , V2 ,.. . , 1/1' , V ' ,. . . , V,, that each induces a path. If e V E(V) for each i, then V1 , V2 ,. .. , V, is an n-path-partition of G - e. Thus we have p(G — e) p(G)+l.

To prove that p(G - c) p(G) - 1, assume that p(G - e) p(G) —2. Let V1 , V2.... , be a (p(G) - 2)-path-partition of C - e. If e = uv and u, v E V for an i, then it is possible to partition V(G) in p(G) - 1 subsets V1 , V2 .... , V', V",. . . , V (Q)2 such that each induces a path, a contradiction. If e = uv, u e V1 and v E V, with i j, then V1 , V2 ,. . ., V,(Q)2 is a (p(G) - 2)-path-partition of C, again a contradiction.

Examples of a graph C such that p(G - e) = p(G) + 1:

(i) C = P,, n 2.

(ii) C=K2 (P 1 ,Pj2 ) where !?1 ^!3,€2 ^!4 and eEE(Pj for i=1,2.

(iii) C = B,, , h odd and e a level 0 edge. 68

Examples of a graph C such that p(G - e) = p(C) - 1:

(i) G=C.

(ii) G = n odd.

Examples of a graph G such that p(G - e) =

(i) G=K, n>2 and neven.

(ii) G = K2(P 1 ,P 2 ) where £ = 2 and £2 > 5 and e the edge of P.

(iii) C = K2 (P11 ,Pt2 ) where

a) £1>3and1?2>4or and edge e = uv joins vertex u of P€1 and vertex v of P2. b) £ = 2 and £2 > 5

5.2 p-criticality

If G is a graph such that p(G) = k and p(C - v) = k - 1 for every v E V(G), then we say that C is k- minus- critical.

k An example of a graph that is k-minus-critical for k even is C. and for k odd is K2k+l.

We now prove that no graph G satisfies p(G - v) = Ic + 1 for every v E V(G): Let G be a graph and p(C) = n. Suppose V1 , V2 ,.. . , V is a partition of V(C) into subsets that each induces a path. Let v be an endvertex of (V) , 1 < i < n. Then p(C - v) :5 p(G).

Therefore it is not plausible to define a graph C to be k-plus-critical in a similar way we defined a graph to be k-minus-critical. If C is a graph such that p(G) = k and p(G - e) = k + 1 or k - 1 for every e E E(G), then we say that C is k-plus-minimal or k-minus-minimal respectively.

An example of a k-plus-minimal graph is kK2 and an example of a k-minus-minimal graph is where k is an even natural number.

If C is a graph such that p(G - v) = p(G) for every v E V(G), then we say that G is

vertex-neutral.

An example of a vertex-neutral graph C with p(G) = k, is C = kK2.

We can define a graph C to be edge-neutral, similarly.

An example of such a graph C with p(G) = k, is G = 2Kk , k an even natural number and k>4.

Proposition 5.3 If G is a graph with p(G) = k, then C contains a k- minus- critical subgraph.

Proof Let G be a graph such that p(C) = k. NA; is a subgraph of G that is k-minus-critical.

Proposition 5.4 Every graph C of order at least k + 1 with p(G) = k contains a k-plus- minimal subyraph.

Proof Let G be a graph such that p(G) = k. Note'that JE(G)I > 1. C has a subgraph isomorphic to Nk+l + e, where e = uv, and u,v E V(G).

This subgraph is k-plus-minimal. 0 70

Figure 5.1: C'3 9K5 - -

Suppose that C is a graph with components K, p(K1 ) = k, and k = k.

Then it is easy to see that C is a k-minus-critical graph if and only if each component Ki is k1-minus-critical.

Thus, we only have to investigate the concept k- minus- critical for connected graphs. We need the following definition and ensuing three lemma's to prove a similar result for the blocks of a graph.

Let C1 and C2 be graphs. Let C(G1 .G2 ) denote the class of all graphs obtained by identifying a vertex of C1 with a vertex of G2 . We name the resulting vertex of C E C(C1 • G2 ) the identified vertex.

Consider the next example:

Let C1 = C3 and C2 = K5 . Then C1 • C2 is the graph in Figure 5.1 (p 70) with v the identified vertex.

Lemma 5.1 If p(GI ) ^! bi and p(G) b + b2 - 1 for some graph G E C(G 1 • C2 ), then p(C2) < b2.

Proof Let C e C(G1 .C2 ) and let v be the identified vertex of C. Suppose V1 , V2,... , Vb,+b2_1 is a (b1 + b2 - 1)-path-partition of C, with v E V1 . Then each of V2 ,. . . , V6+b2 _1 is 71

contained in either V(G1 ) or in V(G2). We may suppose that V2 ,.. . , V are in V(G1)

and the others in V(G2). Then V1 fl V(G1 ), 172,.. . , V. is an r-path-partition of C1 and

V1 fl V(G2),V.+1,... ,Vb1 +b2 _ 1 is a (b1 + b2 - r)-path-partition on V(G2 ). Thus p(G2)

bi +b2 —rp(G 1 )>b1 . 0

Lemma 5.2 Let C1 be a b1 -minus-critical graph and C2 a b2 -minus-critical graph. Then

G E C(G1 • C2 ) is a (b1 + b2 -1)-minus-critical graph.

Proof Let v be the identified vertex of C. Since p(Gi ) = b1 and p(G2 - v) = b2 - 1, it

follows that p(G) < b + b2 - 1. Suppose now that p(G) b + b2 - 2. Then, by Lemma

5. 1, p(G2 ) :5 b2 - 2 + 1 = b2 - 1 which is a contradiction. Thus p(G) ^: b1 + b2 - 1 and

consequently p(G) = b1 + b2 - 1.

Furthermore, p(C - u) = b1 + b2 - 2 for each vertex u of C:

We first consider u = v: Since v is the identified vertex, C - v consists of two graphs that are

isomorphic to G1 - v and G2 - v. Since G1 is a b1 -minus-critical graph, p(Gi - v) = b1 - 1.

Similarly p(G2 - V) = b2 -1. Thus p(G - v) = b1 + b2 -2.

Let v u E V(C1 ). Consider the subgraph of C—u induced by the vertices V(C1 ) - u. This graph is Ci —u and p(Ci — u) = b1 -1. Also p(Gz—v) = b2 -1. Thus p(G—u) b1 +b2 -2. By

Theorem 5.1 we also have that b1 --b2 -2 p(C—u) b+b2 and thus p(C—u) = b1+b2-2.

Similarly, p(C - u) = b1 + b2 - 2 holds for v u e V(C2). 0

Lemma 5.3 Let C1 and C2 be graphs with p(C1 ) = b (i = 1, 2) and let C E C(C1 • C2 ) be a k-minus-critical graph with k = b1 + b2 - 1. Then C, is a b-minus-critical graph for i = 1, 2. 72

Proof Case (i): Suppose v is the cutvertex of C. Then C - v is the union of the two

components C1 - v and C2 - v. Since G is a k- minus- critical graph, p(G - v) = k - 1 so

that p[(G1 - v) U (C2 - v)] = p(G1 - v) + p(G2 - v) = k - 1 and consequently p(G1 - v) =

k — i — p(G2 —v) < k - b2 = b, —1. Since p(GI ) —1 < p(G1 - v) p(Gi )+ 1 by Theorem 5. 1,

it follows that p(Gi — v)= Si - 1.

Similarly, p(G2 - v) = b2 - 1.

Case (ii): Let v u E V(G1 ). Since C is a k-minus-critical graph, it follows that p(C—u) =

k - 1 = b + b2 - 2. Since p(G2 ) = b2 , we have that p(Gi - u) S Si - 2 + 1 = bi - 1 by

Lemma 5.1. Again it is easy to show that p(Gi - u) = - 1.

Thus C1 is a b1 -minus-critical graph.

Similarly C2 is a b2-minus-critical graph. IN

Lemma 5.4 Suppose p(Gi ) ^: bi and p(G2 ) > b2. Then p(G) > fri + S2 - 1, G E C(C1 • G2).

Proof Assume that p(G) < b1 + b2 - 2. Since p(Gi ) ^! b, it follows from Lemma 5.1 that

p(G2 ) < b2 - 1. This is a contradiction. U

We use the notation C 1 • G2 . ... . G,r for a graph in C(( ((Ci • C2 ) • G3 )) • Ca).

Lemma 5.5 Suppose p(G) = bi ,i = 1,2, ... ,n. Then p(G1.G2.•.U) b1—n+1.

Proof Apply Lemma 5.4 in an inductive argument. 0 73

Theorem 5.5 Let C be a connected graph consisting of cyclic blocks B, with p(B 1 ) = b, for

i=1,2,...,nwithn>2 and letk=b1—n+1.

Then C is k- minus- critical if and only if each of the blocks Bi is a b2 -minus-critical graph.

Proof Suppose C is a k- minus- critical graph. We prove that each block Bi is b-minus-

critical (i = 1,2,... ,n) by using induction on n.

If n = 2, then it follows from Lemma 5.3.

Suppose the following holds: If C is a ( b1 - t + 0-minus-critical graph, then each block

Bi is b1-minus-critical, i = 1)2,.. . ,t.

t+1 Suppose that C is a ( b1 - t)-minus-critical graph. Let B+1 be an endblock of G and v

be the cutvertex which belongs to Denote the graph which results if all the vertices of

B +1, except v, are removed from G, by H.

i+1 Since p(C) b—t and p(B+ 1 ) = b+1 it follows from Lemma 5.1 that p(H) b—t+1. =

Since p(B) b(i 1, 2,.. . , t) it follows from Lemma 5.5 that p(Ht ) = p(Bi • B2 . • sBg) >

b - t + 1.

Thus we have p(H) = b - t + 1.

Directly from Lemma 5.3 follows We now prove that block B+1 is bt+i-minus-critical and i+1 that the graph H1 is ( b1 - t + 1)- minus-critical.

Since H1 is a (b1 - t + 0-minus-critical graph, it follows from the induction hypothesis that each block B, i = 1, 2 1 . . . 7 t is b1-minus-critical too. 74

To prove the converse, Suppose that each block I3 ol graph C' is h1 - i nirl us-cri I,ical. We prove that C is a k-minus-critical graph where k = b - it + 1, by using JiLductioli on n.

If n = 2, then it follows from Lemma. 5.2.

Suppose for ii. = I the following holds: if C is any connected graph consisting oft blocks and if

each block Bi of C is b-niin(Is-critical where 1 < i < 1, then C is a ( —t + I)-minus-critical graph.

Let it = I + I and suppose that C is a connected graph with I. -I- I blocks and that block

13, is b! -iniiius-critical, I < i < I -j- I. Let I3 he an endblock of C and V be the cutvertex

wh ich belongs to B +1. I)eiiotc the graph which iesuhts if all the vertices of I-3t+i, except v,

are removed from C, by C'. By the induction hypothesis, we have that C' is a ( b - I + 1)-

+1 minus-critical graph. It now foilows froiii Lemma 5.2 that C is a ( b - I)-minus-critical graph. 11

We conclude with the following conjecture:

Conjecture 5.6 Let C be a connected c/rap/i consisiiiuj of cyclic blocks Bi wit/i. p(B1 ) = b for i= 1,2,...,n with n>2 and let

Then C is k-nijnus-minimal if and only if each of the blocks 13i is a b-minus-minimal graph. Bibliography

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arboricity . 4 canonical 57 cartesian product ...... 8 complete £-partite graph ...... 14 d-dimensional hypbercube ...... 10 k-dimensional mesh ...... 10 edge-neutral ...... 69 excess degree...... 13 induced path number...... 1 induced tree number ...... 14 induced Li-tree number...... 14 island number...... 6 level i edges...... 7 linear arboricity ...... 4 linear forest ...... 5 m-linear-forest-partitionable ...... 56 linear vertex arboricity...... 2 k-minus-critical ...... 2 k-minus-minimal ...... 69 partition classes ...... 5 path number ...... 5 r-path-partition ...... 5

Ic-plus-minimal ...... 69 m-lva-saturated ...... 2 uniquely m-linear-forest-partitionable . 2 vertex- arboricity ...... 5 vertex-neutral...... 69