ANTIMAGIC LABELING OF GRAPHS
Oudone Phanalasy
A thesis submitted in total fulfilment of the requirement for the degree of Doctor of Philosophy
School of Electrical Engineering and Computer Science Faculty of Engineering and Built Environment The University of Newcastle NSW 2308, Australia
August 2013 Certificate of Originality
This thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or writ- ten by another person, except where due reference has been made in the text. I give consent to this copy of my thesis, when deposited in the University Library, being made avail- able for loan and photocopying subject to the provisions of the Copyright Act 1968.
(Signed)
Oudone Phanalasy
i Publications Arising from This Thesis
1. M. Baˇca,M. Miller, O. Phanalasy and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a,Super d-antimagic labelings of disconnected plane graphs, Acta Mathematica Sinica (English Series), 26(12) (2010), 2283-2294. 2. O. Phanalasy, M. Miller, L. Rylands and P. Lieby, On a relationship between completely separating systems and antimagic labelling of regular graphs, Lec- ture Notes in Computer Science, 6460 (2011), 238-241. 3. J. Ryan, O. Phanalasy, M. Miller and L. Rylands, On antimagic labelling for generalized web and flower graphs, Lecture Notes in Computer Science, 6460 (2011), 303-313. 4. M. Miller, O. Phanalasy and J. Ryan, All graphs have antimagic total label- ings, Electr. Notes in Discrete Mathematics, 38 (2011), 645-650. 5. L. Rylands, O. Phanalasy, J. Ryan and M. Miller, Construction for antimagic generalized web graphs, AKCE Int. J. Graphs Comb., 8(2) (2011), 141-149. 6. M. Miller, O. Phanalasy, J. Ryan and L. Rylands, Antimagicness of some families of generalized graphs, Austral. J. Combin., 53 (2012), 179-190. 7. M. Baˇca,L. Brankovic, M. Lascs´akov´a,O. Phanalasy and A. Semaniˇcov´a- Feˇnovˇc´ıkov´a,On d-antimagic labellings of plane graphs, Electr. J. Graph Theory and Applications, 1(1) (2013), 28-39. 8. M. Baˇca,M. Miller, O. Phanalasy and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a,Construc- tion antimagic labelings for some families of regular graphs, J. Algorithms and Computation, 44 (2013), 1-7. 9. M. Miller, O. Phanalasy, J. Ryan and L. Rylands, Sparse graphs with vertex antimagic edge labelings, AKCE Int. J. Graph Comb., 10(2) (2013), 193-198.
ii 10. S. Arumugam, M. Miller, O. Phanalasy and J. Ryan, Antimagic labelling of generalized pyramid graphs, Acta Mathematica Sinica (English Series), In press. 11. L. Rylands, O. Phanalasy, J. Ryan and M. Miller, An application of com- pletely separating systems to graph labeling, Lecture Notes in Computer Sci- ence, Accepted. 12. M. Miller, O. Phanalasy, J. Ryan and L. Rylands, A note on antimagic label- ings of trees, Bulletin of ICA, Submitted. 13. O. Phanalasy, Antimagic labelling of generalized sausage graphs, Utilitas Math., Submitted. 14. J. W. Daykin, C. S. Iliopoulos, M. Miller and O. Phanalasy, Antimagicness of generalized corona and snowflake graphs, JCMCC, Submitted. 15. M. Baˇca,M. Miller, O. Phanalasy, J. Ryan, A. Semaniˇcov´a-Feˇnovˇc´ıkov´aand Anita A. Sillasen, Totally antimagic graphs, In preparation. 16. M. Baˇca,O. Phanalasy, J. Ryan and A. Semaniˇcov´a-Feˇnovˇc´ıkov´a,Antimagic labelling of join graphs, In preparation. 17. M. Miller, O. Phanalasy and J. Ryan, All graphs have vertex antimagic and edge antimagic total labelings, In preparation. 18. D. Buset, M. Miller, O. Phanalasy and J. Ryan, Antimagicness for family of generalized antiprism graphs, In preparation.
iii Further Publications Produced During my Candidature
1. O. Phanalasy, I. T. Roberts and L. Rylands, Covering separating systems and application to search theory, Austral. J. Combin., 45 (2009), 3-14. 2. O. Phanalasy, M. Miller, C. S. Iliopoulos, S. P. Pissis and E. Vaezpour, Con- struction of antimagic labeling for the Cartesian product of regular graphs, Mathematics in Computer Science, 5(1) (2011), 81-87.
iv Acknowledgements
I would like to thank my supervisor Professor Mirka Miller for giving me a chance to do research. Her invaluable guidance, encouragement, constant support and friendship throughout my PhD candidature have combined to make my research experience worthwhile.
I would like to thank my co-supervisor Dr Joe Ryan for his constant and invalu- able support and many enjoyable and useful discussions during my candidature. Throughout my candidature he was always there to help me not only with research but also non-academic problems.
My special thanks go to my external supervisor Associate Professor Leanne Rylands for her friendship. I am thankful that our earlier collaboration could continue during my PhD candidature. I have learned a lot from my external supervisor Dr Andrea Semaniˇcov´a-Feˇnovˇc´ıkov´a.Working on problems together with Andrea and Professor Martin Baˇcacontributed greatly to my training as a researcher.
During my PhD candidature I have benefitted from discussions with Professor Brian Alspach, Professor Subramanian Arumugam, Dr Paulette Lieby and Dr Yuqing Lin. In particular, I would like to thank Dr Ian Roberts who as my M.Sc. supervisor first inspired me to become a researcher and introduced me to combinatorics of finite sets.
My thanks also go to the School of Electrical Engineering and Computer Science, the University of Newcastle and the Graduate Studies Office who provided excellent
v facilities and research environment.
No acknowledgement would be complete without a recognition of the contributions of my sisters, brothers and especially my wife Kipmala Xayavong and my daughters, Donemala Phanalasy, Phoutmala Phanalasy, Donesanty Phanalasy and Oumala Phanamaly for their love and encouragement. I could not have finished my PhD studies without their continuous support.
vi Dedication
To the memory of my parents Noulith and Thongdam Phanalasy, and my mentor and benefactor Professor Beth Southwell.
vii Contents
Declaration i
Publications Arising from This Thesis ii
Further Publications Produced During my Candidature iv
Acknowledgements v
Dedication vii
Abstract 1
Chapter 1 Introduction 2
Chapter 2 Basic Terminology in Graph Theory 6
2.1 Operations on Graphs ...... 7
I EDGE LABELING 10
Chapter 3 Basic Concepts and Literature Review 11
viii 3.1 Completely Separating Systems ...... 11
3.2 Antimagic Labeling of Graphs ...... 14
Chapter 4 Application of Combinatorics to Antimagic Labeling of Graphs 17
4.1 Relationship between Completely Separating Systems and Labeling of Graphs ...... 17
4.2 Modification of Completely Separating Separating Systems . . . . . 21
4.2.1 Edge Deletion with no Isolated Vertex ...... 21
4.2.2 Edge Switching ...... 21
4.2.3 Splitting of Roberts’ Construction ...... 23
4.3 Generalized Antiprism Graphs ...... 24
4.4 Conclusion ...... 41
Chapter 5 Antimagicness of some Families of Graphs 43
5.1 Generalized Web and Flower Graphs ...... 45
5.2 Generalized Sausage Graphs ...... 55
5.3 Generalized Corona and Snowflake Graphs ...... 61
5.4 Join of Graphs ...... 69
5.5 Trees and Unicyclic Graphs ...... 78
5.6 Conclusion ...... 83
II TOTAL LABELING 85
Chapter 6 Basic Concepts and Literature Review 86
ix 6.1 Total Labeling ...... 86
6.2 d-Antimagic Labeling ...... 89
Chapter 7 Total and Totally Antimagic Labeling of Graphs 91
7.1 Total Labeling of Graphs ...... 91
7.1.1 Vertex Antimagic Total Labeling ...... 91
7.1.2 Edge Antimagic Total Labeling ...... 96
7.2 Totally Antimagic Total Labeling of Graphs ...... 99
7.2.1 Join Graph ...... 100
7.2.2 Corona Graphs ...... 108
7.2.3 Union of Graphs ...... 112
7.3 Conclusion ...... 114
Chapter 8 Antimagic Labeling of Plane Graphs 115
8.1 Super d-Antimagic Labeling of Type (1, 1, 0) ...... 115
8.1.1 Edge Antimagic Labeling of Paths ...... 115
8.1.2 Partitions with Determined Differences ...... 117
8.1.3 d-Antimagic Labeling for Certain Families of Plane Graphs . 119
8.2 Super d-Antimagic Labeling of Type (1, 1, 1) ...... 129
8.2.1 Super 1-Antimagic Labeling of Type (1, 1, 1) ...... 129
8.2.2 Super d-Antimagic Labeling of Type (1, 1, 1) ...... 134
8.2.3 Some Classes of Plane Graphs ...... 144
x 8.3 Conclusion ...... 146
Chapter 9 Summary 147
9.1 Open Problems ...... 148
References 150
xi List of Tables
n Sk 4.1 Value (i, t) corresponding to value m for A E(Gj), 1 k n. 29 G − j=1 ≤ ≤ n Sk 4.2 Value (i, t) corresponding to value m for T E(Gj), 1 k n, G− j=1 ≤ ≤ n even...... 36
n Sk 4.3 Value (i, t) corresponding to value m for T E(Gj), 1 k n, G− j=1 ≤ ≤ n odd...... 40
xii List of Figures
2.1 The Cartesian product C4 P3...... 8 ×
2.2 The join P2 + C3...... 8
2.3 The graphs G and G e...... 8 −
2.4 The corona graph P3 C3...... 9
3.1 Antimagic labeling of the cycle C6 and the complete graph K4.... 15
4.1 The graph G = (V,E) with edge labeling...... 19
4.2 The graph B(6, 4) with antimagic edge labeling...... 20
4.3 The graph G D with antimagic edge labeling...... 22 −
4.4 The graph K4 with antimagic edge labeling...... 22
4.5 An array CSS obtained by edge switching and corresponding graph 2 2 K4 ./ G with antimagic edge labeling...... 23
4.6 The array (CSS) L1 and graph G(L1) with antimagic edge labeling. 24
4.7 The array (CSS) L2 and graph G(L2) with antimagic edge labeling. 24
4.8 The generalized antiprism A3 with antimagic edge labeling. . . . . 26 C4
4.9 The generalized toroidal antiprism T 4 with antimagic edge labeling. 34 C4
xiii 5.1 Illustration of the construction of the generalized flower graph FL(Kp, m, n, p), for p 2, m 2, n 3, and m even...... 46 ≥ ≥ ≥
5.2 The generalized flower graph FL(C4, 2, 3, 4) with antimagic edge la- beling...... 49
5.3 Illustration of the construction of the complete generalized flower graph CFL(G, m, n, (m + n 2)p), for p 2, m 3, n 3, and m − ≥ ≥ ≥ odd...... 53
5.4 The complete generalized flower graph CFL(4K2, 2, 2, 16) with an- timagic labeling...... 54
5.5 Illustration of antimagic labeling of S(Kp, m), p 2, m 2 and m ≥ ≥ even...... 56
5.6 A labeling of the path Pm+2, m 1...... 59 ≥ 5.7 Illustration of antimagic labeling of CMS(G, m), m 3 and m even. 60 ≥
5.8 The complete mixed generalized sausage CMS(K2, 3) with antimagic edge labeling...... 62
5.9 Antimagic labeling of the generalized corona graph P3 (2K1,K1 ∪ K2,C3)...... 64
5.10 Antimagic labeling of the generalized snowflake graph Sf(K1,K2,C3). 66
5.11 The complete 4-partite graph K2,2,2,2 with antimagic labeling. . . . 77
5.12 The tree with VAE labeling obtained from d = (4, 3, 3, 2, 1, 1, 1, 1, 1, 1) by Construction 1...... 80
5.13 The tree with VAE labeling obtained from d = (4, 3, 3, 2, 1, 1, 1, 1, 1, 1) by Construction 2...... 80
7.1 A VAT labeling for G...... 92
7.2 An EAT labeling for G...... 96
xiv 7.3 A VAT but not an EAT labeling for W4...... 100
7.4 An EAT but not a VAT labeling for W4...... 100
7.5 A TAT labeling for W4...... 100
8.1 A super 9-antimagic labeling of type (1, 1, 0) for F6...... 121
8.2 A super 10-antimagic labeling of type (1, 1, 0) for E5...... 123
8.3 A super 0-antimagic labeling of type (1, 1, 0) for L4...... 126
8.4 A super 20-antimagic labeling of type (1, 1, 0) for H5...... 128
8.5 A super 1-antimagic labeling of type (1, 1, 1) for L4...... 132
8.6 A super 1-antimagic labeling of type (1, 1, 1) for 2L4...... 132
8.7 A super 1-antimagic labeling of type (1, 1, 1) for G...... 133
8.8 A super 1-antimagic labeling of type (1, 1, 1) for 2G...... 133
8.9 A super 5-antimagic labeling of type (1, 1, 1) for L4...... 141
8.10 A super 5-antimagic labeling of type (1, 1, 1) for 2L4...... 141
xv Abstract
The concept of labeling of graphs has attracted many researchers to this branch of research since the concept was introduced. It is becoming popular, partly because of mathematical challenges, and partly also because of the wide range of applications in other branches of science.
A labeling of a graph is a map from one or more graphs elements to sets of numbers, for instance, labeling of edges, vertices, both edges and vertices, or edges, vertices and faces, if a labeling is applied to plane graph. Correspondingly, we distinguish edge labeling, vertex labeling, total labeling, or d-labeling. In this thesis we deal with edge labeling to contribute new results towards settling the Hartsfield and
Ringel conjecture that ‘Every graph different from K2 is antimagic’. We also prove that every graph has a vertex antimagic total labeling, and an edge antimagic total labeling if the graph contains no isolated vertex. We introduce the notion of totally antimagic total labeling and provide several initial results for such a labeling. Furthermore, we study super d-antimagic labeling and obtain several new results.
Nevertheless, many open problems remain in antimagic labeling of graphs and we conclude the thesis by listing some of those that arose from our study.
1 Chapter 1
Introduction
Graph theory can be considered to have its origin in 1736 when Euler explained the K¨onigsberg bridge problem, that is “Is it possible to walk across the seven bridges that span the Pregel river, dividing the K¨onigsberg town into four areas in such a way that we cross each bridge exactly once?” It took 200 years until the first book on graph theory was written by K˝onigin 1936. Since then graph theory has been developed and applications continue to grow rapidly, largely due to the usefulness of graphs as models for computation and optimization, road maps, social relationships, databases, the flow of control in a computer program, communication networks, circuit design and many more.
The basic notion of a graph consists of a set V of points, nodes, or vertices, and a set E of links, or edges, each edge connecting two members of V . In the real world the set V can be used to represent all kinds of objects, such as cities, people, computers, and so on. The set E represents connections between cites (for exam- ple, roads, flights, telephone lines between cities), relationships between people (for example, collaborators, relatives or friends), and so on. These objects and connec- tions between two objects can be modeled as points and lines between two points, respectively.
In this thesis, we deal with graph labeling. Graph labeling was first introduced in the late 1960s. Since then it has attracted many researchers to this branch of graph theory. Graph labeling is becoming increasingly popular, partly because it contains many interesting mathematical challenges, and partly also because of the wide range of applications to other branches of science, for instance, circuit design,
2 Chapter 1. Introduction bioinformatics, x-ray crystallography, coding theory, cryptography (secret sharing schemes), astronomy, circuit design and communication design, see [28, 29]. Over 1000 papers on various types of graph labeling have been published during the last 50 years. For a detailed survey, see [42].
This thesis deals with some particular types of graph labelings. It consists of two separate parts.
In the first part of this thesis, we study in particular the Hartsfield and Ringel conjecture [48] that was proposed in 1990, namely, that every connected graph, except K2, is antimagic. More precisely, in our terminology such a labeling is called a vertex antimagic edge labeling, that is, an edge labeling (i.e., a bijection from the set of edges of a graph with q edges to the set 1, 2, . . . , q ), in which all vertex { } weights are pairwise distinct, where the weight of a vertex is the sum of all the edge labels incident with that vertex. During the subsequent 20 years, various particular classes of graphs have been proved to be antimagic. However, so far the general Hartsfield and Ringel conjecture remains open. In this thesis we prove that for every degree sequence pertaining to a regular graph, there exists an antimagic labeling for a regular graph with that degree sequence. We also prove some families of non regular graphs to be antimagic. Moreover, we prove that for any degree sequence pertaining to a tree, there is at least one antimagic labeling for a tree with the given degree sequence, and with a slight modification we also obtain an antimagic labeling of a unicyclic graph as well.
In the second part of this thesis we focus on labeling of type (v, e, f), where v, e, f have value 1 if the vertices, edges and faces are labeled, and 0 otherwise; for instance, a labeling of type (1, 1, 1) of a plane graph is a labeling such that vertices, edges and faces are all labeled. A labeling of type (1, 1, 0) is called a total labeling. Vertex antimagic total labeling was introduced in [10]. A vertex antimagic total labeling of a graph is a total labeling such that all vertex weights are pairwise distinct, where the vertex weight of a vertex is the sum of the label of that vertex and all the edge labels incident with that vertex. We also study edge antimagic total labeling. An edge antimagic total labeling of a graph is a total labeling such that all edge weights are pairwise distinct, where the edge weight of an edge is the sum of the label of that edge and the labels of the vertices incident with that edge. The notion
3 Chapter 1. Introduction of (a, d)-edge antimagic total labeling ((a, d)-EAT labeling) was introduced in [76]. A total labeling of a graph G with p vertices is called an (a, d)-EAT labeling of G if the set of the edge weights is W = a, a + d, . . . , a + (p 1)d , for some { − } integers a > 0 and d 0. We introduce the notion of totally antimagic total ≥ labeling of graphs, that is, a total labeling in which both vertex weights and edge weights are simultaneously distinct. We prove that all graphs have vertex antimagic total labeling, and edge antimagic total labeling except when the graph contains an isolated vertex. We also provide some initial results on totally antimagic total for several families of graphs. Apart from vertex/edge antimagic total and totally antimagic total labelings of graphs we also investigate d-antimagic labeling of types (1, 1, 0) and (1, 1, 1) for some families of plane graphs, that is, labelings of types (1, 1, 0) and (1, 1, 1) in which all face weights are pairwise distinct, where the face weight is the sum of all edges labels and vertex labels surrounding that face (and together with the face label when the type is (1, 1, 1)). The concept of d-antimagic labeling of plane graphs was defined in [24].
The thesis is structured as follows.
Chapter 2: Basic Terminology in Graph Theory. In this chapter we introduce the graph theoretical notation and terminology that will be used throughout the thesis.
Chapter 3: Basic Concepts and Literature Review. In this chapter we provide an historical overview of some results on vertex antimagic edge labelings.
Chapter 4: Application of Combinatorics to Antimagic Labeling of Graphs. We present in this chapter vertex antimagic edge (VAE) labeling of some families of regular graphs by applying completely separating systems on finite sets. We also extend the results to VAE labeling of non-regular graphs.
Chapter 5: Antimagicness of some Families of Graphs. We provide VAE labeling of some families of non-regular graphs. At the end of this chapter we construct VAE labeling for certain trees and unicyclic graphs.
Chapter 6: Basic Concepts and Literature Review. In this chapter we give an overview on antimagic total labeling and super d-antimagic labeling of plane graphs.
4 Chapter 1. Introduction
Chapter 7: Total and Totally Antimagic Labelings of Graphs. We present in this chapter results on VAT and EAT labelings for all graphs, and super (a, d)- VAT (resp., (a, d)- EAT) for some graphs. Totally antimagic labeling of some families graphs are also included.
Chapter 8: Antimagic Labeling of Plane Graphs. We first examine the existence of super d-antimagic labeling of type (1, 1, 0) for several families of plane graphs. Finally, we present super d-antimagic labelings of type (1, 1, 1) for some families of disconnected plane graphs.
Chapter 9: Summary. In this chapter we summarize the results presented in the thesis and give a list of open problems arising form the thesis.
All original results are indicated by symbol , and the ends of proofs are marked ♦ by the symbol .
5 Chapter 2
Basic Terminology in Graph Theory
A graph G = (V,E) is a mathematical structure consisting of a finite nonempty set V (or V (G) when we need to emphasize the graph G) of objects called nodes or vertices together with a set E (or E(G)) of unordered pairs of vertices; the set E can be empty. The elements of E(G) are called edges. If u and v are vertices in V then the edge with the endpoints u and v is denoted by uv. The order of a graph G is the number of vertices of G and the size is the number edges of G. A general reference for graph theoretic notions is [88].
A graph is called simple if there is no loop (an edge that has both endpoints equal) and no multiple edge (more than one edge between the same two vertices).
A plane graph is a graph that can be drawn on a plane in such a way that there is no edge crossing, that is, edges intersect only at their common vertices.
In a graph G = (V,E), a vertex u V is said to be adjacent to a vertex v V if ∈ ∈ there is an edge uv between u and v. The degree of a vertex v, denoted by deg(v), is the number of edges that have v as an endpoint. The largest and smallest of the vertex degrees of G are denoted by ∆(G) and δ(G), respectively.
A k-regular graph is a graph whose vertices all have degree equal to k.
The path of length n 1, Pn, is the graph with n vertices v1, v2, . . . , vn and n 1 − − edges distinct v1v2, v2v3, . . . , vn−1vn. A graph G = (V,E) is said to be connected if there is a path between any two vertices of V . Otherwise, a graph is said to be disconnected.
6 Chapter 2. Basic Terminology in Graph Theory
Let G = (V,E) be a simple connected graph. The distance between two vertices u and v in G, denoted by d(u, v), is the length of a shortest path between u and v.
The diameter of G is diam(G) = maxu,v∈V d(u, v), the greatest distance between any two vertices of G.
The cycle of length n, Cn, is the graph with n vertices x1, x2, . . . , xn and n edges x1x2, x2x3, . . . , xnx1.
A tree is a connected graph without any cycle.
The wheel Wn is the graph obtained from the cycle, Cn, by connecting each vertex of the cycle with a further vertex by an edge. The vertex of degree n is called the central vertex. Each edge incident to the central vertex is called a spoke. The edges not incident to the central vertex are called rim edges.
The complete graph on n vertices, Kn, is a graph such that every pair of vertices is joined by an edge.
A bipartite graph G is a graph whose vertex set V can be partitioned into two subsets U and W , such that any edge of G has one endpoint in U and one endpoint in W . More generally, a s-partite graph G is a graph whose vertex set V can be partitioned into s subsets Vi, 1 i s , such that edge of G has one endpoint in ≤ ≤ Vi and one endpoint in Vj, i = j. 6
The complete bipartite graph, Km,n, is a bipartite graph such that every vertex in V is joined to every vertex in W , and vice versa. Define similarly complete s-partite graph, denoted by Km1,m2,...,ms .
2.1 Operations on Graphs
The Cartesian product G H of the graphs G and H is the graph whose vertex × set is V (G H) = V (G) V (H) and edge set is E(G H) = (V (G) E(H)) × × × × ∪ (E(G) V (H)), see Figure 2.1 for an example. × The join G + H of graphs G and H is the graph with V (G + H) = V (G) V (H) ∪ and E(G + H) = E(G) E(H) uv : u V (G) and v V (H) . For example, see ∪ ∪ { ∈ ∈ } Figure 2.2.
7 Chapter 2. Basic Terminology in Graph Theory
= ×
Figure 2.1: The Cartesian product C4 P3. ×
+ =
Figure 2.2: The join P2 + C3.
If e is an edge of a graph G = (V,E), then the edge deletion graph G e is the − graph obtained from the graph G by deleting the edge e from the edge set E. That is, V (G e) = V (G) and E(G e) = E(G) e . More generally, if D E(G), − − − { } ⊆ then the result of deleting all edges in D is denoted by G D, see Figure 2.3 for − an example.
e
G G e − Figure 2.3: The graphs G and G e. − The disjoint union of two graphs G(V,E) and G0(V 0,E0) is the graph G G0 whose ∪ vertex set and edge set are the disjoint unions V V 0 and E E0, respectively. ∪ ∪ 8 Chapter 2. Basic Terminology in Graph Theory
If G = (V,E) and H = (V 0,E0) are graphs, the corona of G with H, denoted by G H, is the graph obtained by taking one copy of G and V copies of H and | | joining the i-th vertex of G with an edge to every vertex in the i-th copy of H, see
Figure 2.4. In particular, when G = Cn and H = K1, then the corona is called a crown (i.e., a cycle with a pendent edge attached at each vertex).
Figure 2.4: The corona graph P3 C3.
These are just some initial definitions that will be used throughout the thesis.
More specific terminology, definitions and notation used in a particular part or chapter will be defined therein.
9 Part I
EDGE LABELING
10 Chapter 3
Basic Concepts and Literature Review
In this chapter we provide an historical overview of some important results in the area of combinatorics on finite sets, particularly completely separating systems on finite sets, that will be applied to construct vertex antimagic edge (VAE) labeling for some families of graphs. We also present a brief overview of the known results of VAE labeling for families of graphs.
3.1 Completely Separating Systems
Combinatorics deals with questions such as: Given a collection of finite objects, for example, natural numbers, graphs, vectors, sets, and so on, satisfying certain restrictions, how large or how small can the collection be? Combinatorics renders many services to both pure and applied mathematics. It is being realized more and more that combinatorics has connections with many areas of mathematics and the- oretical computer science, such as algebra, probability, search theory, cryptography, coding theory, graph theory, and so on.
We focus on a particular area of separating systems which were first introduced in 1961 by R´enyi [66] in the context of solving certain problems in information theory.
Let be a collection of subsets of [n] = 1, 2, . . . , n . An element a [n] is said to C { } ∈ be separated from b [n] in if there is a member (set) in which contains a but ∈ C C not b.A separating system (or SS) on [n] is a collection of subsets of [n] in which C for any pair of elements a, b [n], either a is separated from b or b is separated ∈ from a.
11 Chapter 3. Basic Concepts and Literature Review
In 1969, Dickson [38] introduced completely separating systems (CSSs). A com- pletely separating system (CSS) on the finite set [n], or (n)CSS, is a collection of subsets of [n] in which for each pair of elements a = b [n], a is separated from 6 ∈ b and b is separated from a in . For example, in the collection 1, 2 , 1, 3 , 1 C {{ } { }} is separated from 2 by 1, 3 but 2 is not separated from 1. Hence the collection { } 1, 2 , 1, 3 is not a CSS. {{ } { }} The sets in the (n)CSS are usually called blocks and the elements of these sets are usually called points. Let h and k be positive integers and let be a (n)CSS. If h C ≤ A k, for all A , then is said to be a (n, h, k)CSS. If h = k, then is said to be | | ≤ ∈ C C C a (n, k)CSS. A d-element in a collection of sets is an element which occurs in exactly d sets in the collection. For any n, h, k fixed positive integers, R(n),R(n, h, k) and R(n, k) are defined as follows: R(n) = min : is a (n)CSS ; R(n, h, k) = {|C| C } min : is a (n, h, k)CSS ; and R(n, k) = min : is a (n, k)CSS .A(n)CSS {|C| C } {|C| C } for which = R(n) is a minimal (n)CSS; (n, h, k)CSS for which = R(n, h, k) |C| |C| is a minimal (n, h, k)CSS; and a (n, k)CSS for which = R(n, k) is a minimal |C| (n, k)CSS.
Let = C1,C2,...,Cm be a completely separating system on [n]. The dual of C { } A is the collection = X1,X2,...,Xn of subsets of [m] such that Xi = j : i C A { } { ∈ Cj, 1 j m for all i [n]. We often omit the brackets and commas when writing ≤ ≤ } ∈ a block, for example, the set a, b, c is often written as abc. The collection = { } C 123, 145, 246, 356 is a minimal (6, 3)CSS; the dual of is 12, 13, 14, 23, 24, 34 . { } C { } An antichain on [n] is a collection of subsets of [n] such that for any distinct A A, B , A B. ∈ A 6⊂ Dickson [38] showed that R(n) log n. Spencer [77] obtained a sharper result ∼ 2 by exploiting the duality of CSSs and antichains and by applying the well-known r Sperner’s theorem (the maximum size of an antichain on [r] is r ), that is, b 2 c
r R(n) = min r : n . { r ≥ } b 2 c Spencer [77] did not explicitly construct any minimal CSS although his proof pro- vided sufficient information to guide any reader who wishes to do so.
12 Chapter 3. Basic Concepts and Literature Review
Separating systems have found applications in information theory [40, 61], search theory [63,69] and results have found uses in the theory of antichains [67].
Subsequently, several variants have been explored in [62, 64, 65, 67], among oth- ers. Ramsay and Roberts [64], and Roberts [67] have explored minimal (n)CSSs, (n, h, k)CSSs and (n, k)CSSs and gave a method for the construction of minimal (n, k)CSSs. The special case of the construction given below is a tool for the study of antimagic labelings of graphs, so it is restated here.
Roberts’ construction [67]
Assume that k 2, n k+1 and k 2n, and let R = R(q, k) = 2n/k. An (R k)- ≥ ≥ 2 | × array M is constructed, where each row of M forms a subset of [n] and the R rows of M form an (n, k)CSS. Let eij denote the element of M in row i and column j. Initialize all elements of M to zero. For e from 1 to n, in order, include e in the two positions of M defined by
min min eij : eij = 0 , j i { }
min min eij : eij = 0 . i j { }
That is, e is placed in the first row of M containing 0, in the first 0-valued place in that row, e is then also placed in the first column of M containing 0, in the first 0-valued place in that column. Each of the integers 1 to n appears in M in two positions, and the array M is the array of an (n, k)CSS. This concludes Roberts’ construction.
Using Roberts’ construction, we obtain the following array of the (12, 4)CSS
1 2 3 4 1 5 6 7 2 6 8 9 3 7 10 11 4 8 10 12 5 9 11 12
13 Chapter 3. Basic Concepts and Literature Review
Although the Roberts’ construction produces an (n, k)CSS for every n and k, with k 2, n k+1 and k 2n, note that it is still an open problem to catalogue all ≥ ≥ 2 | non-isomorphic (n, k)CSSs.
3.2 Antimagic Labeling of Graphs
Graph labeling was first introduced in 1967 by [70]. Let G = (V,E) be a graph. Rosa called an injection f : V 0, 1,..., E a β-valuation if, when each edge uv → { | |} is labeled f(u) f(v) , the resulting edge labels are distinct. Golomb [28] called | − | such a labeling graceful and this is now the most popular term.
The idea of antimagic graph labeling followed from that of a magic graph labeling. The idea of magic graph labeling was first introduced by Sedl´aˇcek[73]. A graph G is magic (more specifically, vertex magic) if it has an edge labeling, with range the real numbers, such that the sums of the edge labels incident with a vertex are all equal to the same integer called the magic constant, independent of the choice of the vertex.
The idea of a magic labeling has been generalized and used as an inspiration by many authors, for example, [50–52] and also [57,58,78,79]. For more details see [84]. Instead of labeling edges of a graph, it is possible to label vertices, or both edges and vertices. Correspondingly, we distinguish ‘edge labeling’, ‘vertex labeling’ and ‘total labeling’. In the next two chapters of Part I we deal only with ‘edge labeling’.
An edge labeling is a bijection
l : E(G) 1, 2,..., E(G) . → { | |}
The weight of a vertex x, x V (G) is defined as ∈ X wt(x) = l(xy) where the sum is taken over all vertices y adjacent to x.
14 Chapter 3. Basic Concepts and Literature Review
We say that a graph has a vertex magic edge labeling if the vertex weight is the same (equal to the magic constant) for all vertices of the graph. At the other extreme in terms of magicness, we have the situation whereby all the vertex weights are pairwise distinct. We call such a labeling a ‘vertex antimagic edge labeling’, or simply, ‘antimagic labeling’, and we say that a graph G is antimagic if there exists an antimagic labeling of G. The notion of an antimagic labeling of a graph was introduced by Hartsfield and Ringel [48] in 1990. An antimagic labeling of a graph G with q edges, is a bijection from the set of edges to the set of positive integers 1, 2, . . . , q such that all the vertex weights are pairwise distinct, where the weight { } of a vertex is defined in the same way as for magic graphs. Figure 3.1 shows an antimagic labeling of the cycle C6 and the complete graph K4.
3 1 1 4 2 5 2 6 3
6 4 5
Figure 3.1: Antimagic labeling of the cycle C6 and the complete graph K4.
Note that most edge labelings of a given graph will be neither magic nor antimagic. However, there is the Hartsfield and Ringel conjecture [48], proposed in 1990.
Conjecture 3.1 [48] Every connected graph, except K2, is antimagic.
During the subsequent 20 years, various classes of graphs have been proved to be antimagic. However, so far the general Hartsfield and Ringel conjecture remains open.
While in general the Hartsfield and Ringel conjecture remains open, research has been conducted in two main directions: some researchers investigate antimagic la- belings with restrictions placed on the weights, while others stay with the idea of plain antimagicness but restrict their attention to particular classes of graphs. For example, Bodendiek and Walther [30] introduced the concept of (a, d)-antimagic la- beling, later also referred to more precisely as (a, d)-vertex antimagic edge labeling. This is an antimagic edge labeling such that the vertex weights form an arithmetic
15 Chapter 3. Basic Concepts and Literature Review progression starting at a and with difference d. Clearly, any (a, d)-vertex antimagic edge labeling is an antimagic labeling.
The second direction of research into antimagic labeling has been proving the an- timagicness of particular families of graphs. Alon et al. [3] used probabilistic meth- ods and some techniques from analytic number theory to show that the conjecture is true for all graphs having minimum degree at least Ω(log V (G) ). They also | | proved that if G is a graph with order V (G) 4 and maximum degree ∆(G), | | ≥ V (G) 2 ∆(G) V (G) 1, then G is antimagic. Alon et al. [3] also have | | − ≤ ≤ | | − shown that all complete multipartite graphs, except K2, are antimagic. Hefetz [49] used the combinatorial nullstellensatz to prove that a graph with 3k vertices, where k is a positive integer, and admits a K3-factor is antimagic. Various papers on the antimagicness of particular classes of graphs have been published, for example, see [14,17,18,25]. It was proved that the cycle Cn, path Pn, n 3, wheel Wn, star ≥ Sn, complete graph Kn, n 3, and the complete bipartite K2,n are antimagic [48]. ≥ Wang [85] proved that the toroidal grids Cn Cn Cn and graphs of 1 × 2 × · · · × s the form G Cn, if G is a k-regular graph with k > 1, are antimagic. Cheng [33] × proved that lattice grids and prisms are antimagic. All Cartesian products of two or more regular graphs of bounded degrees have been proved to be antimagic in [34]. Cranston [35] proved that every regular bipartite graph (complete or not) is an- timagic. Wang and Hsiao [87] proved that the lattice grid Pm Pn, m n 2, is × ≥ ≥ antimagic. They also proved that the generalized prism grid graph G Pn, n 2, × ≥ and the generalized toroidal grid graph G Cn, where G is a d-regular graph, are × antimagic. Recently, Zhang and Sun [89] proved that the Cartesian product of two or more paths is antimagic, if there is at least one path with three or more edges; they also proved that the Cartesian product of an antimagic graph and a connected graph is antimagic. For more details see the dynamic survey [42].
However, to date there are not many results on the antimagicness of non-regular graphs and, especially, of trees [32,48].
16 Chapter 4
Application of Combinatorics to Antimagic Labeling of Graphs
In this chapter we present a powerful relationship between the combinatorics of finite sets and labeling of graphs, more precisely, a relationship between completely separating systems and edge labeling of graphs. Based on this relationship we produce a family of regular vertex antimagic edge labeled graphs and then we extend our results to construct vertex antimagic edge labeling for regular and non-regular graphs.
4.1 Relationship between Completely Separating Systems and Labeling of Graphs
The following two theorems provide a relationship between completely separating systems and edge labeling of graphs.
Theorem 4.1 Let V = v1, . . . , vp be a collection of subsets of [q]. If V is a ♦ { } (q)CSS in which each member is a 2-element, and E is the set of all unordered pairs
vi, vj , where vi vj = , then G = (V,E) is a simple graph, where V = p and { } ∩ 6 ∅ | | E = q. Also, G has an edge labeling l given by l(vi, vj) = vi vj. | | ∩
Proof. Let V = v1, v2, . . . , vp be a (q)CSS. Since V is a (q)CSS consisting of { } 2-elements, an element e of [q] must be an element of exactly two members of V . Take V to be the set of vertices of a graph. Define the edges to be the pairs of
17 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
vertices vj, vl for which there is, in the (q)CSS V , an element e common to both. { } This e is the label of that edge. The set of edges E can be identified with the set [q]. If G = (V,E) is not a simple graph, then there exists either a loop or multiple edge. This means that either there exists at least one member of V containing a duplicate element e of [q] or there is a pair of sets which both contain ei and ej, i = j. Both scenarios contradict V being a CSS. 6
Note that if V = v1, . . . , vp is a (q, k)CSS, then G is a k-regular graph together { } with an edge labeling.
Theorem 4.2 Let G = (V,E) be a simple graph with V = p, E = q with an ♦ | | | | edge labeling l : E [q]. For v V , let Sv be the set of labels of edges incident with → ∈ v. Then the collection of sets Sv v V is a (q)CSS consisting of 2-elements. { | ∈ }
Proof. Let G = (V,E) be a graph having an edge labeling. As each edge has a unique label from the set [q], E can be identified with [q]. Let V = v1, v2, . . . , vp . { } For each vertex vj V , identify vj with Sv . Then vj [q], and V is a collection ∈ j ⊆ of subsets of [q]. As each edge is incident with two distinct vertices, each e [q] ∈ appears in exactly 2 sets in V .
Assume that V is not a completely separating system on E. Then there exist at least two numbers ej, el [q] which are not separated from each other. As each ∈ element of [q] is a 2-element, this means that there exist at least two members vj and vl, j = l, both containing ej and el. Hence G contains multiple edges which 6 contradicts the fact that G is a simple graph. Therefore, V is a (q)CSS.
Note that in Theorem 4.2, if G is a k-regular graph, that is, each vertex of G has degree k, then V is a (q, k)CSS.
Let an edge labeling of a 3-regular graph G = (V,E), where V = 8 and E = 12 as | | | | shown in Figure 4.1. Then we have the array L of the edge labeling of G = (V,E).
18 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
3 9
12 8 711 4 10 12 5 6
Figure 4.1: The graph G = (V,E) with edge labeling.
1 2 3 1 5 6 2 4 5 3 8 9 L = 4 8 10 6 11 12 7 9 11 7 10 12
The rows of the array L form a (12, 3)CSS. This completely separating system has not been obtained by Roberts’ construction. Note that this edge labeling is not antimagic. However, if we swap labels 10 and 11, we would obtain an antimagic labeling of G.
Based on the relationship beteween CSSs and edge labeled graphs we present results concerning the antimagicness of some regular and non-regular graphs obtained from the construction of completely separating systems.
Throughout this chapter when we write G(V,E,L) (or simply, G(L)) to denote a graph G = (V,E) (or G) with the array L as an edge labeling.
Theorem 4.3 Let L be the array of a (q, k)CSS obtained by Roberts’ construc- ♦ tion. Then the k-regular graph G(V,E,L), where V = p = 2q/k, E = q, is | | | | antimagic.
Proof. Use Roberts’ construction to obtain V = v1, v2, . . . , vp , a (q, k)CSS. It is { } clear that V consists of 2-elements and vi is the block i of the array L. Let ei,j be the element in block i and column j of L. Given any pair of blocks vi and vl with i < l, for 1 j k, either ei,j is less than el,j or there exists at most one pair ≤ ≤ P P ei,j = el,j. Therefore, wt(vi) = e∈vi e must be less than wt(vl) = e∈vl e.
19 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
Thus for every value p = 2q/k, Roberts’ construction defines an antimagic k-regular graph G = (V,E) with V = p and E = q; write B(p, k) for this labeled graph. | | | | This family of graphs includes all cycles Cn = B(n, 2) and all complete graphs Kn = B(n, n 1), for n 3. Figure 4.2 shows the antimagic graph B(6, 4) corresponding − ≥ to the (12, 4)CSS in Section 3.1.
1 3 5 7 11 2 6 9 12 4 10 8
Figure 4.2: The graph B(6, 4) with antimagic edge labeling.
Disjoint unions of regular graphs in the family B(n, k) are proved to be antimagic in the following lemma.
Lemma 4.1 Let Gj(Lj), 1 j s, be an antimagic graph, where Lj is the ♦ ≤ ≤ array of a (qj, kj)CSS obtained using Roberts’ construction. Then the disjoint union Ss H = j=1 Gj is antimagic.
Proof. Let Gj(Lj), 1 j s be a k-regular graph with pj vertices and qj edges. ≤ ≤ We may assume that kj kj+1. We construct the array A of edge labels of H as ≤ follows.
Pj−1 (1) Relabel the edge labels in the array Lj, 1 j s, by adding qt to each ≤ ≤ t=1 of the original edge labels; (2) Form the array A as shown below.
L1 L2 . .
Ls
20 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex below.
4.2 Modification of Completely Separating Separating Systems
In this section, based on the construction in the previous section we provide further constructions of antimagic edge labeling of graphs.
4.2.1 Edge Deletion with no Isolated Vertex
Let G = (V,E) be a graph and D E. We define an edge deletion subgraph ⊂ G D as a subgraph of G obtained from G by deleting all edges in D and isolated − vertices if any exist. Note that when G is labeled, E can be identified with [ E ] = | | 1, 2,..., [E] . { | |}
Theorem 4.4 Let G(V,E,L) be a graph, where L is the array of edge labels ♦ obtained by Roberts’ construction. Let D = [t] E and 1 t E 2. Then ⊂ ≤ ≤ | | − G D is antimagic. −
Proof. We construct the array L0 of edge labels of G D by subtracting t from − each edge label in L. Note that whenever there exist non positive edge labels in L0 then they are deleted and the vertices (rows) with no entries are also erased. Under this process, it easy to check that the weight of each vertex (row) in the array L0 is less than the weight of the vertex (row) below.
Recall the antimagic graph G = B(6, 4) from Figure 4.2. Let D = 1, 2, 3 . Then { } we have the antimagic graph G D as shown in Figure 4.3. Note that the deleted − edges are shown by dashed lines.
4.2.2 Edge Switching
Let Gj(Lj), for 1 j 2, be a labeled kj-regular graph with qj edges, where Lj is ≤ ≤ the array of edge labels obtained by Roberts’ construction. Assume that k1 k2. ≤ We construct the edge switching of n1 < k1 edges in G1 with n2 < k2 edges in G2 as follows.
21 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
... 1 2 . 2 3 4 . 3 5 6 4 . 4 7 8 8 3 1 5 7 9 6 2 6 8 9 9 1 7 5
Figure 4.3: The graph G D with antimagic edge labeling. −
(1) Replace the edge labels in L2 with the new labels obtained by adding q1 to each of the original edge labels;
(2) Switch the last largest n1 labels in the last row of L1 with the first smallest n2 labels in the first row of L2.
n1 n2 Denoted by G1 ./ G2 the graph obtained from this edge switching construction.
n1 n2 To guarantee the antimagicness in G ./ G , the conditions k1 k2, 1 n2 1 2 ≤ ≤ ≤ k2 and k1 k1 n1 + n2 k2 n2 + n1 k2 must hold. Note that when k1 = k2 b 2 c ≤ − ≤ − ≤ then n1 and n2 must be equal.
n1 n2 It is easy to prove that the weights of all vertices in G1 ./ G2 are pairwise distinct. n1 n2 It is clear that G1 ./ G2 is connected.
As an example, let K4 and its antimagic labeling obtained using Roberts’ construc- tion as shown in Figure 4.4 and let G = B(6, 4) from Figure 4.2. Then we have the 2 2 graph K4 ./ G and its antimagic labeling in Figure 4.5.
1
1 2 3 4 3 1 4 5 2 4 6 5 2 3 5 6
6
Figure 4.4: The graph K4 with antimagic edge labeling.
22 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
1 4 3
2 1 2 3 1 4 5 2 4 6 5 3 7 8 6 5 6 9 10 7 11 12 13 8 12 14 15 8 7 9 13 16 17 9 10 14 16 18 11 11 15 17 18 13 17 12
15 18 10 16 14
Figure 4.5: An array CSS obtained by edge switching and corresponding graph 2 2 K4 ./ G with antimagic edge labeling.
By repeating this process, we can switch the edges of more such kj-regular graphs kj+1 Gj, 1 j m, with kj kj+1 and nj < kj, 1 nj+1 and kj ≤ ≤ ≤ ≤ ≤ b 2 c ≤ n1 n2 kj nj +nj+1 kj+1 nj+1 +nj kj+1, for 1 < j < m 1, then (((G ./ G ) ./ − ≤ − ≤ − 1 2 n3 nm 1 nm G3 ) ./ . . . ./ Gm−−1 ) ./ Gm is antimagic.
4.2.3 Splitting of Roberts’ Construction
Let G(V,E,L) be a labeled graph with V = p and E = q, where L is an array of | | | | edge labels using Roberts’ construction. We split L into two subarrays L1 and L2 (not rectangular) as follows.
(1) Choose any positive integer q1, 1 q1 < q. Take L1 to be the array containing ≤ edge labels from 1 to q1 and L2 containing edge labels from q1 + 1 to q; (2) Replace each edge label ei in L1 with a new edge label q1 + 1 ei; − (3) Replace each edge label ei in L2 with a new edge label ei q1. −
By the construction of the array L1, it is clear that the weight of each vertex (row) in the array is greater than the weight of the vertex (row) below. Similarly, for the
23 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
array L2, the weight of each vertex (row) in the array is less than the weight of the vertex (row) below.
Note that if L1 or L2 is not the array of edge labeling of K2 (that is, q1 = 1 or q1 = q 1), then we have two non-regular antimagic graphs corresponding to the − edge labelings L1 and L2, otherwise we have only one non-regular antimagic graph.
For an example of the splitting Roberts’construction, recall the 4-regular antimagic graph B(6, 4) from Figure 4.2. If we choose q1 = 7, then we obtain the CSS L1 and its corresponding antimagic G(L1) in Figure 4.6, and L2 and its corresponding antimagic graphs G(L2) in Figure 4.7. Note that the solid dots and dark lines represent the vertices and edges of G(Lj), j = 1, 2, respectively, while the dashed lines and circles in the graphs only show how the construction works.
5 7 7 6 5 4 3 7 3 2 1 1 6 2 .. 6 2 5 1 .. 4 ... 3 ... 4
Figure 4.6: The array (CSS) L1 and graph G(L1) with antimagic edge labeling.
...... 1 2 4 .. 3 4 . 1 3 5 2 . 2 4 5 5 3 1
Figure 4.7: The array (CSS) L2 and graph G(L2) with antimagic edge labeling.
4.3 Generalized Antiprism Graphs
Let us start with two definitions that will be used throughout this section.
24 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
Based on the definition of generalized antiprism from [25] we extend the concept to a more general one. Let G be any regular graph with m vertices. A generalized n antiprism A is a graph obtained by completing the generalized prism G Pn, m G × ≥ 3 and n 2, by edges vi,j+1vi+1,j : 1 i m 1, 1 j n 1 vm,j+1v1,j : ≥ { ≤ ≤n − n≤ ≤ − } ∪ { 1 j n 1 . That is, the vertex set of AG is V (AG) = V (G Pn) = vi,j : 1 ≤ ≤ − } n n × { ≤ i m, 1 j n and the edge set of A is E(A ) = E(G Pn) vi,j+1vi+1,j : ≤ ≤ ≤ } G G × ∪ { 1 i m, 1 j n 1 , where i is taken modulo m. The generalized antiprism n≤ ≤ ≤ ≤ − } n Am in [25] is a special case of AG when G = Cm. Throughout this section we use n n n layer n outer layer ACm instead of Am. A copy of G in AG is called a of AG. An is a layer that contains all vertices with degree d 2 while each vertex in each inner − layer has degree d, for example, see Figure 4.8.
n A generalized toroidal antiprism TG is a graph obtained from the generalized an- n tiprism AG by joining the two outer layers of the antiprism with the edges in the same way as joining between two consecutive layers of the antiprism, see Figure 4.9 as an example.
Theorem 4.5 Let G be any antimagic Cm or Km, m 3, obtained by Roberts’ ♦ ≥ construction. Then the generalized antiprism An , n 2, is antimagic. G ≥
Proof. Assume that G has m vertices and q edges. Let Lj, 1 j n, be the ≤ n ≤ array of the edge labels of Gj, where Gj is the j-th copy of G in AG, n 2. Let l ≥l Tl, 1 l 2(n 1), be the (m 1)-array of edges ei, 1 i m, where ei are the ≤ ≤n − × ≤ ≤ edges of AG that do not belong to any copy Gj. We construct the array A of edge labels of An , n 2, as follows. G ≥
(1) Replace the edge labels in the array Lj, 1 j n, with new labels by adding ≤ ≤ 2(j 1)m + (j 1)q to each of the original edge labels; − −l (2) Label the edge e , 1 i m, in row i of the array Tl, 1 l 2(n 1), with i ≤ ≤ ≤ ≤ − l q +(l 1)m+2i 1, for l 1 mod 2, and l q +(l 2)m+2i, for l 0 mod 2; 2 − − ≡ 2 − ≡ (3) Form the array A as shown below. For n = 2, L1 T1 T2 ∗ ∗ T1 T2 L2
25 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
for n = 3, L1 T1 T2 L2 T3 T4 ∗ ∗ ∗ ∗ T1 T2 T3 T4 L3 and for n 4, ≥
L1 T1 T2 L2 T3 T4 ∗ ∗ T1 T2 L3 T5 T6 ∗ ∗ T3 T4 L4 T7 T8 ...... ∗ ∗ T2(n−3)−1 T2(n−3) Ln−1 T2(n−1)−1 T2(n−1) ∗ ∗ ∗ ∗ T2(n−2)−1 T2(n−2) T2(n−1)−1 T2(n−1) Ln
∗ l l+1 l+1 l+1 l+1 t ∗ l l l l l+1 t where Tl = (e1 e1 e2 . . . em−2 em−1) and Tl+1 = (e2 e3 e4 . . . em em ) , for l 1 mod 2 (see, for example, the array of edge labels in Figure 4.8). ≡
By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below.
We illustrate the generalized antiprism A3 and its antimagic edge labeling in Fig- C4 ure 4.8.
1 2 5 6 1 3 7 8 1 525 6 2 2 4 9 10 3 4 11 12 17 18 13 14 17 18 7 9 13 15 19 20 19 13 14 21 14 16 21 22 26 27 20 15 16 22 15 16 23 24 5 7 17 19 25 26 8 10 23 24 6 9 18 21 25 27 8 11 20 23 26 28 31128 12 4 10 12 22 24 27 28
Figure 4.8: The generalized antiprism A3 with antimagic edge labeling. C4
The following result is derived from the above theorem by deleting all edges of some layers from the original graphs.
26 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
n n Sk Corollary 4.1 Let Gj be a j-th copy of G in AG.Then the graph AG j=1 E(Gj), ♦ 2 − 1 k n, other than A E(G1) E(G2), is antimagic. ≤ ≤ G − ∪
Proof. Delete the arrays L1,L2,...,Lk from the array A of the construction as given in the proof of Theorem 4.5. Then subtract jq from each label (entry) of the array Tl, for 2j 1 l 2j when 1 j < k; and kq from each label (entry) − ≤ ≤ ≤ of the array Lj, and Tl, for 2k 1 l 2(n 1) when k j n 1, of the − ≤ ≤ − ≤ ≤ − resulting array. Note that we use the same notations as the one given in the proof n Sk of Theorem 4.5. We obtain the array of edge labels of the graph A E(Gj), G − j=1 1 k n, as below. ≤ ≤ Case 1: 1 k n 1 ≤ ≤ −
T1 T2 T3 T4 ∗ ∗ T1 T2 T5 T6 ∗ ∗ T3 T4 T7 T8 ...... ∗ ∗ T2(n−3)−1 T2(n−3) Ln−1 T2(n−1)−1 T2(n−1) ∗ ∗ ∗ ∗ T2(n−2)−1 T2(n−2) T2(n−1)−1 T2(n−1) Ln
It is clear that the antimagicness property is preserved.
Case 2: k = n
Subcase 2.1: n = 3 T1 T2 T3 T4 ∗ ∗ ∗ ∗ T1 T2 T3 T4
It is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below with some exceptions. These are the weights of the vertices ∗ ∗ ∗ ∗ (rows) in the last two subarrays T3T4 and T1 T2 T3 T4 that need to be verified.
We have
wt(rm+i) = 4m + 4i 1, for 1 i m; − ≤ ≤
27 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
wt(r2m+1) = 4m + 8;
wt(r2m+t) = 4m + 8t 2, for 2 t m 1; and − ≤ ≤ −
wt(r3m) = 12m 4. − It is clear that
(i) wt(rm+i) = wt(r2m+1), for 1 i m; 6 ≤ ≤ (ii) wt(rm+i) = wt(r2m+t), for 1 i m and 2 t m 1; 6 ≤ ≤ ≤ ≤ − (iii) wt(rm+i) = wt(r3m), for 1 i m. 6 ≤ ≤
Subcase 2.2: n 5 ≥
T1 T2 T3 T4 ∗ ∗ T1 T2 T5 T6 ∗ ∗ T3 T4 T7 T8 ...... ∗ ∗ T2(n−3)−1 T2(n−3) T2(n−1)−1 T2(n−1) ∗ ∗ ∗ ∗ T2(n−2)−1 T2(n−2) T2(n−1)−1 T2(n−1)
It is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below with some exceptions. These are the weights ∗ ∗ of the vertices (rows) in the last two subarrays T2(n−3)−1T2(n−3)T2(n−1)−1T2(n−1) ∗ ∗ ∗ ∗ and T2(n−2)−1T2(n−2)T2(n−1)−1T2(n−1) that need to be verified.
We have
wt(r ) = 8mn 24m + 7; (n−2)m+1 − wt(r ) = 8mn 24m + 8i 2, for 2 i m 1; (n−2)m+i − − ≤ ≤ − wt(r ) = 8mn 16m 3; (n−1)m − − wt(r ) = 8mn 20m + 8; (n−1)m+1 − wt(r ) = 8mn 20m + 8t 2, for 2 t m 1; and (n−1)m+t − − ≤ ≤ − 28 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
wt(rmn) = 8mn 12m 4. − − Note that inside each subarray the weight of each vertex (row) is less than the weight of the vertex (row) below. We consider five cases.
(i) It is clear that wt(r ) < wt(r ), wt(r ) = wt(r ), (n−2)m+1 (n−1)m+1 (n−1)m 6 (n−1)m+1 and wt(r(n−1)m) < wt(rmn). (ii) Assume that wt(r ) = wt(r ). Hence we have 2m 4i + 5 = 0, (n−2)m+i (n−1)m+1 − for 2 i m 1, which is a contradiction. ≤ ≤ − (iii) Assume that wt(r ) = wt(rmn). Then we have 6m 4i 1 = 0, for (n−2)m+i − − 2 i m 1; this is a contradiction. ≤ ≤ − (iv) Assume that wt(r ) = wt(r ). Then we have 4m 8t 1 = 0, (n−1)m (n−1)m+t − − for 2 t m 1. It is a contradiction. ≤ ≤ − (v) Assume that wt(r ) = wt(r ). Then we have m + 2(t i) = 0. (n−2)m+i (n−1)m+t − Therefore,
(a) If m is odd, we get a contradiction. (b) If m is even, then we have the value of pair (i, t) corresponding to the value m as shown in Table 4.1.
m (i, t) 4 no solution 6 (5,2) 8 (7,3); (6,2) 10 (9,4); (8,3); ( 7,2) . . . . m (m 1, m 1); (m 2, m 2); ... ;( m + 3, 3); ( m + 2, 2) − 2 − − 2 − 2 2
n Sk Table 4.1: Value (i, t) corresponding to value m for A E(Gj), 1 k n. G − j=1 ≤ ≤
Since there is no pair (i, t), for m = 4, wt(r ) = wt(r ), for (n−2)m+i 6 (n−1)m+t 2 i m 2 and 2 t m 1. ≤ ≤ − ≤ ≤ − When m 6, there exist m−4 pairs of vertices (rows) corresponding ≥ 2 to (i, t) in which wt(r(n−2)m+i) = wt(r(n−1)m+t). So we need a small
29 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
change to meet the antimagicness property. This can be done in the ∗ ∗ ∗ ∗ subarray T2(n−2)−1T2(n−2)T2(n−1)−1T2(n−1) (particularly, in the subarray ∗ ∗ T2(n−2)−1T2(n−2)) as follows:
When m−4 0 mod 2, we swap the edge labels 2 ≡ 2(n 3)m + 2 and 2(n 3)m + 4, − − 2(n 3)m + 6 and 2(n 3)m + 8, . − − . 2(n 3)m + 2( m−4 1) and 2(n 3)m + 2( m−4 ); − 2 − − 2
respectively, when m−4 1 mod 2, we swap 2 ≡
2(n 3)m + 2 and 2(n 3)m + 4, − − 2(n 3)m + 6 and 2(n 3)m + 8, . − − . 2(n 3)m + 2( m−4 ) and 2(n 3)m + 2( m−4 + 1). − 2 − 2
∗ ∗ This process also effects the subarray T2(n−2)−5T2(n−2)−4T2(n−2)−1T2(n−2) (particularly, in the subarray T2(n−2)−1T2(n−2)) that contains these edge labels but it does not create any new pair of vertices (rows) with the same weight since the difference between the weight of two consecutive vertices (rows) in that subarray before swapping is at least 7. In particular, when
n = 6 and m = 8, wt(r38) = wt(r42) and wt(r39) = wt(r43). After swapping the edge labels 2(n 3)m + 2 = 50 in r42 (also in r25) and − 2(n 3)m + 4 = 52 in r43 (also in r26), then the antimagicness property − holds.
Subcase 2.3: n = 4
T1 T2 T3 T4 ∗ ∗ T1 T2 T5 T6 ∗ ∗ ∗ ∗ T3 T4 T5 T6
30 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
(i) m 6 ≥ The same argument as given above together with: When m−4 0 mod 2, we swap the edge labels 2 ≡ 2(n 3)m + 1 and 2(n 3)m + 3, − − 2(n 3)m + 5 and 2(n 3)m + 7, . − − . 2(n 3)m + 2( m−4 1) 1 and 2(n 3)m + 2( m−4 ) 1; − 2 − − − 2 −
respectively, when m−4 1 mod 2, we swap 2 ≡
2(n 3)m + 1 and 2(n 3)m + 3, − − 2(n 3)m + 5 and 2(n 3)m + 7, . − − . 2(n 3)m + 2( m−4 ) 1 and 2(n 3)m + 2( m−4 + 1) 1. − 2 − − 2 −
(ii)3 m 5 ≤ ≤ These small cases can be verified by hand calculation.
Alternatively, we can construct the array of edge labels of An Sk E(Gj), 1 G − j=1 ≤ k n, without using the array of the edge labels of An . The details are omitted ≤ G here since it is similar to the construction as given in the proof of Theorem 4.5. The rest of the argument is the same as that given in the above proof.
n Recall that Theorem 4.5 gives antimagicness for every antiprism AG, for G = Cm,Km, for m 3 and n 2. We can extend this to a further result of an- ≥ ≥ timagicness for generalized toroidal antiprism graphs.
Theorem 4.6 Let G be either an antimagic Cm or Km, m 3, obtained by ♦ ≥ Roberts’ construction. Then, for n 3, the generalized toroidal antiprism T n is ≥ G antimagic.
Proof. Assume that G has m 3 vertices and q edges. Let Lj, 1 j n, be the ≥ n ≤ ≤ array of edge labels of the j-th copy of G in T , for n 3 . Let Tl, 1 l 2n, G ≥ ≤ ≤ be the (m 1)-array of edges el, 1 i m, where el are the edges of T n that do × i ≤ ≤ i G
31 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
n not belong to any copy of G. We construct the array A of the edge labels of TG, for n 3. We consider two cases. ≥ Case 1: n is even
l (1) Label the edge e , 1 i m, in row i of the array Tl, 1 l 2n, with i ≤ ≤ ≤ ≤ ( l 1)q + (l 1)m + 2i 1, for l 1 mod 2, and ( l 1)q + (l 2)m + 2i, 2 − − − ≡ 2 − − for l 0 mod 2; ≡ (2) Replace the edge labels in the array Lj, 1 j n, with new labels by adding ≤ ≤ 2jm + (j 1)q to each of the original edge labels; − (3) Form the array A as shown below.
T1 T2 L1 T3 T4 ∗ ∗ T1 T2 L2 T5 T6 ∗ ∗ T3 T4 L3 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 Ln−1 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n Ln
By the construction of the array A, it is clear that the weight of each vertex (row) is less than the weight of the vertex (row) below with some exceptions. These are the weights of the last row (rm) and the first row (rm+1) of the subarrays T1T2L1T3T4 ∗ ∗ and T1 T2 L2T5T6, respectively, that need to be verified.
Let eg,h be the edge label at row g and column h in the array A.
We first consider G = Cm. In this case, we have the edge labels in rows rm and rm+1 as shown below.
rm : 2m 1 2m . . . q + 2m q + 4m 1 q + 4m − − rm+1 : 1 3 . . . q + 4m + 2 2q + 4m + 1 2q + 4m + 2
Since em,1 + em,2 + em,4 + em,5 + em,6 = 3q + 14m 2 < 5q + 13m + 6 = em+1,1 + − em+1,2 + em+1,4 + em+1,5 + em+1,6 and em,3 < em+1,3, hence wt(rm) < wt(rm+1). It follows immediately when G = Km.
Cases 2: n is odd
32 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
The construction of Case 1 cannot provide the antiprism property when n is odd. ∗ ∗ However, we can modify the second subarray T1 T2 L2T5T6 of the construction to ∗ ∗ meet that property. Let Eh, 1 h m, be row h of T1 T2 in the subarray ∗ ∗ ≤ ≤ T T L2T5T6, that is, E1 = (1 3), Eh = (2+2(h 2) 5+2(h 2)), for 2 h m 1, 1 2 − − ≤ ≤ − and Eh = (2 + 2(h 2) 5 + 2(h 2) 1), for h = m. When m 0 mod 2, we swap − − − ≡ E2 and E3, E4 and E5,..., Em−2 and Em−1, (resp., when m 1 mod 2, we swap ≡ E2 and E3, E4 and E5,..., Em−1 and Em). Then we have the resulting subarray ∗ ∗ t E L2T5T6, where E = (E1 E3 E2 ...Em−1Em−2Em) when m 0 mod 2, (resp., ∗ t ≡ E = (E1 E3 E2 ...EmEm−1) when m 1 mod 2). Finally, we have the array of n ≡ the edge labels of TG as shown.
T1 T2 L1 T3 T4 ∗ E L2 T5 T6 ∗ ∗ T3 T4 L3 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 Ln−1 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n Ln
Since, for 2 f m 1, the difference between wt(Ef ) and wt(Ef+1) is at most 4 ≤ ≤ − and the difference between wt(rf ) and wt(rf+1) of the subarray L2T5T6 is at least 5, ∗ the weights of the vertices (rows) in the subarray E L2T5T6 are pairwise distinct.
Note that when n is odd, the construction of Case 1 as that given in the proof of Theorem 4.6 provides another antimagic graph, but slightly different to the one obtained in Case 2 above (it is not antimagic generalized antiprism graph).
The generalized toroidal antprism T 4 and its antimagic edge labeling are illustrated C4 in Figure 4.9.
n n Sk Corollary 4.2 Let Gj be a j-th copy of G in T . Then the graph T E(Gj), ♦ G G− j=1 1 k n, is antimagic. ≤ ≤
Proof. Remove the array Lj, 1 j k, from the construction as given in the ≤ ≤ proof of Theorem 4.6. Then, in the resulting array, we subtract jq from each label
(entry) of the array Tl, for 2j + 1 l 2j + 2, when 1 j k; and kq from each ≤ ≤ ≤ ≤
33 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
1 2 9 10 13 14 9 3 4 9 11 15 16 15 13 5 6 10 12 17 18 3 1 7 8 11 12 19 20 43845 37 2 1 3 21 22 25 26 34 33 26 25 2 5 21 23 27 28 16 41 39 14 4 7 22 24 29 30 6 8 23 24 31 32 29 22 21 27 11 47 46 10 13 15 33 34 37 38 30 24 23 28 14 17 33 35 39 40 19 42 40 17 16 19 34 36 41 42 32 31 18 20 35 36 43 44 36 35 25 27 37 39 45 46 74448 43 5 8 6 26 29 38 41 45 47 28 31 40 43 46 48 20 18 30 32 42 44 47 48 12
Figure 4.9: The generalized toroidal antiprism T 4 with antimagic edge labeling. C4 label (entry) of the array Lj, and Tl, for 2k + 3 l 2n, when k + 1 j n. ≤ ≤ ≤ ≤ Note that we use the same notations as the one given in the proof of Theorem 4.6.
n Sk To check the antimagicness of the graph T E(Gj), 1 k n, we consider G − j=1 ≤ ≤ two cases.
Let ri be the i-th row in the array.
Case 1: n is even
Subcase 1.1: k = 1
T1 T2 T3 T4 ∗ ∗ T1 T2 L2 T5 T6 ∗ ∗ T3 T4 L3 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 Ln−1 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n Ln
It is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below.
34 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
Subcase 1.2: 2 k n 1 ≤ ≤ −
T1 T2 T3 T4 ∗ ∗ T1 T2 T5 T6 ∗ ∗ T3 T4 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 Ln−1 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n Ln
It is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below with some exceptions. These are the weights of the vertices ∗ ∗ (rows) in the first two subarrays T1T2T3T4 and T1 T2 T5T6 that need to be verified.
We have
wt(ri) = 4m + 8i 2, for 1 i m, − ≤ ≤
wt(rm+1) = 8m + 7,
wt(rm+t) = 8m + 8t 2, for 2 t m 1, and − ≤ ≤ −
wt(r2m) = 16m 3. −
(i) It is clear that wt(ri) = wt(rm+1), for 1 i m. 6 ≤ ≤ (ii) Assume that wt(ri) = wt(rm+t). Then we have m + 2(t i) = 0. Therefore, −
(a) If m is odd, we get a contradiction. (b) If m is even, we have the value (i, t) corresponding the value m in Table 4.2.
When m 4, there exist m−2 pairs of vertices (rows) corresponding to ≥ 2 (i, t) in which wt(ri) = wt(rm+t). So we need to make a small change to meet the antimagicness property. When m−2 0 mod 2, we swap the 2 ≡ labels 2 and 4, 6 and 8, ... , m 4 and m 2, (resp., when m−2 1 mod 2, − − 2 ≡ we swap the labels 2 and 4, 6 and 8, ... , m 2 and m). Since the two − consecutive vertices (rows) in the first subarray have their weights different
35 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
m (i, t) 4 (4,2) 6 (6,3); (5,2) 8 (8,4); (7,3); (6,2) 10 (10,5); (9,4); (8,3); (7,2) . . . . m (m, m ); (m 1, m 1); ... ;( m + 3, 3); ( m + 2, 2) 2 − 2 − 2 2 n Sk Table 4.2: Value (i, t) corresponding to value m for TG j=1 E(Gj), 1 k n, n even. − ≤ ≤
by 8, this process guarantees that there is no new pair of vertices (rows) with the same weight in that subarray after swapping. For instance, for
any n even and m = 8, wt(r6) = wt(r10) = 78, wt(r7) = wt(r11) = 86 and wt(r8) = wt(r12) = 94. So the edge labels 2 and 4, 6 and 8 must be ∗ ∗ swapped in T1 T2 T5T6 (also in T1T2T3T4) to make these rows have different weights.
(iii) By (ii), we have wt(rm) = wt(r m ) < wt(r2m). m+ 2
Subcase 1.3: k = n T1 T2 T3 T4 ∗ ∗ T1 T2 T5 T6 ∗ ∗ T3 T4 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n
It is clear that the weight of each vertex (row) in the array of the labeling of the graph is less than the weight of the vertex (row) below with some exceptions. These ∗ ∗ are the weights of vertices (rows) in the first two subarrays T1T2T3T4 and T1 T2 T5T6; ∗ ∗ ∗ ∗ ∗ ∗ and in the last two subarrays T2n−5T2n−4T2n−1T2n and T2(n−1)−1T2(n−1)T2n−1T2n that need to be verified. However, a modification of the weights in the first two subarrays has been done in Subcase 1.2, so only the latter that needs to be verified here.
We have
36 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
wt(r ) = 8mn 16m + 7, (n−2)m+1 − wt(r ) = 8mn 16m + 8i 2, for 2 i m 1, (n−2)m+i − − ≤ ≤ − wt(r ) = 8mn 8m 3, (n−1)m − − wt(r ) = 8mn 12m + 8, (n−1)m+1 − wt(r ) = 8mn 12m + 8t 2, for 2 t m 1, and (n−1)m+t − − ≤ ≤ −
wt(rmn) = 8mn 4m 4. − −
(i) It is clear that wt(r ) < wt(r ) < wt(rmn), wt(r ) = (n−2)m+1 (n−1)m+1 (n−1)m 6 wt(r(n−1)m+1) and wt(r(n−1)m) < wt(r(mn). (ii) It is clear that wt(r ) = wt(r ), for 2 t m 1. (n−1)m 6 (n−1)m+t ≤ ≤ − (iii) Assume that wt(r ) = wt(r ). Then we have m + 2(t i) = 0. (n−2)m+i (n−1)m+t − Therefore,
(a) If m is odd, this is a contradiction. (b) If m is even then we have the value (i, t) corresponding to the value m as the one given in Table 4.1.
Since there is no (i, t) when m = 4, wt(r ) = wt(r ), for (n−2)m+i 6 (n−1)m+t 2 i m 1 and 2 t m 1. ≤ ≤ − ≤ ≤ −
When m 6, there exist m−4 pairs of vertices (rows) corresponding ≥ 2 to (i, t) in which wt(r(n−2)m+i) = wt(r(n−1)m+t). So we need to make a small change to meet the antimagicness property. This can be done ∗ ∗ ∗ ∗ in the subarray T2(n−1)−1T2(n−1)T2n−1T2n (particularly, in the subarray ∗ ∗ T2(n−1)−1T2(n−1)) by swapping the edge labels:
When m−4 0 mod 2, 2 ≡
2(n 2)m + 2 and 2(n 2)m + 4, − − 2(n 2)m + 6 and 2(n 2)m + 8, . − − .
37 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
2(n 2)m + 2( m−4 1) and 2(n 2)m + 2( m−4 ); − 2 − − 2
respectively, when m−4 1 mod 2, 2 ≡
2(n 2)m + 2 and 2(n 2)m + 4, − − 2(n 2)m + 6 and 2(n 2)m + 8, . − − . 2(n 2)m + 2( m−4 ) and 2(n 2)m + 2( m−4 + 1). − 2 − 2
∗ ∗ This process also effects the subarray T2(n−1)−5T2(n−1)−4T2(n−1)−1T2(n−1) (particularly, in the subarray T2(n−1)−1T2(n−1)) that contains these edge labels but it does not create any new pairs of vertices (rows) with the same weight since the difference between the weights of two consecutive vertices (rows) in that subarray before swapping is at least 7. For instance, when
n = 6 and m = 10, wt(r47) = wt(r52) = 374, wt(r48) = wt(r53) = 382 and wt(r49) = wt(r54) = 390. After swapping the edge labels 2(n 2)m + − 2 = 82 in r52 (also in r31) and 2(n 2)m + 4 = 84 in r53 (also in r32), m−4 − m−4 2(n 2)m+2( = 86 in r54 (also in r33) and 2(n 2)m+2( +1) = 88 − 2 − 2 in r55 (also in r34), then the antimagicness property holds.
Case 2: n is odd
Subcase 2.1: k = 1
We have T1 T2 T3 T4 ∗ E L2 T5 T6 ∗ ∗ T3 T4 L3 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 Ln−1 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n Ln
38 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
It is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below.
Subcase 2.2: 2 k n 1 ≤ ≤ −
We have T1 T2 T3 T4 ∗ E T5 T6 ∗ ∗ T3 T4 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 Ln−1 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n Ln
By the construction of the array, it is clear that the weight of each vertex (rows) in the array is less than the weight of the vertex (row) below with some exceptions. 1 2 These are the weight of the vertices (rows) of the first two subarrays T T T3T4 and ∗ E T5T6 that need to be verified.
For m = 3, it is simple to check that all weights of the vertices in these subarrays are pairwise distinct. We only here consider when m 4. ≥
It is not hard to check that the vertices (rows) containing Eh and Eh+1, for 2 h ≤ ≤ m 2 and m 0 mod 2 (resp., Eh and Eh+1, for 2 h m 3 and m 1 mod 2) − ≡ ≤ ≤ − ≡ have the same even weight. So we need to make a small change:
(i) If m is even and m−2 2 mod 4, we swap the edge labels 5 and 6, 13 and 4 ≡ 14,... , 2m 3 and 2m 2, (resp. when m−2 0 mod 4, we swap the edge − − 4 ≡ labels and 5 and 6, 13 and 14,... , 2m 7 and 2m 6). We have the weights − − of the vertices (rows) in the resulting subarray are pairwise distinct. (ii) If m is odd and m−1 0 mod 4, we swap the edge labels 5 and 6, 13 and 4 ≡ 14,... , 2m 5 and 2m 4, (resp. when m−1 2 mod 4, we swap the edge − − 4 ≡ labels 5 and 6, 13 and 14,... , 2m 9 and 2m 8). We have the weights − − of the vertices (rows) in the resulting subarray are pairwise distinct, except, m−1 wt(r2m−1) and wt(r2m) are equal and odd, for 0 mod 4. 4 ≡
39 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
Let ET 5T 6 be the resulting subarray of (i) and (ii). It is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below with some exceptions. These are the weighs of the vertices (rows) in the first two subarrays T 1T 2T 3T 4 and ET 5T 6 that need to be verified.
We have
wt(ri) = 4m + 8(i 1) + 6, for 1 i m, − ≤ ≤ m−2 wt(rm+2t) = 8m + 16(t 1) + 18, for 1 t if m 0 mod 2, and, − ≤ ≤ 2 ≡ for 1 t m−3 if m 1 mod 2. ≤ ≤ 2 ≡
It is obvious that wt(rm) < wt(r2m−1).
Assume that wt(ri) = wt(rm+2t). Then m + 1 = 2(i 2t). We have value (i, t) − corresponding to the value m in Table 4.3.
m (i, t) 5 (5,1) 7 (6,1) 9 (7,1) 11 (8,1) . . . . m+5 m ( 2 , 1)
n Sk Table 4.3: Value (i, t) corresponding to value m for TG j=1 E(Gj), 1 k n, n odd. − ≤ ≤
This shows that when m 3, all weights of vertices in the first two subarrays 1 2 3 4 5 6 ≥ T T T T and E T T are pairwise distinct, except when m is odd, wt(r m+5 ) = 2 wt(rm+2).
To have all weights of the vertices (rows) in these two subarrays T 1T 2T 3T 4 and ET 5T 6 pairwise distinct we need a small modification by swapping 3 and 4, 2m 1 − and 2m, for m−1 0 mod 4, (resp. 3 and 4, for m−1 2 mod 4). 4 ≡ 4 ≡
40 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
Subcase 2.3: k = n
We have T1 T2 T3 T4 ∗ E T5 T6 ∗ ∗ T3 T4 T7 T8 ...... ∗ ∗ T2n−5 T2n−4 T2n−1 T2n ∗ ∗ ∗ ∗ T2(n−1)−1 T2(n−1) T2n−1 T2n
It is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below with some exceptions. These are the weight of vertices ∗ (rows) in the first two subarrays T1T2T3T4 and E T5T6, and the last two subar- ∗ ∗ ∗ ∗ ∗ ∗ rays T2n−5T2n−4T2n−1T2n and T2(n−1)−1T2(n−1)T2n−1T2n that need to be verified. However, a modification of the weights in the first two subarrays has been done in Subcase 2.2, and for the latter, it is similar to Subcase 1.3.
n Sk Note that an alternative construction of the array of edge labels of T E(Gj), G− j=1 1 k n, can be obtained directly without using the array of the edge labels of n≤ ≤ TG. The details are omitted here since they are similar to the construction as given in the proof of Theorem 4.6. The rest of the argument is the same as that given in the above proof.
4.4 Conclusion
We have proved that for every degree sequence pertaining to a regular graph dif- ferent from K2, there exists an antimagic labeling of the regular graph with that degree sequence by applying Roberts’ construction. We have also modified Roberts’ construction to construct antimagic labeling for some families of non regular graphs.
Antimagic labelings for generalized antiprisms with layers Cm or Km (built by using Roberts’ construction) are constructed.
We conclude with a challenge to prove or disprove the following open problem.
41 Chapter 4. Application of Combinatorics to Antimagic Labeling of Graphs
n Open Problem 4.1 Is it possible to find vertex antimagic labelings for all AG, n 2, and T n, n 3, where G is any regular graph? ≥ G ≥
42 Chapter 5
Antimagicness of some Families of Graphs
In this chapter we deal with antimagic labeling of families of generalized flower and web graphs, generalized sausage graphs, generalized corona and snowflake graphs, and join graphs. Finally, we prove that for any degree sequence pertaining to a tree, there exists an antimagic labeling of a tree with that degree sequence. We also prove that for any degree sequence of a unicyclic graph, there is an antimagic labeling of a unicyclic graph with that degree sequence.
Let G be a k-regular graph with p vertices and q edges.
The generalized web graph WB(G, m, n) is the graph obtained from the Cartesian product G Pm by adjoining a vertex, the apex, to each vertex of one end copy of × G and then taking p copies of Pn (n 2) and merging an end vertex of a different ≥ copy of Pn with each vertex of the furthermost copy of G from the apex. When G is a cycle, WB(G, m, 2) is simply called the web graph.
The generalized flower graph with p petals (or simply, generalized flower graph) FL(G, m, n, p) is the graph obtained from the generalized web graph WB(G, m, n) by connecting each of the p pendant vertices to the apex with an edge. A petal of
FL(G, m, n, p) is the subgraph Cm+n that contains the new edge, see Figure 5.2 for an example. The complete generalized flower graph with (m+n 2)p petals (or sim- − ply, complete generalized flower) CFL(G, m, n, (m + n 2)p) is the graph obtained − from the generalized web graph WB(G, m, n) by adding (m+n 2)p edges to ensure − that the apex is adjacent to every other vertex. Petals in CFL(G, m, n, (m+n 2)p) − are defined as for the generalized flower graph and there are m + n 2 types of −
43 Chapter 5. Antimagicness of some Families of Graphs
petals: C3,C4,...,Cm+n, for an example, see Figure 5.4. When G is a cycle, CFL(G, m, n, (m + n 2)p) is simply called the complete flower graph. − The generalized sausage graph, denoted by S(G, m), is the graph obtained from the
Cartesian product graph G Pm, m 1 (G P1 = G), by joining each vertex of × ≥ × each end of the G Pm to a further vertex with an edge; and the two new vertices × are called apexes. In particular, when m = 1, each vertex of the graph G joins to two vertices with two edges. The complete mixed generalized sausage, denoted by CMS(G, m), is the graph obtained from the generalized sausage by joining each vertex of each copy of G, except the two nearest copies of G to the apexes, to each apex with an edge, and each corresponding pair of vertices of the two nearest copies of G to the apexes with an edge, see Figure 5.8 as an example.
We extend the definition of corona graph presented in Chapter 2 to a more general one in which the p instances of the graph H are not required to be all the same. The generalized corona graph of the graph G with the graphs H1,H2,...,Hp, denoted by G (H1,H2,...,Hp), is the graph obtained by joining the j-th vertex of G with an edge to every vertex in the graph Hj, 1 j p. The corona graph G H is a ≤ ≤ special case of G (H1,H2,...,Hp) when Hj = H, for 1 j p. Note that when ≤ ≤ G is disconnected, then G (H1,H2,...,Hp) is also disconnected. See Figure 5.9 for an example of a generalized corona graph.
A snowflake graph is constructed recursively from a sequence of graphs, by adjoining copies of a graph to its neighboring graph in the sequence, in a similar way to the corona graph construction; the generalized form does not require all the graphs to be the same. So consider a sequence of graphs H1,H2,...,Hm, and let Hj, 1 j m, contain nj vertices. The generalized snowflake graph of G, denoted ≤ ≤ by Sf(H1,H2,...,Hm,G), is defined to be the graph obtained from G and Hj, 1 j m, by joining the f-th vertex of G to each vertex of the f-th copy of Hm, ≤ ≤ for 1 f p, and then joining the g-th vertex of all p copies of Hm to each vertex ≤ ≤ of the g-th copy of Hm−1, for 1 g pnm; continue iterating until the process of ≤ ≤ joining the h-th vertex of all copies of H2 to each vertex of the h-th copy of H1, for 1 h n2n3 . . . nmp, is reached. See Figure 5.10 for an example of a generalized ≤ ≤ snowflake graph.
44 Chapter 5. Antimagicness of some Families of Graphs
We first label the edges of G by allocating integers 1, 2, . . . , q randomly. Then calculate the weights of vertices and order the vertices so that wt(vi) wt(vi+1), ≤ 1 i p 1. This ordering results in an array L of edge labels of G. It is ≤ ≤ − considered as the original labeling and will be applied throughout the chapter to produce antimagic labelings for graphs in these families of graphs.
Note that, in Figures 5.1, 5.3, 5.5 and 5.7, the solid dots, dashed circles, circles, lines represent the vertices, union of isolated vertices, union of k-regular graphs (not
K1) and set of edges, Th or Tl, respectively.
5.1 Generalized Web and Flower Graphs
We denote by T t the transpose of the array T .
Theorem 5.1 Let G = (V,E) be a k-regular connected graph and G = K1 (resp. ♦ 6 k-regular disconnected graph and G = rK1, r 2). Then the generalized flower 6 ≥ graph FL(G, m, n, p), m 1, n 2, is antimagic. ≥ ≥ Proof. The construction of the generalized flower graph FL(G, m, n, p) uses m copies of G. Let Lj, 1 j m, be the array of edge labels of the j-th copy ≤ ≤ of graph G in FL(G, m, n, p), m 1, n 2. Let Th, 1 h m + n, be the ≥ ≥ ≤ ≤ (p 1)-array of the edges ei, 1 i p, where ei are the edges of FL(G, m, n, p) × ≤ ≤ that do not belong to any copy of G. We construct the array A of edge labels of the generalized flower graph FL(G, m, n, p), m 1, n 2, in two cases as follows. ≥ ≥
Case 1: G = Kp, p 2 ≥
(1) Label the edge ei, 1 i p, in row i of the array Th, 1 h m + n, with ≤ ≤ ≤ ≤ i+(h 1)p, for 1 h n+1, and i+(h 1)p+(h n 1)q, for n+2 h m+n; − ≤ ≤ − − − ≤ ≤ (2) Replace the edge labels in the array Lj, 1 j m, with new labels obtained ≤ ≤ by adding (n + j)p + (j 1)q to each of the original edge labels; − (3) Form the array A as below.
45 Chapter 5. Antimagicness of some Families of Graphs
If m = 1 and n 2, ≥ T1 T2 T2 T3 . . . .
Tn−1 Tn t t T1 Tn+1 Tn Tn+1 L1 More generally, if m 1 and n 2, ≥ ≥
T1 T2 T2 T3 T3 T4 . . . .
Tn−1 Tn t t T1 Tn+1 Tn L1 Tn+2 Tn+1 L2 Tn+3 ......
Tm+n−2 Lm−1 Tm+n Tm+n−1 Tm+n Lm
As an illustration, in Figure 5.1 we present the construction when m is even. The case of m being odd is similar.
T1 T2
T3 Tn+1
Tn L2 L1 Tn+3 Tn+2
Lm Lm 1 Tm+n 1 − Tm+n 2 − − Tm+n Figure 5.1: Illustration of the construction of the generalized flower graph FL(Kp, m, n, p), for p 2, m 2, n 3, and m even. ≥ ≥ ≥
46 Chapter 5. Antimagicness of some Families of Graphs
By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below, except the weight of t t the row T1Tn+1 and the weight of the first row of the subarray TnL1Tn+2 that need to be verified.
t t Let r(n−1)p+1 and r(n−1)p+2 be the row T1Tn+1 and the first row in the subarray TnL1Tn+2, respectively. Let wt(r(n−1)p+1) and wt(r(n−1)p+2) be the weights of r(n−1)p+1 and r(n−1)p+2, respectively. Since the least possible edge labels (that yield the least possible weight) of a vertex in the array L1 are 1 + (n + 1)p, 2 + (n + 1)p, . . . , (p 1) + (n + 1)p, it follows that wt(r ) (1 + (n 1)p) + (1 + (n + − (n−1)p+2 ≥ − 1)p)+ +((p 1)+(n+1)p)+(1+(n+1)p+q) = 2+(n 1)p+ p(p−1) +(n+1)p2+q > ··· − − 2 (n + 1)p2 + p = 1 + + p + (1 + np) + + (p + np) = wt(r ). ··· ··· (n−1)p+1
Case 2: G = Kp, p 2 6 ≥
(1) Label the edge ei, 1 i p, in row i of the array Th, 1 h m + n, with ≤ ≤ ≤ ≤ i + (h 1)p, for 1 h n, and i + (h 1)p + (h n)q, for n + 1 h m + n; − ≤ ≤ − − ≤ ≤ (2) Replace the edge labels in the array Lj, 1 j m, with new labels obtained ≤ ≤ by adding (n + j 1)p + (j 1)q to each of the original edge labels; − − (3) Form the array A as below. If m = 1 and n 2, ≥ T1 T2 T2 T3 . . . .
Tn−1 Tn Tn L1 Tn+1 t t T1 Tn+1 If m 2 and n = 2, ≥ T1 T2 T2 L1 T3 T3 L2 T4 ......
Tm+n−2 Lm−1 Tm+n−1 Tm+n−1 Lm Tm+n t t T1 Tm+n
47 Chapter 5. Antimagicness of some Families of Graphs
and, in general, if m 1 and n 2, ≥ ≥
T1 T2 T2 T3 . . . .
Tn−1 Tn Tn L1 Tn+1 Tn+1 L2 Tn+2 ......
Tm+n−2 Lm−1 Tm+n−1 Tm+n−1 Lm Tm+n t t T1 Tm+n
The illustration of the construction used here is similar to the illustration in Fig- ure 5.1.
By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below.
As an example, using the construction of Case 2 in the proof of Theorem 5.1, we have the array of edge labels of an antimagic labeling for the generalized flower graph FL(C4, 2, 3, 4) in Figure 5.2.
When the array Lj, 1 j m, is removed from the construction as given in the ≤ ≤ proof of Theorem 5.1, we have
Corollary 5.1 The generalized flower graph FL(rK1, m, 2, r), m 1, is an- ♦ ≥ timagic.
When n = 2 and r = 1, we have FL(K1, m, 2, 1) = Cm+2, m 1. The cycle was ≥ proved to be antimagic in [48] and also in Theorem 4.3.
Corollary 5.2 Let G = (V,E), where G = K1 is any k-regular connected graph ♦ 6 (resp. G = rK1, r 2, is any k-regular disconnected graph). Then the generalized 6 ≥ web graph WB(G, m, n), m 1, n 2, is antimagic. ≥ ≥
48 Chapter 5. Antimagicness of some Families of Graphs
1 5 5 2 6 3 7 9 1 4 8 14 5 9 3 17 13 6 10 22 21 7 11 25 8 12 26 18 10 6 9 13 14 17 7 11 19 27 10 13 15 18 28 11 14 16 19 24 23 12 15 16 20 2 17 21 22 25 16 20 15 18 21 23 26 4 12 19 22 24 27 20 23 24 28 8 1 2 3 4 25 26 27 28
Figure 5.2: The generalized flower graph FL(C4, 2, 3, 4) with antimagic edge label- ing.
Proof. In each case in the proof of Theorem 5.1, to construct an array which respects an antimagic labeling of the graph, remove each occurrence of T1 and for h 2, replace Th by Th−1. It is easy to check that in each case this yields an ≥ antimagic labeling of WB(G, m, n), m 1, n 2. ≥ ≥
When n = 2 and r = 1, removing the array Lj, 1 j m from the construction ≤ ≤ used in Corollary 5.2 also works. The reason this works is that K1 has no edges, and hence the array of edge labels is empty. We have WB(K1, m, 2) = Pm+2, m 1. ≥ The path was proved to be antimagic in [48].
Note that when r = 2, WB(2K1, m, 2) = P2m+3, this is a special case of the path Pm+2, m 1. ≥ The antimagicness of complete generalized flower graphs is established by the fol- lowing theorems.
Theorem 5.2 Let G = (V,E) be a k-regular connected graph and G = K1 (resp. ♦ 6 k-regular disconnected graph and G = rK1, r 2). Then the complete generalized 6 ≥ flower graph CFL(G, m, n, (m + n 2)p), m 1, n 2, is antimagic. − ≥ ≥
49 Chapter 5. Antimagicness of some Families of Graphs
Proof. Assume that G has p vertices and q edges. We divide the proof into three cases.
Case 1: n = 2 and m = 1
The proof is the same as that of Theorem 5.1.
By definition, the construction of complete generalized flower graph CFL(G, m, n, (m+ n 2)p) uses m copies of G. Let Lj, 1 j m, be the array of edge labels of − ≤ ≤ the j-th copy of graph G. Let Th, 1 h 2(m + n 2) + 1, be the (p 1)-array ≤ ≤ − × of edges ei, 1 i p, where ei are the edges of CFL(G, m, n, (m + n 2)p) that ≤ ≤ − do not belong to any copy of G. We construct the array A of edge labels of the generalized flower graph CFL(G, m, n, (m + n 2)p), m 1, n 2, as follows. − ≥ ≥
(1) Label the edge ei, 1 i p, in row i of the array Th, 1 h 2(m + n 2) + 1, ≤ ≤ ≤ ≤ − with i + (h 1)p, for 1 h m + 2n 3, and i + (h 1)p + (h m 2n + 3)q, − ≤ ≤ − − − − for m + 2n 3 < h 2(m + n 2) + 1; − ≤ − (2) Replace the edge labels in the array Lj, 1 j m, with new labels obtained ≤ ≤ by adding (m + 2n + j 4)p + (j 1)q to each of the original edge labels; − − (3) To form the array A we consider two cases. The array A is provided for all cases. The details of the proof are given only for the last, most general, subcase. Case 2: n = 2 Subcase 2.1: m = 2 T1 T2 T3 L1 T4 T2 T4 L2 T5 t t t T1 T3 T5 Subcase 2.2: m = 3 T1 T2 T3 T4 L1 T2 T5 L2 T6 T4 T6 L3 T7 t t t t T1 T3 T5 T7
50 Chapter 5. Antimagicness of some Families of Graphs
Subcase 2.3: m 4 ≥
T1 T2 T3 T4 L1 T2 T5 T6 L2 ......
T2m−6 T2m−3 T2m−2 Lm−2 T2m−4 T2m−1 Lm−1 T2m T2m−2 T2m Lm T2m+1 t t t t t T1 T3 ....T2m−3 T2m−1 T2m+1
Case 3: n 3 ≥ Subcase 3.1: m = 1
T1 T2 T2 T3 T4 ......
T2n−4 T2n−3 T2n−2 T2n−2 L1 T2n−1 t t t t T1 T3 ....T2n−3 T2n−1
Subcase 3.2: m = 2
T1 T2 T2 T3 T4 ......
T2n−4 T2n−3 T2n−2 T2n−1 L1 T2n T2n−2 T2n L2 T2n+1 t t t t t T1 T3 ....T2n−3 T2n−1 T2n+1
51 Chapter 5. Antimagicness of some Families of Graphs
Subcase 3.3: m 3 ≥
T1 T2 T2 T3 T4 ......
T2(n−2) T2(n−1)−1 T2(n−1) T2n−1 T2n L1
T2(n−1) T2(n+1)−1 T2(n+1) L2 ......
T2(m+n−2)−6 T2(m+n−2)−3 T2(m+n−2)−2 Lm−2 T2(m+n−2)−4 T2(m+n−2)−1 Lm−1 T2(m+n−2) T2(m+n−2)−2 T2(m+n−2) Lm T2(m+n−2)+1 t t t t t T1 T3 ....T2(m+n−2)−3 T2(m+n−2)−1 T2(m+n−2)+1
The diagram in Figure 5.3 illustrates the construction used here when m is odd. When m is even, the construction is similar and omitted.
By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below, except the weight of the last vertex (row) of the subarray T2(m+n−2)−2T2(m+n−2)LmT2(m+n−2)+1 and t t t t the row T1T3 ...T2(m+n−2)−1T2(m+n−2)+1 that need to be verified.
Let rf be the row f in the array A and wt(rf ) be the weight of the row rf . Let ef,g be the edge label in row f and column g in the array A.
Let r(m+n−1)p and r(m+n−1)p+1 be the last row of the subarray T2(m+n−2)−2T2(m+n−2) t t t t LmT2(m+n−2)+1 and the row T1T3 ...T2(m+n−2)−1T2(m+n−2)+1, respectively.
We have the edge labels of the rows when G = Kp, p 2, as shown below. ≥ r : . 2lp + (m 1)q . . . (2l + 1)p + mq (m+n−1)p − r : 1 . . . p . . . . (2l 1)p + (m 2)q . . . (2l + 1)p + mq (m+n−1)p+1 − − where l = m + n 2. − Since Pp e + e = p(p+1) + (2l 1)p + (m 2)q = g=1 (m+n−1)p+1,g (m+n−1)p+1,(m+n−2)p 2 − − 2lp + (m 1)q = e and e > − (m+n−1)p,(m+n−2)p (m+n−1)p+1,(m+n−2)p−1
52 Chapter 5. Antimagicness of some Families of Graphs
T2 T1
T4 T2(n 1) T3 − L2 T2(n+1) 1 T2(n+1) −
T2(m+n) 10 . . . − T2(m+n) 7 Lm 1 − − T2(m+n) 6 −
Lm T2(m+n) 3 − T2(m n) 4 − − T 2(m+n) 5 Lm 2 − − T2(m+n) 8 − ...... T2(m+2) T L3 2(n+1)+1
T2n
L1 T2n 1 −
Figure 5.3: Illustration of the construction of the complete generalized flower graph CFL(G, m, n, (m + n 2)p), for p 2, m 3, n 3, and m odd. − ≥ ≥ ≥ e and e > e for (m + n 2)p + 1 (m+n−1)p,(m+n−2)p−1 (m+n−1)p+1,g (m+n−1)p,g − ≤ g (m + n 1)p 1, wt(r ) > wt(r ). This also guarantees ≤ − − (m+n−1)p+1 (m+n−1)p that wt(r ) > wt(r ), when G = Kp, p 2. Therefore, for any (m+n−1)p+1 (m+n−1)p 6 ≥ k-regular graph G = rK1, r 1, wt(r ) > wt(r ). 6 ≥ (m+n−1)p+1 (m+n−1)p
When the array Lj, 1 j m, is removed from the construction as given in the ≤ ≤ proof of Cases 1 and 2 of Theorem 5.2, the following corollary emerges.
Corollary 5.3 The complete generalized flower graph CFL(rK1, m, 2, mr), m ♦ ≥ 1, is antimagic.
Using the construction of Subcase 2.1 of Theorem 5.2, we have the array A of edge labels of antimagic labeling and the corresponding antimagic complete generalized
flower graph CFL(4K2, 2, 2, 16) as shown in Figure 5.4.
53 Chapter 5. Antimagicness of some Families of Graphs
1 9 2 10 3 11 4 12 5 13 6 14 7 15 8 16 17 25 29 18 25 30 19 26 31 20 26 32 21 27 33 22 27 34 23 28 35 24 28 36 9 29 37 41 10 30 37 42 11 31 38 43 12 32 38 44 13 33 39 45 14 34 39 46 15 35 40 47 16 36 40 48 1 ... 8 17 ... 24 41 42 43 44 45 46 47 48
37 9 10 1 2 8 41 42 3 29 25 30 48 43 16 11 17 18 36 31 24 19 40 28 26 38 23 20 35 32 22 21 15 12 47 44 34 27 33 7 46 45 4 6 5 14 13 39
Figure 5.4: The complete generalized flower graph CFL(4K2, 2, 2, 16) with an- timagic labeling.
54 Chapter 5. Antimagicness of some Families of Graphs
5.2 Generalized Sausage Graphs
Theorem 5.3 Let G = nK1, n 1, be any connected or disconnected k-regular ♦ 6 ≥ graph. Then the generalized sausage graph S(G, m), m 1, is antimagic. ≥
Proof. Let Lj, 1 j m, be the array of edge labels of the j-th copy of the graph ≤ ≤ G in S(G, m), m 1. Let Tl, 1 l m + 1, be the (p 1)-array of the edges ei, ≥ ≤ ≤ × 1 i p, where ei are the edges of S(G, m), m 1, that do not belong to any ≤ ≤ ≥ copy of G. We construct the array A of edge labels of S(G, m), m 1, as follows. ≥
Case 1: G = Kp, p 2 ≥
(1) Label the edge ei, 1 i p, in row i of the array Tl, 1 l m + 1, with ≤ ≤ ≤ ≤ i + (l 1)p, for 1 l 2; and i + (l 1)p + (l 2)q, for 3 l m + 1; − ≤ ≤ − − ≤ ≤ (2) Replace the edge labels in the array Lj, 1 j m, with new labels obtained ≤ ≤ by adding (j + 1)p + (j 1)q to each of the original edge labels; − (3) Form the array A as shown below. For m = 1, t T1 t T2 T1 T2 L1 and for m 2, ≥ t T1 t T2 T1 L1 T3 T2 L2 T4 ......
Tm−1 Lm−1 Tm+1 Tm Tm+1 Lm
The diagram in Figure 5.5 illustrates the antimagic labeling used here.
By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below.
Case 2: G = Kp, p 1 6 ≥ 55 Chapter 5. Antimagicness of some Families of Graphs
T1 T3 Tm 1 Tm+1 Tm Tm 2 T4 T2 . . . − −. . . t t T1 L1 Lm 1 Lm Lm 2 L2 T2 − −
Figure 5.5: Illustration of antimagic labeling of S(Kp, m), p 2, m 2 and m even. ≥ ≥
(1) Replace the edge labels in the array Lj, 1 j m, with new labels obtained ≤ ≤ by adding (j 1)(p + q) to each of the original edge labels; − (2) Label the edge ei, 1 i p, in row i of the array Tl, 1 l m + 1, with ≤ ≤ ≤ ≤ i + (l 1)p + lq, for 1 l m; and i + (l 1)p + mq, for l = m + 1; − ≤ ≤ − (3) Form the array A into two cases as shown below. Subcase 2.1: m = 1
L1 T1 T2 t T1 t T2
By the construction of the array A, it is clear that the weight of each vertex (row) is less than the weight of the vertex (row) below, except the weight of the last row of t t the subarray L1T1T2 and the weight of the row T1 and T2 that need to be verified. This we do in three subcases.
Let wt(rf ) be the weight of the row rf .
Subcase 2.1.1: G = 2K2 (p = 4, q = 2 and k = 1)
t We have wt(rp) = wt(T1) = 18. However, by swapping the edge labels 7 and 10, then all weights of vertices are pairwise distinct.
Subcase 2.1.2: p = q = 4 and k = 2
t t It is simple to check that wt(rp) = 27, wt(T1) = 26 and wt(T2) = 42.
Subcase 2.1.3: p, q > 4, k > 1
Since the largest possible edge labels of the last row in the array L1 are q (k k(k−1) − − 1), q (k 2), . . . , q 1 and q, wt(rp) (3p + k + 2)q 2 < 3p + (k + 2)q < p(p+1)− − t − ≤ t −t 2 + pq = wt(T1). It is obvious that wt(T1) < wt(T2).
56 Chapter 5. Antimagicness of some Families of Graphs
Subcase 2.2: m 2 ≥ L1 T1 T2 T1 L2 T3 ......
Tm−2 Lm−1 Tm Tm−1 Lm Tm+1 t Tm t Tm+1 By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below, except the weights of t the last row in the subarray Tm−1LmTm+1 and the row Tm that need to be verified.
Let ef,g be the edge label in row f and column g in the array A. We have the largest possible edge labels in the last row (that is the row rmp) of the array Lm t and the row Tm as shown below.
rmp : ... (q 1) + (m 1)(p + q) q + (m 1)(p + q)(m + 1)p + mq − − − T t : . . . m(p + q) 2 m(p + q) 1 m(p + q) m − −
We have emp,p−2 + emp,p−1 + emp,p (3m 1)p + 3mq 1 3mp + 3mq 3 = ≤ − − ≤ − emp+1,p−2 + emp+1,p−1 + emp+1,p. Since emp,g < emp+1,g (in case there is no emp,g, t we assume emp,g = 0), 1 g p 2 and p 2, wt(rmp) < wt(T ). ≤ ≤ − ≥ m
When the array Lj, 1 j m, is removed from the construction as given in ≤ ≤ the proof of Case 1 of Theorem 5.3, the sausage graph degenerates into the path
S(K1, m) = Pm+1 if n = 1, The path was proved to be antimagic originally in [48]. We have the following corollary when n 2. ≥
Corollary 5.4 The generalized sausage graph S(nK1, m), m 1 and n 2, is ♦ ≥ ≥ antimagic.
Proof. For n = 2, S(2K1, m) is a circle C2m+2. It was proved to be antimagic in [48] and also in Theorem 4.3.
We next prove for n 3. We first label the edges of the path Pm+2 as shown in the ≥ diagram in Figure 5.6. We label the edge ei, 1 i m + 1, of Pm+2 labels with ≤ ≤
57 Chapter 5. Antimagicness of some Families of Graphs i. This ensures that the weights of the vertices with degree 2 are pairwise distinct.
To build the graph S(nK1, m) we use n copies of Pm+2. Let Lj, 1 j n, be the ≤ ≤ array of j-th copy of Pm+2, where the weights of the vertices with degree 2 are in the ascending order. We construct the array A of edge labels of S(nK1, m), m 1 ≥ and n 2, as follows. ≥
(1) Replace the label i of the edge ei in the array Lj, 1 j n, by adding ≤ ≤ (j 1)(m + 1) to each of the original edge labels; − (2) Form the array A as shown below.
0 L1 0 L2 . . 0 Ln A1 A2
where L0 , 1 j n, is the array of the edge labels of the vertices of degree 2 of j ≤ ≤ j-th copy of Pm+2, A1 = (m m + (m+ 1) m + 2(m+ 1) . . . m+ (n 1)(m+ 1)) − and A2 = (m + 1 2(m + 1) 3(m + 1) . . . n(m + 1)).
We omit details of the proof when n 3 and m = 1. For the case n 3 and m = 2, ≥ ≥ we only need a small change from the case of n = 3 and m = 2 by swapping the labels 1 and 2; and the rest of the proof is omitted here since it is similar to the following case.
We now consider the case n 3 and m 3. By the construction of the array A, it ≥ ≥ clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below, except the weight of the last four rows that need to be verified.
0 Let rnm−1 and rnm be the last two rows of the subarray Ln. We have wt(rnm−1) = m + (n 1)(m + 1) + (m 2) + (n 1)(m + 1) = 2(n + 1)m 4, wt(rnm) = − − − − n(n−1)(m+1) n(m + 1) + (m 1) + (n 1)(m + 1) = 2n(m + 1) 2, wt(A1) = nm + 2 n−(n+1)(m+1)− − and wt(A2) = 2 .
58 Chapter 5. Antimagicness of some Families of Graphs
For n = 3, we have wt(rnm−1) = 6m + 2, wt(rnm) = 6m + 4, wt(A1) = 6m + 3 and wt(A2) = 6m + 6.
n2m−3nm+n2−5n+4 For n 4, we have wt(A1) wt(rnm) = > 0. Therefore, ≥ − 2 wt(rnm−1) < wt(rnm) < wt(A1) < wt(A2).
m m 2 m 4 1 m 3 m 1 m +1 − − ...... − −
Figure 5.6: A labeling of the path Pm+2, m 1. ≥
We extend Theorem 5.3 to more general cases in the following theorem and corollary.
Let T = (1 2 . . . p 1 p)t. We define the reverse of the array T as T ↑ = (p p − − 1 ... 2 1)t.
Theorem 5.4 Let G = nK1, n 1, be any connected or disconnected k-regular ♦ 6 ≥ graph. Then the complete mixed generalized sausage graph CMS(G, m), m 3, is ≥ antimagic.
Proof. Let Lj, 1 j m, be the array of edge labels of the j-th copy of the graph ≤ ≤ G in CMS(G, m), m 3. Let Tl, 1 l 3m, be the (p 1)-array of the edges ≥ ≤ ≤ × ei, 1 i p, where ei are the edges of CMS(G, m), m 3, that do not belong to ≤ ≤ ≥ any copy of G. We construct the array A of CMS(G, m), m 3, as follows. ≥
(1) Replace the edge labels in the array Lj, 1 j m, with new labels obtained ≤ ≤ by adding (j 1)(p + q) to each of the original edge labels; − (2) Label the edges ei, 1 i p, in row i of the array Tl, 1 l 3m, with ≤ ≤ ≤ ≤ i + (l 1)p + lq, for 1 l m, and i + (l 1)p + mq, for m + 1 l 3m; − ≤ ≤ − ≤ ≤ (3) Form the array A as shown below. For m = 3, L1 T1 T3 T5 T6 ↑ L2 T2 T4 T5 T7 ↑ L3 T3 T4 T8 T9 t t t T1 T7 T8 t t t T2 T6 T9
59 Chapter 5. Antimagicness of some Families of Graphs
More generally, for m 4, ≥
L1 T1 T3 Tm+2 Tm+3 ↑ L2 T2 T4 Tm+2 Tm+4 L3 T3 T5 Tm+5 Tm+6 L4 T4 T6 Tm+7 Tm+8 ......
Lm−2 Tm−2 Tm T3m−5 T3m−4 Lm−1 Tm−1 Tm+1 T3m−3 T3m−2 ↑ Lm Tm Tm+1 T3m−1 T3m t t t t t t t T1 Tm+4 Tm+5 ...T3m−7 T3m−5 T3m−3 T3m−1 t t t t t t t T2 Tm+3 Tm+6 ...T3m−6 T3m−4 T3m−2 T3m
The diagram in Figure 5.7 illustrates the antimagic labeling used here.
Tm+7 Tm+6
T3m 3 Tm+2 T ↑ − 3m
L1 L2 T1 T3 T5 Tm 1 Tm+1 Tm T6 T4 T2 . . . − . . . L3 Lm 1 Lm L4 Tm+5 − Tm+8
T3m 1 T3m 2 − − Tm↑ +4 Tm+3
Figure 5.7: Illustration of antimagic labeling of CMS(G, m), m 3 and m even. ≥
By the construction of the array A, it is clear that the weight of each vertex (row) is less than the weight of the vertex (row) below, except some special cases that need to be verified.
Since for the case m = 3 is quite simple, we omit the details. We next verify some special cases when m 4. ≥
Let ef,g be the label in row f and column g in the array A.
(i) Rows rp and rp+1
60 Chapter 5. Antimagicness of some Families of Graphs
We have the edge labels of rows rp and rp+1 as shown.
rp : ... (m + 2)p + mq (m + 3)p + mq rp+1 : ... 1 + (m + 1)p + mq (m + 4)p + mq
Since ep,mp−1 + ep,mp = (2m + 5)p + 2mq < (2m + 5)p + 2mq + 1, for all p and q, and ep,g < ep+1,g, for 1 g mp 2, wt(rp) < wt(rp+1). ≤ ≤ − (ii) Rows r(m−1)p and r(m−1)p+1 We have e + e = (4m 1)p + 2mq < 4mp + 2mq + 1 = (m−1)p,mp−2 (m−1)p,mp − e(m−1)p+1,mp−2 + e(m−1)p+1,mp, for all p and q; and e(m−1)p,g < e(m−1)p+1,g, for 1 g mp 3, and g = mp 1. Then wt(r ) < wt(r ). ≤ ≤ − − (m−1)p (m−1)p+1 (iii) Rows rmp and rmp+1 Since m 4 and p 2, it is clear that rmp < rmp+1. ≥ ≥ (iv) Rows rmp+1 and rmp+2 t t Let A and B be the sum of all the edge labels in subarrays T1Tm+4 and t t T2Tm+3, respectively. It is easy to check that A < B and emp+1,g < emp+2,g, for 2p + 1 g mp. Hence wt(rmp+1) < wt(rmp+2). ≤ ≤
Figure 5.8 shown the complete mixed generalized sausage graph CMS(K2, 3) with antimagic edge labeling obtained using the construction as given in the proof of Theorem 5.4.
The same construction as the one given in the proof of Theorem 5.4 also works when the array Lj, 1 j m, is removed. Then we have ≤ ≤
Corollary 5.5 The complete mixed generalized sausage graph CMS(nK1, m), ♦ m 3, n 1, is antimagic. ≥ ≥
5.3 Generalized Corona and Snowflake Graphs
Theorem 5.5 Let G be any connected or disconnected graph with p vertices ♦ and no isolated vertex. Let Hj, 1 j p, be any connected or disconnected ≤ ≤ graph with nj vertices, where ∆(Hh) δ(Hj), nh nj, for 1 h < j p, and ≤ ≤ ≤ ≤
61 Chapter 5. Antimagicness of some Families of Graphs
17 14 8 10 1 2 8 12 14 2 5 1 3 9 13 15 4 5 10 12 17 4 6 11 13 16 18 12 21 1 7 4 7 8 10 18 21 19 13 20 7 9 11 19 20 2 3 16 17 18 19 5 6 14 15 20 21 3 6 9 11 16 15
Figure 5.8: The complete mixed generalized sausage CMS(K2, 3) with antimagic edge labeling. n1 + δ(G) ∆(Hp) + 1. Then the generalized corona graph G (H1,H2,...,Hp) ≥ is antimagic.
Proof. Assume that G has p vertices and q edges, and Hj has nj vertices and qj edges. There are three stages in the construction, each performed consecutively:
first labeling each of the graphs Hj (not necessarily antimagically); then labeling the induced edges between these graphs and G; and finally labeling G. So let Tj, 1 j p, be the (nj 1)-array of the edges ei, 1 i nj, where ei are the edges ≤ ≤ × ≤ ≤ of G (H1,H2,...,Hp) that do not belong to neither G nor Hj, 1 j p. We ≤ ≤ construct the array A of the edge labels of G (H1,H2,...,Hp) as follows.
j−1 (1) Label the edges of Hj, 1 j p, with a + Σ qr : a = 1, 2, . . . , qj , and let ≤ ≤ { r=1 } Lj be the array of edge labels of Hj, in which wtHj (v) = 0 if v is an isolated vertex and wtH (vg) wtH (vh), for 1 g < h nj; j ≤ j ≤ ≤ (2) Label the edge ei, 1 i nj, in row i of the array Tj, for 1 j p, with p j−1 ≤ ≤ ≤ ≤ i + Σr=1qr + Σr=1nr; p (3) Label the edges of G with a + Σ (qr + nr), a = 1, 2, . . . , q and let LG be the { r=1 } array of incident edges of G, in which wtG(vg) wtG(vh), for 1 g < h p; ≤ ≤ ≤
62 Chapter 5. Antimagicness of some Families of Graphs
(4) Form the array A as shown below.
L1 T1 L2 T2 . . . .
Lp Tp
TLG
where T is the (p 1)-array of the arrays T t, 1 j p. × j ≤ ≤
The condition that G has no isolated vertex ensures that, when edges are induced between G and the Hj to make the corona graph, that the forbidden graph K2 is not created.
By the construction of the array A, it is easy to check that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below. From the condition ∆(Hh) δ(Hj), nh nj, for 1 h < j p, the number of rows ≤ ≤ ≤ ≤ (vertices) and columns (adjacencies) in successive Lj type matrices increases, and similarly for the Tj arrays. The property n1 + δ(G) ∆(Hp) + 1 ensures that there ≥ are at least as many columns (adjacencies) in the array LG as the array Lp plus the one adjacency in Tp. Hence, since the edges are labeled incrementally during the three stages of the construction, we can see that the weights satisfy the antimagic property.
The following corollary is easily obtained from Theorem 5.5 by deleting Lj and Tj related to Hj = K0 from the construction.
Corollary 5.6 (i) If H1 = K0 (null graph), then the generalized corona graph ♦ G (H1,H2,...,Hp) is antimagic.
(ii) If Hj = K0, for some j, 1 j p, and G is antimagic with labeling on p ≤ ≤ a + Σ (qr + nr), a = 1, 2, . . . , q , then the generalized corona graph G { r=1 } (H1,H2,...,Hp) is antimagic.
Note that the crown graph is a special case of the generalized corona graph G
(H1,H2,...,Hp) when G = Cn, p = n and Hj = mK1, for m 1, 1 j n; it is ≥ ≤ ≤ simply denoted by Cn mK1. Consider also the corona graph K1 H1. When the
63 Chapter 5. Antimagicness of some Families of Graphs
array LG is removed from the construction as given in the proof of Theorem 5.5, it proves that K1 H1 is antimagic. The reason why this works is that K1 has no edges, and hence the array LG of edge labels is empty.
The construction as given in Theorem 5.5 also produces antimagic labeling of trees when G = Tm (a tree on m vertices) and Hj = njK1, 1 j m. To see this, ≤ ≤ observe that the conditions in the theorem are satisfied trivially, since the degrees
∆(Hh) = δ(Hj) = 0, and so n1 + δ(G) 1. However, this only establishes antimag- ≥ icness of certain types of trees: for instance, the construction does not apply to all caterpillars. Hence the conjecture for trees still remains open.
By using a construction such as the one given in the proof of Theorem 5.5, we can construct an antimagic labeling of P3 (2K1,K1 K2,C3), as shown in Figure 5.9. ∪ 5 6 2 4 7 11 1 8 3 1 9 5 10 12 7 2 3 10 2 4 11 13 14 8 6 9 1 3 4 12 5 6 13 7 8 9 14 10 11 12 13 14
Figure 5.9: Antimagic labeling of the generalized corona graph P3 (2K1,K1 ∪ K2,C3).
We now consider the antimagic property for generalized snowflake graphs.
Theorem 5.6 Let G be any connected or disconnected graph with p vertices and ♦ no isolated vertex. Let Hj, 1 j m, be any connected or disconnected kj-regular ≤ ≤ graph with nj vertices, where nh−1+kh nj−1+kj < nm+δ(G), for 2 h < j m. ≤ ≤ ≤ Then the generalized snowflake graph Sf(H1,H2,...,Hm,G) is antimagic.
Proof. Assume that G has p vertices and q edges, and Hj has nj vertices and qj edges, for 1 j m. Let sj = nj+1nj+2 . . . nmp, for 1 j m 1 and ≤ ≤ ≤ ≤ − sm = p; that is, sj is the number of copies of the graph Hj in each iteration of the generalized snowflake graph. There are three stages in the construction: first
64 Chapter 5. Antimagicness of some Families of Graphs
labeling consecutively each of the sj copies of Hj (not necessarily antimagically); then labeling the induced edges between these graphs and G; and finally labeling the graph G.
We begin by choosing any edge labeling of Hj on 1, 2, . . . , qj if Hj = njK1, and { } 6 let L be the array of the edge labels of the lj-th copy of Hj, for 1 j < m, (j,lj) ≤ 1 lj sj, in which wtH (vg) wtH (vh), for 1 g < h nj; and we consider ≤ ≤ j ≤ j ≤ ≤ that edge labeling as the original edge labels of Hj. If some Hj = njK1, then there is no edge labeling and we leave the array L empty. Let T , for 1 j < m, (j,lj) (j,lj) ≤ 1 lj sj, be the (nj 1)-array of the edges ei, 1 i nj, where ei are the edges ≤ ≤ × ≤ ≤ of Sf(H1,H2,...,Hm,G) that do not belong to neither Hj nor G. We construct the array A of the edge labels of Sf(H1,H2,...,Hm,G) as follows.
(1) For 1 j m, 1 lj sj, leave L empty if there exist Hj = njK1, and ≤ ≤ ≤ ≤ (j,lj) if Hj = njK1, replace the edge labels of L(j,lj) with new labels obtained by 6 j−1 adding Σ sr(qr + nr) + (lj 1)qj to each of the original edge labels; r=1 − (2) Label the edge ei, 1 i nj, in row i of the array T(j,lj), for 1 j m, ≤j−1≤ ≤ ≤ 1 lj sj, with i + Σr=1sr(qr + nr) + sjqj + (lj 1)nj; ≤ ≤ m − (3) Label the edges of G with a + Σ sr(qr + nr): a = 1, 2, . . . , q and let { r=1 } LG be the array of the edge labels of G, in which wtG(vg) wtG(vh), for ≤ 1 g < h p; ≤ ≤ (4) Form the array A as follows.
L1 T1 ∗ T1 L2 T2 ∗ T2 L3 T3 ...... ∗ Tm−2 Lm−1 Tm−1 ∗ Tm−1 Lm Tm ∗ Tm LG
t t where Lj = (L(j,1)L(j,2) ...L(j,sj)) , Tj = (T(j,1)T(j,2) ...T(j,sj)) and T ∗ = (T t T t ...T t )t. j (j,1) (j,2) (j,sj)
65 Chapter 5. Antimagicness of some Families of Graphs
As for Theorem 5.5, if the condition that G has no isolated vertex ensures that
K2 is not created when generating the snowflake graph. By the construction of the array A, it is easy to check that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below. Since each Hj is kj-regular, we have kj + nj−1 + 1 labels contributing to the weight at each vertex in Hj. Since the labels are assigned incrementally, these weights also increase. The condition nh−1 + kh nj−1 + kj < nm + δ(G) ensures that there are at least as many labels ≤ being summed at each vertex for each successive layer of the snowflake graph, all with incremental labeling. Hence the weights satisfy the antimagic property.
By applying the same construction as the one given in the proof of Theorem 5.6 we have an antimagic labeling of the generalized snowflake graph Sf(K1,K2,C3), as shown in Figure 5.10.
1 2 6 3 9 4 15 1 5 5 14 17 6 10 1 7 10 2 7 11 18 7 3 8 12 11 4 8 13 13 16 5 9 14 4 2 12 6 9 15 8 10 11 16 17 12 13 16 18 3 14 15 17 18
Figure 5.10: Antimagic labeling of the generalized snowflake graph Sf(K1,K2,C3).
Similarly to Theorem 5.5, if G = K1, then LG is effectively removed from the construction as given in the proof of Theorem 5.6, and we obtain the following corollary.
Corollary 5.7 The generalized snowflake graph Sf(H1,H2,...,Hm,K1) is an- ♦ timagic.
66 Chapter 5. Antimagicness of some Families of Graphs
Note that with the same conditions on the degree sequences, both Theorem 5.5 and Theorem 5.6 may produce antimagic labelings of trees that are different from the ones constructed in Theorem 5.16.
Our main results are that generalized corona graphs and generalized snowflake graphs are antimagic, and hence special cases of these graphs such as some trees also share this property. In view of the conjecture by Hartsfield and Ringel [48], namely that all connected graphs (excepting K2) are antimagic, we conclude with some proposals for future lines of research.
Open Problem 5.1 Given an antimagic labeling l0 : E0 1, 2,..., E0 ♦ −→ { | |} on a subgraph G0 = (V 0,E0) of a graph G = (V,E), give necessary and suffi- cient conditions for extending the sublabeling l0 to an antimagic labeling l : E −→ 1, 2,..., E on the entire graph G = (V,E). { | |}
0 For instance, consider the cycle G = C4 and let G = P3. Trivially, an antimagic labeling of P3 (with labels on the path: 1,2) can be extended to an antimagic labeling of C4 (with labels on the cycle: 1,2,4,3). On the other hand, if we consider 0 G = C5 and G = P4 with the particular labeling of P4: 1,3,2, then this cannot be extended to an antimagic labeling of C5.
We believe that a fruitful line of enquiry towards settling the conjecture would be to consider recursive modular graph decomposition [41]: for instance, first find the unique maximal proper modules, and then replace each of these modules by a single vertex to yield a prime (indecomposable) structure; then, in a manner similar to the constructions as given in this section, label the modules suitably followed by labeling the primal structure.
Alternatively, we propose increasing the known classes of graphs with antimagic labelings by starting with a class of graphs which satisfies this property and, where possible, applying compounding to the graph, that is replacing single vertices with graphs in such a way that the resulting graph is also antimagic. We will illustrate this with any graph G which need not to be antimagic, and a collection of distinct graphs Hi, each of which is antimagic.
67 Chapter 5. Antimagicness of some Families of Graphs
Lemma 5.1 Let G be any graph with vertices vi, 1 i p. Let graphs Hi ♦ ≤ ≤ with ni vertices, 1 i p, be antimagic. Further suppose that ∆(Hi) δ(Hi+1), ≤ ≤ ≤ 1 i p 1, and ∆(Hp) δ(G). Then the compounding of G, given by merging ≤ ≤i − ≤ vertex u in Hi, 1 i p, 1 j ni, with the corresponding vi in G, is antimagic. j ≤ ≤ ≤ ≤
Proof. The proof is similar to the proofs of Theorems 5.5 and 5.6. First, each graph
Hi is labeled antimagically, with each set of labels being assigned incrementally i i relative to Hi−1. Let wtH (u ) and wtH (u ), 1 i p, be the minimum and i 1 i ni ≤ ≤ maximum weights under the labeling of Hi. Assume that G and Hi have q and qi edges, respectively. We next label each edge of G with an integer from 1 + Pp Pp Pp { qi, 2 + qi, . . . , q + qi randomly. Then calculate the weights of i=1 i=1 i=1 } vertices and order the vertices so that wtG(vi) wtG(vi+1), 1 i p 1. ≤ ≤ ≤ − i i+1 The conditions in the lemma force, for 1 i p 1, wtHi (uni ) < wtHi+1 (u1 ) i i+1 ≤ ≤ − and wtHi (uni ) < wtHi+1 (uni+1 ).
i We merge the vertex uni in Hi with the vertex vi of G, 1 i p. After compound- i≤ ≤ ing, the weights of vertices in each Hi different from uni are unaltered. The new weight of each vertex vi, 1 i p, in G inside the compounding graph is given by 0 i≤ ≤ 0 0 wt (vi) = wtG(vi)+wtH (u ). It is clear that wt (vi) < wt (vi+1), for 1 i p 1. i ni ≤ ≤ − Hence the compounding graph is antimagic.
The construction in the proof of Lemma 5.1 can be applied to obtain an antimagic compounding graph of any graph G with a collection of graphs Hi = K2 that are not antimagic.
Observation 5.1 Let G be any graph with p vertices and Hi = K2, 1 i p. ♦ ≤ ≤ Then the compounding of G with Hi is antimagic.
When the collection of Hi contains some isolated vertices, that is, K1 then we have
Observation 5.2 Let wt(vi) < wt(vi+1), 1 i a and wt(vi) wt(vi+1), ♦ ≤ ≤ ≤ a + 1 i p 1; and let Hi = K1, 1 i b. If b a, then the compounding of ≤ ≤ − ≤ ≤ ≤ G with Hi is antimagic.
In other words we can lift some of the conditions δ(Hi) 1 in Lemma 3 for simple ≥ cases of compounding; further, since compounding a graph with isolated vertices
68 Chapter 5. Antimagicness of some Families of Graphs leaves the original graph essentially unchanged, the above observation shows that we do not always need the number of graphs Hi to be the same as the number of vertices in the graph G.
5.4 Join of Graphs
Theorem 5.7 For any two positive integers m and n, n 2m + 1, the complete ♦ ≥ bipartite graph mK1 + nK1 = Km,n is antimagic.
Proof. Let Tl, 1 l n, be the (m 1)-array of edges ei, 1 i m, of Km,n. ≤ ≤ × ≤ ≤ We construct the array A of edge labels of the graph Km,n as follows.
(1) Label the edge ei, 1 i m, in row i of the array Tl, 1 l n, with ≤ ≤ ≤ ≤ i + (l 1)m; − (2) Form the array A as shown below.
t T1 t T2 . . t Tn T1 T2 ...Tn
By the construction of the array A, it is clear that the weight of each vertex (row) in the array is less than the weight of the vertex (row) below, except the weights of t the row Tn and the first row in the subarray T1T2 ...Tn that need to be verified.
Let rn+1 be the first row of the subarray T1T2 ...Tn. Then we have the edge labels t of the row Tn and rn+1 as shown below.
T t : 1 + (n 1)m 2 + (n 1)m . . . m + (n 1)m n − − − rn+1 : 1 1 + m ...... 1 + (n 1)m −
t m(m+1)+2(n−1)m2 2n+n(n−1)m Since n 2m+1, we have wt(T ) = < = wt(rn+1). ≥ n 2 2
Corollary 5.8 For 1 m 10 and m n 2m, the complete bipartite graph ♦ ≤ ≤ ≤ ≤ Km,n, except K2 = K1,1, is antimagic.
69 Chapter 5. Antimagicness of some Families of Graphs
Proof. The same construction as in the proof of Theorem 5.7 also works, except for some cases that need to be modified a little. For m = 3 and n = 3, 6, we swap the labels 1 and 2. For m = 5 and n = 5, 7, we swap the labels 2 and 3, and 4 and 5, respectively. When m = n = 7, we swap the labels 3 and 4. For m = n = 9, we swap the labels 4 and 5.
Combining Theorem 5.7 and Corollary 5.8, we have the following results. Note that when m = 1, 2 and n 2, this result was proved in [48]; it is a special case of the ≥ following corollary.
Corollary 5.9 For 1 m 10 and n 1, the complete bipartite graph Km,n, ♦ ≤ ≤ ≥ except K2 = K1,1, is antimagic.
Theorem 5.8 Let G be any connected or disconnected graph. Then the join ♦ G + nK1, for n 2 V (G) + 1, is antimagic. ≥ | |
Proof. Let G be a graph of order p and size q. We first choose any labeling of G.
Let L be the p-row array of edge labels of G in which wtL(vi) = 0 if vi is an isolated vertex and 0 wtL(vi) wtL(vj), for 1 i < j p. Let Tl be the (p 1)-array of ≤ ≤ ≤ ≤ × edges ei, 1 i p, where ei are the edges of G + nK1 that do not belong to G. We ≤ ≤ construct the array A of edge labels of the graph G + nK1, n 2p + 1, as follows. ≥
(1) Label edge ei in row i, 1 i p of the array Tl, 1 l n, with q +i+(l 1)p; ≤ ≤ ≤ ≤ − (2) Form the array A as shown below.
t T1 t T2 . . t Tn LT1 T2 ...Tn
By the construction of the array A, it is clear that the weight of each vertex (row) of the array is less than the weight of the vertex (row) below, except the weight of t the row Tn and the weight of the first row of the subarray LT1T2 ...Tn that need to be verified. However, if rn+1 is the first row of the subarray LT1T2 ...Tn, then t by the proof of Theorem 5.7 it follows that wt(Tn) < wt(rn+1).
70 Chapter 5. Antimagicness of some Families of Graphs
Applying Theorem 5.8 we have
Corollary 5.10 The complete multi-partite graph Kn ,n ,...,n , for ni 2ni−1 + ♦ 1 2 r ≥ 1, 2 i r, is antimagic. ≤ ≤
Theorem 5.9 Let G = (V,E) be any connected or disconnected graph with ♦ V (G) 5 and V (G) ∆(G) + 2. Then the join G + 2K1 is antimagic. | | ≥ | | ≥
Proof. Let G be a graph of order p and size q. We first choose any labeling of G.
Let L be the p-row array of edge labels of G in which wtL(vi) = 0 if vi is an isolated vertex and 0 wtL(vi) wtL(vj), for 1 i < j p. Let Tl, 1 l 2, be the ≤ ≤ ≤ ≤ ≤ ≤ (p 1)-array of edges ei, 1 i p, where ei are the edges of G + 2K1 that do not × ≤ ≤ belong to G. We construct the array A of edge labels of G + 2K1 as follows.
(1) Label the edge ei, 1 i p, in row i of the array Tl, 1 l 2, with ≤ ≤ ≤ ≤ i + q + (l 1)p; − (2) Form the array A as shown below.
LT1 T2 t T1 t T2
By the construction of the array A, it is clear that the weight of each vertex (row) is less than the weight of the vertex below (row), except the weights of the last row t of the subarray LT1T2 and the row T1 that need to be verified.
Let ef,g be the edge label in row f and column g of the array A and let wt(rf ) be the weight of the vertex f (row f).
We first consider the case V (G) = ∆(G) + 2. Let rp and rp+1 be the last row of | | t the subarray LT1T2 and the row T1, respectively. We have the possible largest edge labels of rp and the edge labels of rp+1 as shown below.
rp : q ∆ + 1 . . . q 2 q 1 q q + p q + 2p − − − rp+1 : q + 1 . . . q + p 4 q + p 3 q + p 2 q + p 1 q + p − − − −
71 Chapter 5. Antimagicness of some Families of Graphs
Pp Pp Then we have ep,g = 5q + 3p 3 and ep+1,g = 5q + 5p 10. Since g=p−4 − g=p−4 − p 5 and ep,g < ep+1,g, for 1 g p 5, wt(rp) < wt(rp+1). It clearly follows ≥ ≤ ≤ − that wt(rp) < wt(rp+1) when V ∆(G) + 2. | | ≥ Applying the construction used in the proof of Theorem 5.9, we have
Corollary 5.11 The join (((G + 2K1) + 2K1) + ... ) + 2K1 is antimagic. ♦
Theorem 5.10 Let G be any either connected or disconnected graph. Then the ♦ join G + Kp , for p1 1, except K2 = K1 + K1, is antimagic. 1 ≥
Proof. Let G be a graph of order p and size q edges, Kp1 has q1 edges. We construct the array A of edge labels of G + Kp1 as follows.
(1) Label the edges of G with 1, 2, . . . , q and let L be the p-row array of edge { } labels of G, in which 0 wtL(vi) wtL(vj), for 1 i < j p; ≤ ≤ ≤ ≤ (2) Let Tl, 1 l p1, be the (p 1)-array of the edges ei, 1 i p, where ≤ ≤ × ≤ ≤ ei are the edges of G + G1 that do not belong to G. Label ei in row i with i + (l 1)p + q; − (3) Label the edges of Kp with pp1 +q +1, pp1 +q +2, . . . , pp1 +q +q1 and let L1 1 { } be the (p1 (p1 1)) of Kp , in which wtL (vi) wtL (vj), for 1 i < j p1; × − 1 1 ≤ 1 ≤ ≤ (4) Form the A as shown below. LT ∗ T L1
∗ t t t t where T = (T1 T2 ...Tp1 ) and T = (T1 T2 ...Tp1 ) .
By the construction of the array A, it is clear that the weight of each vertex (row) of the array is less than the weight of the vertex (row) below.
By induction on n we have
Corollary 5.12 The graph (((G + Kp1) + Kp2) + ... ) + Kpn is antimagic. ♦
Observation 5.3 (i) Given any pair of integers m and n, m > n 2, an an- ♦ ≥ timagic labeling of the complete graph Km can be constructed from any labeling
72 Chapter 5. Antimagicness of some Families of Graphs
(not necessary antimagic) of Kn by applying Corollary 5.12. This labeling of Km might be different from the labeling obtained by [48] and Theorem 4.3.
(ii) When G = K2 and pi = 1, for 1 i n, in Corollary 5.12, an another ≤ ≤ antimagic labeling of the complete graph Kn, n 1 can be built. ≥
The proof of Theorem 5.11 is similar to the proof of Theorem 5.10, so it is omitted here.
Theorem 5.11 Let G be any either connected or disconnected graph. Let G1 ♦ be a k1-regular graph with p1 = k1 + 2 vertices. Then the join H = G + G1 is antimagic.
The next result follows from Theorems 5.10 and 5.11.
Theorem 5.12 Let G be any either connected or disconnected graph. Let G1 be ♦ a graph with p1 vertices and p1 2 deg(vi) p1 1, for 1 i p1. Then the − ≤ ≤ − ≤ ≤ join G + G1 is antimagic.
We conclude this section with VAE labeling of multipartite regular graphs and some related regular graphs.
Lemma 5.2 For m 2, the complete bipartite graph Km,m is antimagic. ♦ ≥
Proof. Let Tl, 1 l m, be the (m 1)-array of the edges ei, 1 i m, of the ≤ ≤ × ≤ ≤ regular complete bipartite graph Km,m. We label the edges ei, 1 i m, in row ≤ ≤ i of the array Tl with i + (l 1)m. Then the array A = T1T2 ...Tm is the array of − edge labels of Km,m, where each row of A is the set of all the labels of the edges incident with a vertex in one partition of vertices while each column is the set of all the labels of edges incident with a vertex in the other partition of vertices. We next prove that all vertex weights of Km,m are pairwise distinct. Let ri and cj for 1 i, j m, be a row (vertex) and a column (vertex) of the array A, respectively. ≤ ≤ m3−m2+2m By the construction of the array A, we have wt(ri) = 2 + (i 1)m and m(m+1) 2 − wt(cj) = + (j 1)m . It is clear that wt(ri) < wt(rf ), for 1 i < f m 2 − ≤ ≤ and wt(cj) < wt(cg), for i j < g m. We finally verify that each vertex weight ≤ ≤ of one partition of the vertices is distinct from each vertex weight in the other partition of the vertices.
73 Chapter 5. Antimagicness of some Families of Graphs
m3−m2+m m3−m2+2m For m even, we have wt(c m ) = < = wt(r1) and wt(rm) = 2 2 2 m3+m2 m3+m2+m < = wt(c m ). Hence all the vertex weights are pairwise distinct. 2 2 2 +1
2 For m odd, suppose wt(ri) = wt(cj), then we have m(m 2jm+2i 1) = 0. Since − − m is a positive integer, m2 2jm + 2i 1 = 0. Therefore, j2 2i + 1 > 0 and must − − − be a square. This leads to i = j; substituting back into the equation we obtain m = 2j 1. In this case the labeling of Km,m is not yet antimagic. To obtain an − m antimagic labeling of Km,m we make a small change by swaping the labels and b 2 c m + 1. b 2 c
Theorem 5.13 For m 1, the complete 3-partite graph Km,m,m is antimagic. ♦ ≥
Proof. Let L be the array obtained from the array A as the one given in the proof of Lemma 5.2 by arranging the vertices of Km,m, in which wtL(vi) wtL(vj), for ≤ 1 i < j 2m, that is, each row in L represents a vertex of Km,m and the entries ≤ ≤ in each row represent the labels of edges incident with that row (vertex). Let Tl, 1 l m, be the (2m 1)-array of the edges ei, 1 i 2m, where ei are the ≤ ≤ × ≤ ≤ edges of Km,m,m that do not belong to Km,m. We construct the array B of the edge labels of Km,m,m as follows.
Case 1: m even
(1) Label the edge ei, 1 i 2m, in row i of the array Tl, 1 l m, with ≤ ≤ ≤ ≤ i + (l 1)2m, for 1 l m ; and i + 2m2 + (l ( m + 1))2m, for m + 1 l m; − ≤ ≤ 2 − 2 2 ≤ ≤ (2) Replace the edge labels in the array L with the new labels obtained by adding m2 to each of the original edge labels; (3) Form the array B as shown below.
74 Chapter 5. Antimagicness of some Families of Graphs
t T1 t T2 . . t T m 2 T1 T2 ...T m LT m T m ...Tm 2 2 +1 2 +2 t T m 2 +1 t T m +2 2. . t Tm
By the construction of the array B, it is clear that the weight of each vertex (row) is less than the weight of the vertex below, except in two cases that need to be verified.
Let ef,g be the edge label in row f and column g of the array B and let wt(rf ) be the weight of the vertex rf (row f).
t (i) wt(T m ) and wt(r m +1) 2 2 m 3 2 3 2 m 2 2m 4m −4m +2m 6m −4m +4m 2 Since Σg=1e m ,g +Σ 3m e m ,g = 4 < 4 = Σg=1e 3m ,g + 2 g= 2 +1 2 2 2m m 3m t Σ 3m e m and e m < e m , for +1 g , wt(T m ) < wt(r m ). g= +1 2 +1,g 2 ,g 2 +1,g 2 2 2 +1 2 ≤ ≤ 2 t (ii) wt(r m +2m) and wt(T m ) 2 2 +1 m 3 2 3 2 2 2m 6m +4m 8m +4m +2m Since Σg=1e m +2m,g + Σ 3m e m +2m,g = 4 < 4 2 g= 2 +1 2 m 2 2m m = Σg=1e m +2m+1,g + Σ 3m e m +2m+1,g and e m +2m,g < e m +2m+1,g, for 2 + 2 g= 2 +1 2 2 2 3m t 1 g 2 , wt(r m +2m) < wt(T m +1). ≤ ≤ 2 2
Case 2: m odd
This case is similar to Case 1 and so the details are omitted here. We only describe how to construct the array Tl, 1 l m, and the array B. ≤ ≤
(1) Label the edge ei, 1 i 2m, in row i of the array Tl, 1 l m, with ≤ ≤ ≤ ≤ i + (l 1)2m, for 1 l m+1 ; and i + 2m2 + m + (l ( m+1 + 1))2m, for − ≤ ≤ 2 − 2 m+1 + 1 l m; 2 ≤ ≤
75 Chapter 5. Antimagicness of some Families of Graphs
(2) Form the array B as shown below.
t T1 t T2 . . t T m+1 2 T1 T2 ...T m+1 LT m+1 T m+1 ...Tm 2 2 +1 2 +2 t T m+1 2 +1 T t m+1 +2 2. . t Tm
By induction on n, we have
Corollary 5.13 For m 1, n 2, the complete n-partite graph Km, m, . . . , m, ♦ ≥ ≥ | {z } n times except K2 = K1,1, is antimagic.
Figure 5.11 illustrates the complete 4-partite graph K2,2,2,2 and the antimagic la- beling obtained by using the construction in the proof of Theorem 5.13.
Note that the complete graph Kn, n 3, was proved to be antimagic in [48] and ≥ also in Theorem 4.3. This is a special case of Corollary 5.13 when m = 1, that is Kn = K1, 1,..., 1. | {z } n times We extend our results to more general regular graphs.
Lemma 5.3 Let G = mK2, m 1. Then G + (2m 1)K1 is antimagic. ♦ ≥ −
t Proof. The array of edge labels of G is L = (1 1 2 2 . . . m m) . Let Tl, 1 l ≤ ≤ 2m 1, be the (2m 1)-array of the edges ei, 1 i 2m, where the ei’s are the − × ≤ ≤ edges of G + (2m 1)K1 that do not belong to mK2. We construct the array B of − the edge labels of G + (2m 1)K1 as follows. −
76 Chapter 5. Antimagicness of some Families of Graphs
7 13
11 10
1 2 3 4 5 6 1 7 8 9 10 19 8 14 15 2 7 11 12 15 20 21 2 23 1 3 8 11 13 16 21 20 5 4 9 12 14 17 22 9 16 5 10 13 14 18 23 12 18 3 19 6 15 16 17 18 24 17 19 20 21 22 23 24
22 6
24 4
Figure 5.11: The complete 4-partite graph K2,2,2,2 with antimagic labeling.
(1) Label the edge ei, 1 i 2m, in row i of the array Tl, 1 l 2m 1, with ≤ ≤ ≤ ≤ − i + (l 1)2m; − (2) Replace the edge labels in the array L with the new labels obtained by adding 2(2m 1)m to each of the original edge labels; − (3) Form the array B as shown below.
t T1 t T2 . . t Tm T1 T2 ...T2m−1 L t Tm+1 t Tm+2 . . t T2m−1
By the construction of the array B, it is clear that the weight of each vertex is less than the weight of the vertex below with two exceptions that need to be verified.
Let ef,g be the edge label in row f and column g of the array B and let wt(rf ) be the weight of the vertex rf (row f).
77 Chapter 5. Antimagicness of some Families of Graphs
t (i) wt(Tm) and wt(rm+1) t 3 2 3 2 We have wt(Tm) = 4m 2m + m < 4m 2m + 2m = wt(rm+1). t − − (ii) wt(r3m) and wt(Tm+1) t We have the edge labels of r3m and Tm+1 as shown below.
r3m : 2m 4m . . . (2m 1)2m (2m 1)2m + m − − T t : 1 + 2m2 2 + 2m2 ... (2m 1)2m2 2m + 2m2 m+1 − 3 2 3 2 t We have wt(r3m) = 4m + 2m m < 4m + 2m + m = wt(T ). − m+1
Theorem 5.14 Let G be a k-regular (connected or disconnected) graph with p ♦ vertices and k 2. Then the p-regular graph G + (p k)K1 is antimagic. ≥ −
Proof. We can use the same construction as in the proof of Theorem 5.13, replacing Km,m by G.
By the induction on n, we obtain
Corollary 5.14 The (k +ns)-regular graph (((G+sK1) + sK1) + ... ) + sK1 is ♦ | {z } n times antimagic, for s = p k. −
Combining Lemma 5.2, Corollary 5.13, Lemma 5.3 and Corollary 5.14, we have
Theorem 5.15 Let G be a k-regular graph, where k 0 and s = p k. Then ♦ ≥ − the (k + ns)-regular graph (((G + sK1) + sK1) + ... ) + sK1 is antimagic. | {z } n times
5.5 Trees and Unicyclic Graphs
We here recall the definition of a Ferrers diagram from [60]. Let r be a positive integer. An n-part partition of r is represented by a sequence of positive integers d = (d1, d2, . . . , dn), where r = d1 + d2 + + dn and d1 d2 dn > 0. ··· ≥ ≥ · · · ≥ The corresponding Ferrers diagram, F (d), consists of r boxes, arranged in n left- justified rows. The number of boxes in row i of F (d) is di, for 1 i n. The ≤ ≤ Ferrers diagram determines a graph with an edge labeling. The i-th row of F (d) corresponds to vertex vi. Vertices vi and vj are joined by edge with label k if k
78 Chapter 5. Antimagicness of some Families of Graphs
appears in both row i and row j of F (d). As the i-th row of F (d) contains di numbers, vertex vi is incident with di edges and so has degree di.
We provide two different constructions of VAE labelings in the proof of the following theorem and then we prove that these two constructions provide two different VAE trees corresponding to the given degree sequence when there are at least seven vertices in Theorem 5.17.
Theorem 5.16 For every degree sequence d = (d1, d2, . . . , dn) of a connected ♦ Pn graph, where n 3 and di = 2(n 1), there exists a vertex antimagic edge ≥ i=1 − labeling of a tree corresponding to that degree sequence.
Proof. Assume that d1 d2 dn 1. Then we can construct the Ferrers ≥ ≥ · · · ≥ ≥ diagram F (d). Let cj, 1 j d1, be the number of boxes in column j. We know ≤ ≤ that c1 = n > c2 c3 cd 1. ≥ ≥ · · · ≥ 1 ≥ Construction 1: We construct an edge labeling of a graph with the degree se- quence d, where entries in each row of F (d) are the labels of edges incident with the vertex of the graph corresponding to that row. We replace the boxes in the first column (from the top) with the labels n 1, n 1, n 2, n 3,..., 2, 1. There − − − − are now n 2 boxes left that need to be replaced by the labels 1, 2, . . . , n 2. We − − replace the box in row i and column j of F (d), for 1 i c2, 2 j di, with Pi−1 ≤ ≤ ≤ ≤ n j when i = 1 and n dk + (i j 1) when 2 i c2. It is easy to check − − k=1 − − ≤ ≤ that no label occurs twice in the same row. Moreover, it is clear that the weight of each vertex (row), that is, the sum of the all labels in that row, is greater than the weight of the vertex below.
Construction 2: This is similar to Construction 1, but we replace the boxes in F (d) column by column. We do the same as in Construction 1 for the first column. There are now n 2 boxes left that need to be replaced by the labels 1, 2, . . . , n 2. − − We replace the box in column j and row i of F (d), for 2 j d1, 1 i cj, with Pj−1 ≤ ≤ ≤ ≤ n (i+1), when j = 2; and n ck (i+1), for 3 j d1. It is easy to check − − k=2 − ≤ ≤ that no label occurs twice in the same row. Moreover, it is clear that the weight of each vertex (row), that is, the sum of all edge labels in that row, is greater than the weight of the vertex below.
79 Chapter 5. Antimagicness of some Families of Graphs
It is easy to show that the graphs obtained from both constructions are connected. As they have n vertices and n 1 edges, they are both trees. Both constructions − result in VAE trees with the correct degree sequence.
Consider the degree sequence d = (4, 3, 3, 2, 1, 1, 1, 1, 1, 1). We first apply Construc- tion 1 to get a VAE labeling of a tree corresponding to that degree sequence as shown in Figure 5.12.
9 8 7 6 9 5 4 8 3 2 7 1 98 7 6 6 5 4 54 32 1 3 2 1
Figure 5.12: The tree with VAE labeling obtained from d = (4, 3, 3, 2, 1, 1, 1, 1, 1, 1) by Construction 1.
By applying Construction 2, we obtain a VAE labeling of a tree corresponding to that degree sequence as shown in Figure 5.13.
9 8 4 1 9 7 3 98 4 1 8 6 2 7 5 6 5 73 62 4 3 2 1 5
Figure 5.13: The tree with VAE labeling obtained from d = (4, 3, 3, 2, 1, 1, 1, 1, 1, 1) by Construction 2.
There is only one star (resp. path) with a given number vertices. Both stars and paths have been proved to be VAE in [48]. Hence we have
80 Chapter 5. Antimagicness of some Families of Graphs
Observation 5.4 For c2 = 1, n 2, for n 3, Constructions 1 and 2 provide ♦ − ≥ the same vertex antimagic edge labeling for Sn and Pn, respectively.
Lemma 5.4 If 2 c2 3 < n 2, then Constructions 1 and 2 provide two ♦ ≤ ≤ − different vertex antimagic edge labelings for the same tree.
0 Proof. Let Tn and Tn be the two VAE trees obtained from Constructions 1 and 2 corresponding to d = (d1, d2, . . . , dn) with d1 d2 dn 1, respectively. ≥ ≥ · · · ≥ ≥ We consider two cases.
Case 1: c2 = 2
In this case, d1 d2 2 and di = 1, for 3 i n. There are two internal vertices ≥ ≥ ≤ ≤ for both Tn and T 0 . The edge label n 1 is incident with the vertex of degree d1 in n − 0 0 Tn and Tn; and also incident with the vertices of degree d2 in Tn and Tn. Hence Tn 0 and Tn are the same tree. It is simple to check that two VAE labelings are different.
Case 2: c2 = 3
In this case, d1 d2 d3 2 and di = 1 for 4 i n. Write vi for the vertex ≥ ≥ ≥ ≤ ≤ 0 which corresponds to the degree di (the i-th row of F (d)). Both Tn and Tn have three internal vertices: v1, v2 and v3 and in both trees the vertex v1 is joined to v2 by edge n 1 and to v3 by edge n 2. The degrees of v1, v2 and v3 are d1, d2 and − − 0 d3 respectively. Therefore Tn = Tn as trees.
The conditions on the degree sequence force d1 > 2. In Tn the vertex v1 is joined to a leaf by edge n 3 and in T 0 the vertex v2 is joined to a leaf by n 3. Therefore − n − the trees have different labelings.
Lemma 5.5 Let 4 c2 n 3 and Tn be a VAE tree obtained by Construc- ♦ ≤ ≤ − tion 1. Then diam(Tn) c2. ≤
Proof. As in the proof of Lemma 5.4, the vertex vi stands for the i-th row of F (d). The conditions on the degree sequence force d1 > 2. There are c2 internal vertices. By Construction 1, vertex v1 has degree d1 > 2 and is adjacent to at least v2, v3 and v4. Hence a path in Tn cannot contain all the internal vertices, it can
81 Chapter 5. Antimagicness of some Families of Graphs
contain at most c2 1 internal vertices and so have length at most c2. Therefore, − diam(Tn) c2. ≤
Lemma 5.6 Let 4 c2 n 3 and T 0 be a VAE tree obtained by Construc- ♦ ≤ ≤ − n 0 tion 2. Then diam(Tn) = c2 + 1.
Proof. As in the proof of Lemma 5.4, the vertex vi stands for the i-th row of F (d). By Construction 2, there are c2 internal vertices: v1, v2, . . . , vc and n c2 leaves. 2 − The vertex v1 is joined exactly with two internal vertices v2 and v3 by the edge labels n 1 and n 2, respectively; the vertex v2 is joined with the vertex v4 by − − the edge label n 3 and the vertex v3 is joined with the vertex v5 by the edge − label n 4, and so on. In general, the vertex vi is joined with the vertex vi+2, for − 2 i c2 2, by the edge label n (i + 1). Hence there is a path in T 0 containing ≤ ≤ − − n 0 all internal vertices. Therefore, diam(Tn) = c2 + 1.
Combining Lemmas 5.5 and 5.6, the following theorem is proved.
Theorem 5.17 Let Tn and T 0 be the VAE trees obtained by Constructions 1 and ♦ n 2, respectively. If 4 c2 n 3, then Constructions 1 and 2 give VAE labelings ≤ ≤ − for two different trees with the same degree sequence.
A unicyclic graph is a connected graph G = (V,E) that contains exactly one cycle. Alternatively, it is a connected graph G = (V,E) such that V = E . The cycle | | | | Cn, n 3, is a special case of unicyclic graphs and it was proved to be VAE in [48] ≥ and also in Theorem 4.3. We modify Construction 1 in the proof of Theorem 5.16 to prove the following theorem.
Theorem 5.18 For every degree sequence d = (d1, d2, . . . , dn), where di 2, ♦ Pn ≥ for some i 3 and di = 2n, there exists a vertex antimagic edge labeling of a ≥ i=1 unicyclic graph corresponding to that degree sequence.
Proof. Assume that d1 d2 dn 1. Then we can construct the Ferrers ≥ ≥ · · · ≥ ≥ diagram F (d). We construct an edge labeling of G satisfying that degree sequence, where entries in each row of F (d) are the labels of edges incident with the vertex of G corresponding to that row. There are 2n boxes in F (d) to be replaced by the
82 Chapter 5. Antimagicness of some Families of Graphs integers from 1, 2, . . . , n . We first replace the boxes in the first column (from the { } top) with the labels n, n, n 1,..., 3, 2. There are now n boxes left that need to − be replaced by the labels 1, 2, . . . , n 1 (label 1 must appear twice). We replace − the box in row i and column j of F (d), for 2 j di, with the label n (j 1) ≤ ≤Pi−1 − − when i = 1; and when 2 i n, with the label n dk + (i j); if the label ≤ ≤ − k=1 − is 0, replace it by 1. That is, the last two boxes are replaced by the label 1. Note that the resulting array is not always the array of an edge labeling. We consider two cases.
Case 1: If there is some di = 2, for some i 3, then the two occurrences of the ≥ label 1 must occur in two different rows and no pair of distinct labels occurs in two different rows. So we have an edge labeling.
Case 2: If there is no di = 2, for i 3, then two occurrences of the label 1 must ≥ occur in the same row, say row i, where i is the last row in which di is the minimal integer greater than 2. We swap one occurrence of the label 1 with the label di 1 − in the first column of F (d) and then reorder the labels in that column in descending order. We now have an edge labeling of G corresponding to that degree sequence.
In both cases it is clear that the weight of each vertex (row), that is, the sum of the all labels in that row, is greater than the weight of the vertex (row) below. It is easy to show that the graph obtained is connected. As G has n vertices and n edges, it is a unicyclic graph.
5.6 Conclusion
In this chapter we have proved that some families of graphs, generalized web and flower graphs, generalized sausage graphs, generalized corona and snowflake graphs, and some families of join of graphs, to be VAE. Complete multi-partite graphs are also proved to be VAE.
At the end of this chapter we focused on trees and unicyclic graphs. We have proved (see Theorems 5.16 and 5.17) that for every degree sequence pertaining to a tree, there exists an antimagic labeling of a tree with that degree sequence; if the number of vertices is least seven then there are at least two VAE labelings for two different trees with that degree sequence. For a degree sequence of a unicyclic graph, there
83 Chapter 5. Antimagicness of some Families of Graphs exists a VAE labeling of a unicyclic graph with that degree sequence, see Theorem 5.18.
With more work one could characterize VAE labeling for every degree sequence pertaining to a non regular graph and one could construct a VAE labeling of a unicyclic graph by applying Construction 2. We propose two open problems as follows.
Open Problem 5.2 For every degree sequence pertaining to a non regular graph different from a tree and a unicyclic graph, there exists an antimagic labeling for a graph corresponding to that degree sequence.
Open Problem 5.3 Given any degree sequence pertaining to a unicyclic graph, is it possible to construct an antimagic unicyclic graph corresponding to that de- gree sequence that is different from the one obtained in the proof of Theorem 5.18? Specifically, can this graph be constructed using Construction 2 in the proof of The- orem 5.16?
84 Part II
TOTAL LABELING
85 Chapter 6
Basic Concepts and Literature Review
In this chapter we provide the terminology, definitions and notation that will be used throughout the second part of the thesis. Terminology, definitions and notation that were presented in the first part of the thesis may be recalled and used. We also provide an overview of some important results in vertex/edge antimagic total labeling of graphs and totally antimagic total labeling of graphs as well as face antimagic labeling of plane graphs.
6.1 Total Labeling
All graphs in this section are simple, finite, undirected and connected , unless stated otherwise. Here we provide terminology, notation and definitions that will be used throughout this part while terminology, definitions and notation specific only to a particular chapter will be defined therein.
A labeling of a graph G = (V,E) is a map from a set of graph elements to a set of integers (usually positive integers). If the domain is V (G), E(G) or V (G) E(G), ∪ then the labeling is called a vertex labeling, an edge labeling or a total labeling of G, respectively. More precisely, a total labeling of a graph G, a bijection f : V (G) ∪ E(G) 1, 2,..., V (G) + E(G) is a total labeling of G. Moreover, if the vertices → { | | | |} are labeled with the smallest possible numbers, that is, f(V ) = 1, 2,..., V (G) , { | |} then the total labeling is called super.
86 Chapter 6. Basic Concepts and Literature Review
For an edge labeling f, the weight of a vertex v is defined by
X wtf (v) = f(uv), u∈N(v) where N(v) is a set of neighbors of v.
For a total labeling f, the weight of a vertex v is defined by
X wtf (v) = f(v) + f(uv) u∈N(v) and the weight of an edge uv E(G) is defined by ∈
wtf (uv) = f(uv) + f(u) + f(v).
Informally, for a total labeling the vertex weight of a vertex is the sum of the label of the vertex and the labels of all the edges incident with the that vertex, while the edge weight of an edge is the sum of the label of the edge and the labels of the vertices incident with that edge.
A labeling f is a called vertex antimagic total (edge antimagic total), abbreviated to VAT (EAT), if all vertex weights (edge weights) are pairwise distinct. A graph that admits VAT (EAT) labeling is called a VAT (EAT) graph. If the vertex weights (edge weights) are all the same then the total labeling is called vertex magic total (edge magic total), abbreviated to VMT labeling (EMT labeling). The concept of VMT and super VMT graph was introduced by MacDougall, Miller, Slamin and Wallis in [57]. There are several results known for regular VMT graphs. A VMT labeling for Kn, for odd n, can be found in [55, 57, 59], and for Kn, with n even, in [43, 47]. A construction for VMT labeling of complete bipartite graphs Km,m is presented in [57]. In [46], it is completely determined which complete bipartite graphs have VMT labelings. The constructions of VMT labelings of certain regular graphs are given in [44,45,53].
In 1990, Hartsfield and Ringel [48] introduced the concept of an antimagic labeling of a graph, that is, in our terminology, a vertex antimagic edge labeling. According to Hartsfield and Ringel, an antimagic labeling of a graph G is an edge labeling, where
87 Chapter 6. Basic Concepts and Literature Review all the vertex weights are required to be pairwise distinct. For an edge labeling, a vertex weight is the sum of the labels of all edges incident with the vertex. Hartsfield and Ringel [48] conjectured that every tree except K2 admits a vertex antimagic edge (VAE) labeling and, moreover, every connected graph except K2 has a VAE labeling. Alon, Kaplan, Lev, Roditty and Yuster [3] showed that this conjecture is true for all graphs having minimum degree Ω(log V (G) ). | | If a VAE labeling satisfies the condition that the set of all the vertex weights is a, a + d, . . . , a + ( V (G) 1)d , where a > 0 and d 0 are two fixed integers, { | | − } ≥ then the labeling is called an (a, d)-VAE labeling. The (a, d)-VAE labeling was defined by Bodendiek and Walther [31] as (a, d)-antimagic labeling. Baˇca,Bertault, MacDougall, Miller, Simanjuntak and Slamin [10] introduced the concept of a (a, d)- VAT labeling.
Moreover, a VAT labeling (resp. (a, d)-VAT labeling) f is called repus if f(V ) = E(G) + 1, E(G) + 2,..., E(G) + V (G) . The basic properties of an (a, d)- {| | | | | | | |} VAT labeling and its relationships to other types of magic-type and antimagic-type labelings are investigated in [10]. In [81], it is shown how to construct super (a, d)- VAT labelings for certain families of graphs, including complete graphs, complete bipartite graphs, cycles, paths and generalized Petersen graphs.
The existence of super (a, d)-VAT labeling for disconnected graphs is examined in [2]. The existence of antimagic labelings for plane graphs is studied in [12].
The edge antimagic total labeling (EAT labeling) of a graph is a total labeling such that all edge weights are pairwise distinct. A graph is EAT if it has an EAT la- beling. The notion of (a, d)-edge antimagic total labeling ((a, d)-EAT labeling) was introduced by Simanjuntak, Bertault and Miller [76] as a natural extension of magic valuation defined by Kotzig and Rosa. Magic valuation is also known as edge magic labeling. Kotzig and Rosa [51] showed that all caterpillars have magic valuations and conjectured that all trees have magic valuations. In [76] Simanjuntak, Bertault and Miller gave constructions of (a, d)-EAT labelings for cycles and paths. Baˇca, Lin, Miller and Simanjuntak [19] presented some relationships between (a, d)-EAT labeling and other labelings, namely, edge magic vertex labeling and edge magic total labeling. A total labeling of a graph G is called an (a, d)-EAT labeling of a graph G if the set of edge weights is W = a, a + d, . . . , a + ( V (G) 1)d , for some { | | − }
88 Chapter 6. Basic Concepts and Literature Review integers a > 0 and d 0. An EAT labeling (resp. (a, d)-EAT labeling) f is called ≥ super if f(V ) = 1, 2,..., V (G) and a EAT labeling (resp. (a, d)-EAT labeling) { | |} f is called repus if f(V ) = E(G) + 1, E(G) + 2,..., E(G) + V (G) . More {| | | | | | | |} results on super (a, d)-EAT labeling for families of graphs can find in [9,15,37,82].
It is natural to ask whether there exist total labelings for graphs that hold simulta- neously VAT and EAT properties. We call such a labeling a totally antimagic total (TAT ). We will construct TAT labelings for some families of graphs in the next chapter.
For further results on graph labelings see [25,42,84].
6.2 d-Antimagic Labeling
Let G = (V,E,F ) be a finite, connected, plane graph without loops and multi- ple edges, where V (G), E(G) and F (G) are its vertex set, edge set and face set, respectively.
A labeling of a graph is any mapping that sends some set of graph elements to a set of numbers or colors. Graph labelings provide valuable information used in several application areas, see [42]. It is interesting to consider labeling the elements of a graph by the elements of a finite field. Properties of finite fields make it possible to consider combinatorial conditions for the labeling (see, for example, [74,75,83]).
A labeling of type (1, 1, 0), is a bijection from the set V (G) E(G) of G to the set ∪ 1, 2,..., V (G) + E(G) . This labeling is also called a total labeling or a labeling { | | | |} of type (1, 1, 0). If we label only vertices (resp. edges) we call such a labeling a vertex (resp. edge) labeling; alternatively, the labeling is said to be of type (1, 0, 0) (resp. type(0, 1, 0)).
A labeling of type (1, 1, 1) is a bijection from the set V (G) E(G) F (G) of G to ∪ ∪ the set 1, 2,..., V (G) + E(G) + F (G) . { | | | | | |} The weight of a face under a labeling is the sum of the labels (if present) carried by that face and the edges and vertices surrounding it.
89 Chapter 6. Basic Concepts and Literature Review
A labeling of a plane graph G is called d-antimagic, if for every number s, the set of s-sided face weights is Ws= as, as + d, as + 2d, . . . , as + (fs 1)d for some integers { − } as and d, d 0, where fs is the number of the s-sided faces. We allow different ≥ sets Ws for different s.
If d = 0 then Lih [54] calls a 0-antimagic labeling magic. This notion of magicality is different from the definition given by Sedl´aˇcekin [73] and also from the definition given by Kotzig and Rosa in [51]. However, a magic edge labeling of a plane graph, in our sense, is equal to a supermagic labeling of the plane dual graph G∗ of G as defined, for instance, in [79].
Lih [54] describes magic (0-antimagic) labelings of type (1, 1, 0) for wheels, friend- ship graphs and prisms. The 0-antimagic labelings of type (1, 1, 1) for grid graphs and honeycomb graphs are given in [4,5], respectively.
The concept of d-antimagic labeling of plane graphs was defined in [24]. The d- antimagic labelings of type (1, 1, 1) for the hexagonal planar maps, generalized Petersen graph P (n, 2) and grids can be found in [8, 14, 17], respectively. Lin et al. in [56] showed that the prism Dn, n 3, admits d-antimagic labelings of type ≥ (1, 1, 1) for d 2, 4, 5, 6 . The d-antimagic labelings of type (1, 1, 1) for Dn and ∈ { } for several d 7 are described in [80]. ≥ A d-antimagic labeling is called super if the smallest possible labels appear on the vertices. The super d-antimagic labelings of type (1, 1, 1) for antiprisms and for d 0, 1, 2, 3, 4, 5, 6 are described in [7], and for disjoint union of prisms for ∈ { } d 0, 1, 2, 3, 4, 5 are given in [1] and for d 6, 7 are given in [6]. ∈ { } ∈ { } In following chapter we mainly investigate the existence of super d-antimagic label- ings of type (1, 1, 1) for disconnected plane graphs. We concentrate on the following problem: If a graph G admits a (super) d-antimagic labeling of type (1, 1, 1), does the disjoint union of m copies of the graph G, denoted by mG, admit a (super) d-antimagic labeling of type (1, 1, 1) as well?
The same problem was studied for strong super edge magic labelings in [22].
90 Chapter 7
Total and Totally Antimagic Labeling of Graphs
In Section 7.1 we deal with VAT and EAT labelings. We prove that all graphs have VAT labelings. We also prove that all graphs have super (resp. repus) VAT labelings. Moreover, we provide super (resp. repus) (a, d)-VAT labelings for some families of graphs. Next we prove that all graphs with no isolated vertex have EAT labelings. Furthermore, we prove that such graphs have super (resp. repus) EAT laabelings. (a, d)-EAT labelings are constructed for some classes of graphs
In Section 7.2 we consider as yet unstudied cases of total labelings hold simul- taneously VAT and EAT for graphs. We call such a labeling a totally antimagic total labeling (or TAT labeling). Some families of graphs are proved to be totally antimagic total.
7.1 Total Labeling of Graphs
7.1.1 Vertex Antimagic Total Labeling
Hereafter, we denote by se(v) the sum of the labels of all edges incident with vertex v, s(e) the sum of the labels of the two vertices of the edge e and [n] the set of positive integers from 1 to n.
In Figure 7.1 we present a vertex antimagic total labeling of a graph G. Such a labeling exists for any graph, as we shall prove in the next theorem.
Theorem 7.1 All graphs G = (V,E) have VAT labelings. ♦
91 Chapter 7. Total and Totally Antimagic Labeling of Graphs
124 16
1 9 2 11 8 14
3 6 5 10
137 15 G
Figure 7.1: A VAT labeling for G.
Proof. Assume G = (V,E) has p vertices and q edges. We first label the edges of G with any q integers in [p+q]. Without loss of generality, assume that se(vi) se(vj), ≤ for 1 i < j p. We next label vertices vi with r and vj with s, where r < s and ≤ ≤ r, s [p + q] but not the labels of the edges. Under this total labeling, it is clear ∈ that wt(vi) < wt(vj), for 1 i < j p. ≤ ≤ In the next theorem we incorporate some constraints in the labeling of edges and vertices.
Theorem 7.2 For every graph G = (V,E), there exist VAT labelings which are ♦ (i) super; (ii) repus; (iii) neither super nor repus.
Proof. Assume G = (V,E) has p vertices and q edges. Let V = V1 V2 and ∪ V1 V2 = , where V1 = p1 and V2 = p2. We first label the edges of G with ∩ ∅ | | | | the labels p1 + 1, p1 + 2, . . . , p1 + q . Without loss of generality, assume that { } se(vi) se(vj), for 1 i < j p. We next label vertex vi with i for 1 i p1 ≤ ≤ ≤ ≤ ≤ and i + q for p1 + 1 i p1 + p2 = p. Under this total labeling, it is clear that ≤ ≤ wt(vi) < wt(vj), for 1 i < j p. Hence G is VAT but neither super nor repus, ≤ ≤ for 0 < p1 < p. If p1 = p, then the set of vertex labels is 1, 2, . . . , p . Therefore, { } G is super VAT. If p1 = 0, then the set of vertex labels is q + 1, q + 2 . . . , q + p , { } Therefore G is repus VAT.
92 Chapter 7. Total and Totally Antimagic Labeling of Graphs
Before presenting our next results we need some definitions. A VAT labeling of G(V,E) is called super even (resp. super odd) if the labels of vertices of the graph are the first p consecutive even positive integers (resp. the first p consecutive odd positive integers) of [ V + E ]. Similarly, a VAT labeling is called repus even (resp. | | | | repus odd) if the labels of vertices of the graph are the last p consecutive even positive integers (resp. the last p consecutive odd positive integers) of [ V + E ]. | | | |
Theorem 7.3 Let G = (V,E) be any graph with V E . Then G has a ♦ | | ≤ | | super-even (resp. a super-odd) VAT labeling.
Proof. Let V = p and E = q. Assume p < q. We first label the edges of G | | | | with the labels 1, 3,..., 2p 1, 2p + 1, 2p + 2, . . . , p + q (resp. 2, 4,..., 2p, 2p + { − } { 1, 2p + 2, . . . , p + q ). Without loss of generality, assume that se(vi) se(vj), for } ≤ 1 i < j p. We next label vertex vi with 2i, 1 i p (resp. with 2i 1, ≤ ≤ ≤ ≤ − 1 i p). Under this total labeling, we have wt(vi) < wt(vj), 1 i < j p. ≤ ≤ ≤ ≤ Similarly, when p = q, we label the edges of G with the labels 1, 3,..., 2p 1 { − } (resp. 2, 4,..., 2p ) and then the vertices of G with the labels 2, 4,..., 2p (resp. { } { } 1, 3,..., 2p 1 ). { − }
Corollary 7.1 Let G = (V,E) be any graph with V E . Then G has a repus ♦ | | ≤ | | even (resp. a repus odd) VAT labeling.
Proof. Let V = p and E = q. We consider two cases. | | | | Case 1: p + q is even
We first label the edges of G with the labels 1, 2, . . . , q p+1, q p+3, . . . , p+q 1 { − − − } (resp. 1, 2, . . . , q p, q p + 2, q p + 4, . . . , p + q ). Without loss of generality, { − − − } assume that se(vi) se(vj), for 1 i < j p. We next label vertex vi with ≤ ≤ ≤ q p + 2i (resp. q p + 2i 1), for 1 i p. Under this total labeling, it is clear − − − ≤ ≤ that wt(vi) < wt(vj), 1 i < j p. ≤ ≤ Case 2: p + q is odd
Similarly, we label the edges of G with the labels 1, 2, . . . , q p, q p+2, . . . , p+q { − − } (resp. 1, 2, . . . , q p + 1, q p + 3, . . . , p + q 1 ) and then the labels of vertices { − − − } of G with q p + 2i 1 (resp. q p + 2i), for 1 i p. − − − ≤ ≤ 93 Chapter 7. Total and Totally Antimagic Labeling of Graphs
When we swap the roles of p and q in Theorem 7.3 and Corollary 7.1, the following theorem and corollary follow.
Theorem 7.4 Let G = (V,E) be any graph with V E . Then G has a ♦ | | ≥ | | VAT labeling with the first E even positive integers (resp. the first E odd positive | | | | integers) of [ V + E ] being the labels of the edges of E. | | | |
Corollary 7.2 Let G = (V,E) be any graph with V E . Then G has a ♦ | | ≥ | | VAT labeling with the last E even positive integers (resp. the last E odd positive | | | | integers) of [ V + E ] being the labels of the edge of E. | | | |
We refer to the definition of magic edge labeling according to Sedl´aˇcek [73], where the labels can be any unique positive integer and the vertex weights are all the same; such the vertex weight is called magic constant. A magic labeling of a graph is called super magic if the edge labels are consecutive positive integers, starting from some particular integer. This should not be confused with super magic from total labelings.
Theorem 7.5 Let G = (V,E) be any k-regular graph with V = p and E = q. ♦ | | | | Then G has a super magic labeling with the magic constant c and the set of edge labels h, h + 1, . . . , q + h 1 if and only if G has a super (c + k(p h + 1) + 1, 1) { − } − VAT labeling.
Proof. Suppose G has a super magic labeling with the magic constant c and the set of edge labels h, h + 1, . . . , q + h 1 . We relabel the edges of G by adding { − } p h+1 to each of the original edge labels. Since G is k-regular then the magicness − property is preserved and the new magic constant is b = c + k(p h + 1). We can − choose any vertex of V to be vi and then label the vertex vi with i, 1 i p. ≤ ≤ Under this total labeling, we have wt(v) v V = b + 1, b + 2, . . . , b + p . { | ∈ } { } Suppose that G has a super (b + 1, 1) VAT labeling. We remove the vertex labels from G. Then we have se(vi) = b, for 1 i p. Since G is k-regular, then it is ≤ ≤ simple to see that the resulting labeling of G is super magic with the magic constant c and the set of edge labels h, h + 1, . . . , q + h 1 . { − }
94 Chapter 7. Total and Totally Antimagic Labeling of Graphs
Theorem 7.6 Let G = (V,E) be any k-regular graph with V = p and E = q. ♦ | | | | Then G has a super magic labeling with magic constant c and the set of edges is h, h + 1, . . . , q + h 1 if and only if G has a repus (c k(h 1) + q + 1, 1) VAT { − } − − labeling.
Proof. Suppose G is super magic with the magic constant c and the set edge labels h, h + 1, . . . , q + h 1 . We relabel the edges of G by subtracting h 1 from { − } − each of the original edge labels. Since G is k-regular then the magicness property is preserved and the new magic constant is b = c k(h 1). We can choose any − − vertex of V to be vi and then label the vertex vi with q + i, 1 i p. Under this ≤ ≤ total labeling, we have wt(v) v V = b + q + 1, b + q + 2, . . . , b + q + p . { | ∈ } { } Suppose that G is a repus (b + q + 1, 1) VAT. We remove the vertex labels from
G. Then we have se(vi) = b, for 1 i p. Since G is k-regular, then it is ≤ ≤ simple that G is super-magic with the magic constant c and the set of edge labels h, h + 1, . . . , q + h 1 . { − } The proof of the following theorem is similar to the proof of Theorem 7.6, so is omitted here.
Theorem 7.7 Let G = (V,E) be any graph with V = p and E = q. Then ♦ | | | | G has a super magic labeling with the magic constant c and the set of edge labels 1, 2, . . . , q if and only if G has a repus (c + q + 1, 1) VAT labeling. { }
Our proof of the “if part” of the following theorem is an alternative proof to the one given in [10].
Theorem 7.8 Let G = (V,E) be a graph with V = p and E = q. Then G ♦ | | | | has an (c, d) vertex antimagic labeling if and only if G has a repus (c + q + 1, d + 1) VAT labeling.
Proof. Assume that G has an (c, d) vertex antimagic labeling. Without loss of generality, let se(vi) = c + (i 1)d, for 1 i p. We next label vertex vi with − ≤ ≤ q + i, for 1 i p. The resulting labeling is a repus (c + q + 1, d + 1)-VAT labeling. ≤ ≤ The converse is straight forward by deleting the labels of vertices.
95 Chapter 7. Total and Totally Antimagic Labeling of Graphs
7.1.2 Edge Antimagic Total Labeling
From now on we assume that all graphs have no isolated vertices and we deal with EAT labelings of such graphs. There is no constraint in choosing the labels of vertices of a graph in the construction of the following theorem; antimagicness can be achieved in such a way that no pair of edges has the same edge weight.
In Figure 7.2 we present an example of an edge antimagic total labeling for a graph G. In fact, every graph admits an edge antimagic total labeling.
54 7
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6 8 12 15
914 16 G
Figure 7.2: An EAT labeling for G.
We prove the following theorem.
Theorem 7.9 All graphs G = (V,E) have EAT labelings. ♦
Proof. Assume G = (V,E) has p vertices and q edges. We first label the vertices of
G with any p integers in [p+q]. Without loss of generality, assume that s(ei) s(ej), ≤ for 1 i < j q. We next label the edges ei with r and ej with s, where r < s ≤ ≤ and r, s [p + q] but not the labels of the vertices. Under this total labeling, it is ∈ clear that wt(ei) < wt(ej), for 1 i < j q. ≤ ≤ When some constraints are added in the labeling of vertices and edges then we have the next theorem.
Theorem 7.10 For every graph G = (V,E), there exist EAT labelings which are ♦
96 Chapter 7. Total and Totally Antimagic Labeling of Graphs
(i) super; (ii) repus; (iii) neither super nor repus.
Proof. Assume G = (V,E) has p vertices and q edges.
(i) We first label the vertices of G with the first p integers of [p + q]. Without
loss of generality, assume that s(ei) s(ej), for 1 i < j q. We next label ≤ ≤ ≤ the edges ei with r and ej with s, where r < s and r, s / [p]. Under this total ∈ labeling, it is clear that wt(ei) < wt(ej), for 1 i < j q. ≤ ≤ (ii) As in (i), we first label the vertices of G with the last p integers of [p + q] and then label the edges of G with the rest of integers that are not used for labeling the vertices. (iii) This is also similar to (i), but neither the first nor last p integers in [p + q] are chosen for the vertex labels. The edges of G are then labeled by the rest of integers in [p + q].
Super even (resp. super odd) and repus even (resp. repus odd) for EAT labeling are defined as for VAT labeling.
Theorem 7.11 Let G = (V,E) be any graph with V E . Then G has a ♦ | | ≤ | | super even (resp. a super odd) EAT labeling.
Proof. Let V = p and E = q. Assume p < q. We first label the vertices of G | | | | with the labels 2, 4,..., 2p (resp. 1, 3,..., 2p 1 ). Without loss of generality, { } { − } assume that s(ei) s(ej), for 1 i < j q. We next label edge ei with 2i 1, for ≤ ≤ ≤ − 1 i p and p + i, for p + 1 i q (resp. with 2i, for 1 i p and p + i, for ≤ ≤ ≤ ≤ ≤ ≤ p + 1 i q). Under this total labeling, we have wt(ei) < wt(ej), 1 i < j p. ≤ ≤ ≤ ≤ Similarly, when p = q, we label the vertices of G with the labels 2, 4,..., 2p (resp. { } 1, 3,..., 2p 1 ) and then the edges of G with the labels 1, 3,..., 2p 1 (resp. { − } { − } 2, 4,..., 2p ) { } It is not hard to prove the following corollary, so we omit its proof here.
Corollary 7.3 Let G = (V,E) be any graph with V E . Then G has a repus ♦ | | ≤ | | even (resp. a repus odd) EAT labeling.
97 Chapter 7. Total and Totally Antimagic Labeling of Graphs
The proof of the next theorem is similar to the proof of Theorem 7.11; use the first E even integers (resp. the first E odd integers) in [p + q] for labeling the edges. | | | |
Theorem 7.12 Let G = (V,E) be any graph with V E . Then G has an ♦ | | ≥ | | EAT labeling with the first E even integers (resp. the first E odd integers) of | | | | [ V + E ] being the labels of the edges of E. | | | |
Similarly, we use the last E even integers (rep. the last E odd integers) in [p + q] | | | | for labeling the edges.
Corollary 7.4 Let G = (V,E) be any graph with V E . Then G has an ♦ | | ≥ | | EAT labeling with the last E even positive integers (resp. the last E odd positive | | | | integers) of [ V + E ] being the labels of the edge of E. | | | |
For the rest of this section we deal with (a, d)-EAT labeling. The path Pn, n 2, ≥ has been proved to be a super (n + 4, 3)-EAT in [19] and the star Sn, n 1, is a ≥ super (a, 2)-EAT, for a = 2n + 4, n + 5 in [82]. We present alternative proofs for these graphs.
Theorem 7.13 (i) The path Pn, n 2, has a super (n + 4, 3)-EAT labeling. ≥ (ii) The star Sn, n 1, has a super (a, 2)-EAT labeling, for a = 2n + 4, n + 5. ≥
Proof.
(i) We first label the vertex vi, 1 i n, of Pn with i and next label the edge ≤ ≤ vivi+1 with n+i. We have wt(vivi+1) = n+3i+1 and wt(vi+1vi+2) = n+3i+4. (ii) Let c and vi, 1 i n be the centre and the other vertices of Sn, respectively. ≤ ≤ For a = 2n + 4, we first label the vertices vi, 1 i n, with i and the center ≤ ≤ with n + 1. We next label the edges cvi, 1 i n, with n + i + 1. The rest ≤ ≤ of proof is not hard to verify.
For a = n + 5, we label the centre with 1 and the vertices vi, 1 i n, with ≤ ≤ i + 1. We next label cvi, 1 i n, with n + 1 + i. It is easy to verify that ≤ ≤ wt(cvi) = (n + 5) + 2(i 1). −
Theorem 7.14 (i) The path Pn, n 2, has a super (or a repus) odd (n, 6)- ♦ ≥ EAT labeling.
98 Chapter 7. Total and Totally Antimagic Labeling of Graphs
(ii) The path Pn, n 2, has a repus (2n + 2, 3)-EAT labeling. ≥ (iii) The star Sn, n 1, has a repus (a, 2)-EAT labeling, for a = 2n + 4, 3n + 3. ≥
Proof.
(i) We first label the vertex vi, 1 i n, of Pn with 2i 1 and we next label ≤ ≤ − the edge vivi+1 with 2i. It is clear that this labeling is both super and repus odd (n, 6)-EAT.
(ii) We first label the vertex vi, 1 i n, of Pn with n + i 1 and we next ≤ ≤ − label the edge vivi+1 with i. It is not hard to check that this labeling is repus (2n + 2, 3)-EAT.
(iii) Let c and vi, 1 i n be the centre and the other vertices of Sn, respectively. ≤ ≤ For a = 2n + 4, we label the centre with n + 1, the vertices vi, 1 i n, with ≤ ≤ n + 1 + i and the edges cvi with i. Then we have wt(cvi) = (2n + 4) + 2(i 1). − For a = 3n+3, we label the centre with 2n+1, the vertices vi, 1 i n, with ≤ ≤ i and the edges cvi with n + i. Then we have wt(cvi) = (3n + 3) + 2(i 1). −
7.2 Totally Antimagic Total Labeling of Graphs
In Section 7.1, all graphs have been proved to be (super) VAT and (super) EAT. Naturally, we can ask whether there exists a graph possessing a labeling that is simultaneously VAT and EAT. We will call such a labeling a totally antimagic total labeling (TAT labeling) and a graph that admits such a labeling a totally antimagic total graph (TAT graph). As is usual, if, moreover, the vertices are labeled with the smallest possible numbers then the labeling is called super. The definition of totally antimagic total labeling is a natural extension of the concept of totally magic labeling defined by Exoo, Ling, McSorley, Phillips and Wallis in [39]. They showed that such graphs appear to be rare. They proved that the only connected totally magic graph containing a vertex of degree 1 is P3, the only totally magic trees are K1 and P3, the only totally magic cycle is K3, the only totally magic complete graphs are K1 and K3, and the only totally magic complete bipartite graph is K1,2.
We present W4 with VAT but not EAT labeling in Figure 7.3. Figure 7.4 illustrates a labeling for W4 that is EAT but without being VAT. The graph W4 with TAT labeling is shown in Figure 7.5.
99 Chapter 7. Total and Totally Antimagic Labeling of Graphs
9 1 3 2 6 12 11 4 10 5 8 7 13
Figure 7.3: A VAT but not an EAT labeling for W4.
3 12 10 2 9 1 6 8 5 7 13 11 4
Figure 7.4: An EAT but not a VAT labeling for W4.
1 7 5 9 12 13 4 10 2 11 8 6 3
Figure 7.5: A TAT labeling for W4.
7.2.1 Join Graph
We say that a labeling g is ordered (sharp ordered) if wtg(u) wtg(v)(wtg(u) < ≤ wtg(v)) holds for every pair of vertices u, v G such that g(u) < g(v). A graph ∈ that admits a (sharp) ordered labeling is called a (sharp) ordered graph.
100 Chapter 7. Total and Totally Antimagic Labeling of Graphs
Recall the definition of the join from Chapter 2. The join G+H of the graphs G and H is the graph with V (G+H) = V (G) V (H) and E(G+H) = E(G) E(H) uv : ∪ ∪ ∪{ u V (G) and v V (H) . In this section we will deal with a totally antimagic ∈ ∈ } total labeling of G + K1. We prove
Theorem 7.15 Let G be an ordered super EAT graph. Then G + K1 is a TAT ♦ graph.
Proof. Let g be an ordered super EAT labeling of G. As g is super, we can denote the vertices of G by the symbols v1, v2, . . . , v|V (G)| such that
g(vi) = i, for i = 1, 2,..., V (G) . | |
Since g is ordered, for i = 1, 2,..., V (G) 1, we have | | −
wtg(vi) wtg(vi+1). ≤
By the symbol u we denote the vertex of G + K1 not belonging to G.
We define a new labeling f of G + K1 such that
f(x) = g(x), x V (G) E(G) ∈ ∪ f(u) = 2 V (G) + E(G) + 1 | | | | f(viu) = V (G) + E(G) + i, i = 1, 2,..., V (G) . | | | | | |
It is easy to see that f is a bijection from V (G + K1) E(G + K1) to the set ∪ 1, 2,..., 2 V (G) + E(G) + 1 . { | | | | }
101 Chapter 7. Total and Totally Antimagic Labeling of Graphs
For the vertex weights under the labeling f we have
|V (G)| X wtf (u) =f(u) + f(uvi) i=1 |V (G)| X = 2 V (G) + E(G) + 1 + V (G) + E(G) + i | | | | | | | | i=1 |V (G)|+1 X = V (G) + E(G) + i | | | | i=1 V (G) + 1 3 V (G) + 2 E(G) + 2 = | | | | | | . 2
For i = 1, 2,..., V (G) we get | | X wtf (vi) =f(vi) + f(viv) + f(viu)
v∈NG(vi) X =g(vi) + g(viv) + V (G) + E(G) + i | | | | v∈NG(vi)
=wtg(vi) + V (G) + E(G) + i | | | | wtg(vi+1) + V (G) + E(G) + i ≤ | | | | Moreover, as g is a super EAT labeling of G, we get X wtf (v ) =g(v ) + g(v v) + 2 V (G) + E(G) |V (G)| |V (G)| |V (G)| | | | | v∈NG(v V (G) ) | | |V (G)|−1 X V (G) + V (G) + E(G) + 1 j + 2 V (G) + E(G) ≤| | | | | | − | | | | j=1 V (G) 1 V (G) + 2 E(G) + 2 =3 V (G) + E(G) + | | − | | | | | | | | 2 Thus, the vertex weights are all different. 102 Chapter 7. Total and Totally Antimagic Labeling of Graphs The edge weights of the edges in E(G) under the labeling f are all different as g is an EAT labeling of G. More precisely, we have wtf (e) = wtg(e), for every e E(G). ∈ Moreover, as g is super, for the upper bound on the maximum edge weight of e E(G) under the labeling f, we have ∈ wtmax(e) = wtmax(e) V (G) + V (G) 1 + V (G) + E(G) f g ≤ | | | | − | | | | = 3 V (G) + E(G) 1. | | | | − For i = 1, 2,..., V (G) , we get | | wtf (uvi) =f(u) + f(uvi) + f(vi) = 2 V (G) + E(G) + 1 + V (G) + E(G) + i + g(vi) | | | | | | | | =3 V (G) + 2 E(G) + 1 + 2i > 3 V (G) + E(G) 1 wtmax(e), | | | | | | | | − ≥ f where e E(G). ∈ It is easy to see that the edge weights are also all different. Thus, f is a TAT labeling of G + K1. Moreover, if G is a regular graph then there exists a special TAT labeling of G+K1. Theorem 7.16 Let G be a regular ordered super EAT graph. Then G + K1 is ♦ a sharp ordered super TAT graph. Proof. Let G be an r-regular graph. Let g be an ordered super EAT labeling of G. As in the proof of the previous theorem, we denote the vertices of G by the symbols v1, v2, . . . , v|V (G)| such that g(vi) = i, for i = 1, 2,..., V (G) | | and by the symbol u we denote the vertex of G+K1 not belonging to G. According to the assumptions, wtg(vi) wtg(vi+1), ≤ 103 Chapter 7. Total and Totally Antimagic Labeling of Graphs for i = 1, 2,..., V (G) 1. | | − We define a new labeling f of G + K1 such that f(vi) = i, i = 1, 2,..., V (G) , | | f(u) = V (G) + 1, | | f(e) = g(e) + 1, e E(G) ∈ V (G) r f(viu) = V (G) + | | + 1 + i, i = 1, 2,..., V (G) . | | 2 | | We prove that f is a super TAT labeling of G + K1. For the vertex weight, |V (G)| X wtf (u) =f(u) + f(uvi) i=1 |V (G)| X V (G) r = V (G) + 1 + V (G) + | | + 1 + i | | | | 2 i=1 V (G) r V (G) V (G) 2 = V (G) V (G) + | | + 2 + | | + | | + 1. | | | | 2 2 2 104 Chapter 7. Total and Totally Antimagic Labeling of Graphs For i = 1, 2,..., V (G) , we get | | V (G) r wtf (vi) = V (G) + | | + r + 1 + wtg(vi) + i | | 2 V (G) r wtf (v ) = 2 V (G) + | | + r + 1 + wtg(v ) ≤ |V (G)| | | 2 |V (G)| V (G) r X =2 V (G) + | | + r + 1 + g(v ) + g(viv) | | 2 |V (G)| v∈NG(vi) r V (G) r X V (G) r 3 V (G) + | | + r + 1 + V (G) + | | + 2 j ≤ | | 2 | | 2 − j=1 V (G) r r r2 =(r + 1) V (G) + | | + 2 + 2 V (G) + 1 | | 2 | | 2 − 2 − V (G) r V (G) 1 V (G) V (G) + | | + 2 + 2 V (G) + | | − 1 ≤| | | | 2 | | 2 − V (G) r 5 V (G) 3 = V (G) V (G) + | | + 2 + | | | | | | 2 2 − 2 This concludes the proof. Let G H denote the disjoint union of graphs G and H. Let mG denote the disjoint ∪ union of m copies of graph G. In the following lemmas we prove that certain families of graphs are totally antimagic total. These include totally disconnected graphs on m vertices, m copies of K2, paths, and cycles. Furthermore, the totally antimagic total labelings of these graphs have useful extra properties. Observation 7.1 For every positive integer m the graph mK1 is sharp ordered ♦ super EAT. Lemma 7.1 For every positive integer m the graph mK2 is sharp ordered super ♦ EAT. Proof. We denote the vertices of mK2 by the symbols vi, i = 1, 2,..., 2m, such that its edge set is E(mK2) = v1v2, v3v4, . . . , v2m−1v2m . { } 105 Chapter 7. Total and Totally Antimagic Labeling of Graphs Let us consider the labeling g of mK2 defined by g(vi) = i, i = 1, 2,..., 2m i+1 g(vivi+1) = 2m + , i = 1, 3,..., 2m 1. 2 − It is easy to see that g is a super EAT labeling of mK2, as the weight of the edge vivi+1, i = 1, 3,..., 2m 1, is − i+1 wtg(vivi+1) = g(vi) + g(vivi+1) + g(vi+1) = (i) + 2m + 2 + (i + 1) 5i+3 = 2m + 2 . For the vertex weights we get 3i+1 2m + 2 , i = 1, 3,..., 2m 1 wtg(vi) = − 3i 2m + 2 , i = 2, 4,..., 2m. This concludes the proof. Lemma 7.2 The graph Pn, n 2, is sharp ordered super EAT. ♦ ≥ Proof. We denote the vertices of Pn by the symbols vi, i = 1, 2, . . . , n, such that E(Pn) = v1v2, v2v3, . . . , vn−1vn . { } It is easy to check that the labeling g, g : V (Pn) E(Pn) 1, 2,..., 2n 1 ∪ → { − } satisfies the above mentioned condition, when n 2i 1, i = 1, 2,..., 2 g(vi) = − 2n + 2 2i, i = n + 1, n + 2, . . . , n − 2 2 n n 1 + 2i, i = 1, 2,..., 2 g(vivi+1) = − 3n 2i, i = n + 1, n + 2, . . . , n 1. − 2 2 − Lemma 7.3 The graph Cn, n 3, is sharp ordered super EAT. ♦ ≥ 106 Chapter 7. Total and Totally Antimagic Labeling of Graphs Proof. We denote the vertices of Cn by the symbols vi, i = 1, 2, . . . , n, such that the edge set of Cn is E(Cn) = v1v2, v2v3, . . . , vnv1 . { } Let us consider the labeling g : V (Cn) E(Cn) 1, 2,..., 2n , defined as follows. ∪ → { } For n even, 1, i = 1 n g(vi) = 2i 2, i = 2, 3,..., 2 + 1 − 2n + 3 2i, i = n + 2, n + 3, . . . , n. − 2 2 For n odd, 1, i = 1 n g(vi) = 2i 2, i = 2, 3,..., 2 − 2n + 3 2i, i = n + 1, n + 2, . . . , n. − 2 2 n n 1 + 2i, i = 1, 2,..., 2 g(vivi+1) = − 3n + 2 2i, i = n + 1, n + 2, . . . , n 1, − 2 2 − g(vnv1) = n + 2. It is a simple mathematical exercise to check that g is a super EAT labeling of Cn with the desired properties. As the join of the complete graphs Kn and K1 is again a complete graph (Kn+1), by Lemma 7.1 and by repeated use of Theorem 7.16, we obtain that the complete graph is totally antimagic total. Corollary 7.5 The complete graph Kn is TAT for n 1. ♦ ≥ 107 Chapter 7. Total and Totally Antimagic Labeling of Graphs Recall the definition from Chapter 2. We can see the wheel with n spokes is iso- morphic to the graph Cn + K1. The following result follows immediately from Lemma 7.3 and Theorem 7.15. Corollary 7.6 The wheel Wn is TAT for n 3. ♦ ≥ The friendship graph Fn is a graph isomorphic to (nK2) + K1. Alternatively, the friendship graph Fn can be obtained from the wheel W2n by removing every second rim edge. Corollary 7.7 The friendship graph Fn is TAT for n 1. ♦ ≥ Proof. The result follows from Lemma 7.1 and Theorem 7.15. If one rim edge is removed from Wn, the resulting graph is called a fan, denoted by Fn. Alternatively, a fan Fn is isomorphic to the graph Pn + K1. From Lemma 7.2 and Theorem 7.15, we get Corollary 7.8 The fan Fn is TAT for n 2. ♦ ≥ Corollary 7.9 The star Sn is TAT for n 1. ♦ ≥ Proof. The star Sn is isomorphic to (nK1) + K1. The result follows from Lemma 7.1 and Theorem 7.15. 7.2.2 Corona Graphs Recall the definition from Chapter 2. If G has order m, the corona of G with a graph H, denoted by G H, is the graph obtained by taking one copy of G and m copies of H and joining the i-th vertex of G with an edge to every vertex in the i-th copy of H. A cycle of order m with n pendant edges attached at each vertex, that is, Cm nK1, is called an n-crown with cycle of order m. Theorem 7.17 Let G be a regular ordered super EAT graph. Then the graph ♦ G nK1 is TAT for every n 1. ≥ 108 Chapter 7. Total and Totally Antimagic Labeling of Graphs Proof. Let G be a regular graph. Let g be an ordered super EAT labeling of G. As g is super, we can denote the vertices of G by the symbols v1, v2, . . . , v|V (G)|, such that g(vi) = i, for i = 1, 2,..., V (G) . | | Since g is ordered, this means that wtg(vi) wtg(vi+1), for i = 1, 2,..., V (G) 1. ≤ | | − We denote the vertices of G nK1 not belonging to G by the symbols ui,j, i = 1, 2,..., V (G) , j = 1, 2, . . . , n, such that ui,jvi E(G nK1). | | ∈ We define a new labeling f of G nK1 in the following way. f(x) = g(x) + 2 V (G) n, x V (G) E(G) | | ∈ ∪ f(ui,j) = (i 1)n + j, i = 1, 2,..., V (G) , j = 1, 2, . . . , n − | | f(ui,jvi) = (i 1)n + j + n V (G) , i = 1, 2,..., V (G) , j = 1, 2, . . . , n. − | | | | It is easy to see that f is a bijection from the vertex set and the edge set of G nK1 to the set 1, 2,..., V (G) + E(G) + 2 V (G) n . For the weight of the edge vivk, { | | | | | | } i = 1, 2,..., V (G) , k = 1, 2,..., V (G) , i = k, under the labeling f, we have | | | | 6 wtf (vivk) = f(vi) + f(vivk) + f(vk) = g(vi) + 2 V (G) n + g(vivk) + 2 V (G) n + g(vk) + 2 V (G) n | | | | | | = wtg(vivk) + 6 V (G) n. | | As g is an EAT labeling, the weights of all the edges e E(G) are different under ∈ the labeling f as well. For the edge ui,jvi, i = 1, 2,..., V (G) , j = 1, 2, . . . , n, we obtain | | wtf (ui,jvi) = f(ui,j) + f(ui,jvi) + f(vi) = (i 1)n + j + (i 1)n + j + n V (G) + g(vi) + 2 V (G) n − − | | | | = 3 V (G) n 2n + i(2n + 1) + 2j. | | − 109 Chapter 7. Total and Totally Antimagic Labeling of Graphs This means that all the edges in G nK1 E(G) have different edge weights. More- \ over, wtf (ui,jvi) = 3 V (G) n 2n + i(2n + 1) + 2j | | − 3 V (G) n 2n + V (G) (2n + 1) + 2n = V (G) + 5 V (G) n ≤ | | − | | | | | | 6 V (G) n, ≤ | | for i = 1, 2,..., V (G) , j = 1, 2, . . . , n. Thus, for all the edges e1 and e2, e1 E(G) | | ∈ and e2 E(G nK1) E(G), ∈ \ wtf (e1) > wtf (e2). Thus the edge weights of all the edges in G nK1 are pairwise different. We have to check that also the vertex weights are different. For the vertex ui,j, i = 1, 2,..., V (G) , j = 1, 2, . . . , n, we get | | wtf (ui,j) = f(ui,j) + f(ui,jvi) = (i 1)n + j + (i 1)n + j + n V (G) − − | | = V (G) n 2n + 2ni + 2j. | | − Thus the vertex weights are all different numbers from the set V (G) n + 2, V (G) n + 4,..., 3 V (G) n . {| | | | | | } For the vertex vi, i = 1, 2,..., V (G) , we get | | 110 Chapter 7. Total and Totally Antimagic Labeling of Graphs n X X wtf (vi) = f(ui,jvi) + f(vi) + f(vivk) j=1 vivk∈E(G) n X = (i 1)n + j + n V (G) + g(vi) + 2 V (G) n − | | | | j=1 X + g(vivk) + 2 V (G) n | | vivk∈E(G) n X =n2(i 1) + j + n2 V (G) + 2 V (G) n − | | | | j=1 X + g(vi) + g(vivk) + 2 V (G) n deg (vi) | | · g vivk∈E(G) n(n + 1) 2 = + n V (G) + 2 V (G) n deg (vi) + 1 2 | | | | g 2 + wtg(vi) + n (i 1). − As G is a regular graph, we have deg (vi) = r, for i = 1, 2,..., V (G) . g | | Thus n(n + 1) 2 2 wtf (vi) = + n V (G) + 2 V (G) n r + 1 + wtg(vi) + n (i 1) 2 | | | | − >3 V (G) n. | | As wtg(vi) wtg(vi+1), for i = 1, 2,..., V (G) 1, this means that also the vertex ≤ | | − weights are all different. Immediately from Theorem 7.17 and Lemmas 7.1 and 7.3, we obtain the following corollaries. Corollary 7.10 The double-star Sn,n = K2 nK1 is TAT for every n 1. ♦ ∼ ≥ Corollary 7.11 The n-crown Cm nK1 is TAT for every n 1, m 3. ♦ ≥ ≥ 111 Chapter 7. Total and Totally Antimagic Labeling of Graphs Note that using Theorem 7.17 and Lemma 7.1, we obtain that the star Sn, Sn ∼= K1 nK1, is TAT for every n 1. ≥ In a similar manner as in the proof of Theorem 7.17 and using Lemma 7.2, we can also prove the following result. Corollary 7.12 The graph Pm nK1 is TAT for every n 1, m 2. ♦ ≥ ≥ 7.2.3 Union of Graphs In this section we will deal with disjoint union of regular TAT graphs. Theorem 7.18 The disjoint union of regular TAT graphs is a TAT graph. ♦ ri|V (Gi)| Proof. Let Gi be a ri-regular graph of order V (Gi) and size E(Gi) = , | | | | 2 i = 1, 2. Let gi, i = 1, 2, be a TAT labeling of Gi. Thus, gi : V (Gi) E(Gi) 1, 2,..., V (Gi) + E(Gi) ∪ → { | | | |} such that wtg (v) = wtg (u), i 6 i for all v, u V (Gi), u = v and ∈ 6 wtg (e) = wtg (h), i 6 i for all e, h E(Gi), e = h. ∈ 6 Without loss of generality we assume that r2 r1. We define a labeling f of G1 G2 ≥ ∪ such that g1(x), x V (G1) E(G1) f(x) = ∈ ∪ g2(x) + V (G1) + E(G1) , x V (G2) E(G2). | | | | ∈ ∪ It is easy to see that f is a total labeling of G1 G2. ∪ 112 Chapter 7. Total and Totally Antimagic Labeling of Graphs For the edge weights under the labeling f we obtain wtg (e), e E(G1) wt (e) = 1 ∈ f wtg (e) + 3 V (G1) + E(G1) , e E(G2). 2 | | | | ∈ As g1 and g2 are edge antimagic labelings, the edge weights of all edges in E(G1) and also in E(G2) under the labeling f are pairwise distinct. Moreover, the maximum weight of an edge e E(G1) is ∈ 3 max max X wt (e) =wt (e) V (G1) + E(G1) + 1 k f g1 ≤ | | | | − k=1 = 3 V (G1) + E(G1) 3. | | | | − Thus, f is an edge antimagic labeling of G1 G2. ∪ For the vertex weights under the labeling f, we get wtg (v), v V (G1) wt (v) = 1 ∈ f wtg (v) + r2 + 1 V (G1) + E(G1) , v V (G2). 2 | | | | ∈ As g1 and g2 are vertex antimagic labelings, the vertex weights of all vertices in V (G1) and also in V (G2) under the labeling f are pairwise distinct. Moreover, the maximum weight of a vertex v1 V (G1) is ∈ r1+1 max max X wt (v1) =wt (v1) V (G1) + E(G1) + 1 k f g1 ≤ | | | | − k=1 r1(r1 + 1) =(r1 + 1) V (G1) + E(G1) . | | | | − 2 As r2 r1, f is also a vertex antimagic labeling of G1 G2. ≥ ∪ Immediately, from the previous theorem, we obtain Corollary 7.13 If G is a regular TAT graph then mG is TAT for every m 1. ♦ ≥ 113 Chapter 7. Total and Totally Antimagic Labeling of Graphs 7.3 Conclusion Finding (a, d)-vertex antimagic and edge antimagic total labelings for some families of graphs are still interesting topics. For more details see [42] and also [25]. We were trying to find total labelings that are simultaneously vertex antimagic total and edge antimagic total. We showed the existence of such labelings for some classes of graphs, such as complete graphs, paths, cycles, stars, double stars, wheels, etc. We also proved that a union of regular totally antimagic total graphs is a totally antimagic total graph. For further investigation we conclude this chapter with the following open problems. Open Problem 7.1 Find other classes of totally antimagic total graphs. Open Problem 7.2 Find some necessary and/or sufficient conditions for a graph to be totally antimagic total. Open Problem 7.3 Characterize totally antimagic total graphs. Another interesting problem related to totally antimagic total labelings is to find total labeling that is simultaneously vertex magic and edge antimagic, or simulta- neously vertex antimagic and edge magic total. We state two open problems. Open Problem 7.4 Find total labeling of some classes of graphs that are simul- taneously vertex magic and edge antimagic. Open Problem 7.5 Find total labeling of some classes of graphs that are simul- taneously vertex antimagic and edge magic. Finding total labelings of graphs with the vertex weights and edge weights being all distinct are also interesting problems. Such a labeling is called a strong totally antimagic labeling. Open Problem 7.6 Characterize strong totally antimagic total graphs. 114 Chapter 8 Antimagic Labeling of Plane Graphs In this chapter we mainly examine the existence of super d-antimagic labelings of type (1, 1, 0) for several families of plane graphs and super d-antimagic labelings of type (1, 1, 1) for disconnected plane graphs. 8.1 Super d-Antimagic Labeling of Type (1, 1, 0) 8.1.1 Edge Antimagic Labeling of Paths Let Pn be the path on n vertices. It is known (see [21]) that Pn is super (a, d)-edge antimagic total if and only if d 3. We denote the vertices of Pn by v1, v2, . . . , vn n ≤ and describe these labelings α : V (Pn) E(Pn) 1, 2,..., 2n 1 in the following d ∪ → { − } way. Note that when d = 0 this labeling is equivalent to super edge magic total labeling. n n (a) The super (2n + + 1, 0)-edge antimagic total labeling α of Pn: d 2 e 0 i+1 n 2 , for i 1 (mod 2) and 1 i n, α0 (vi) = ≡ ≤ ≤ n + i , for i 0 (mod 2) and 2 i n, d 2 e 2 ≡ ≤ ≤ n α (vivi+1) = 2n i, for i = 1, 2, . . . , n 1. 0 − − The common weight for all edges of Pn is lnm n wαn (vivi+1) = 2n + + 1 = C , i = 1, 2, . . . , n 1. 0 2 α,0 − 115 Chapter 8. Antimagic Labeling of Plane Graphs n (b) The super (2n + 2, 1)-edge antimagic total labeling α1 of Pn: n α1 (vi) = i, for i = 1, 2, . . . , n, n α (vivi+1) = 2n i, for i = 1, 2, . . . , n 1. 1 − − The set of edge weights of Pn consists of the consecutive integers n wαn (vivi+1) = 2n + 1 + i = C + i : i = 1, 2, . . . , n 1 . { 1 α,1 − } n n (c) The super (n + + 3, 2)-edge antimagic total labeling α of Pn: d 2 e 2 i+1 n 2 , for i 1 (mod 2), α2 (vi) = ≡ n + i , for i 0 (mod 2), d 2 e 2 ≡ n α (vivi+1) = n + i, for i = 1, 2, . . . , n 1. 2 − The edge weights of Pn constitute the arithmetic progression of difference 2: lnm n wαn (vivi+1) = n + + 1 + 2i = C + 2i : i = 1, 2, . . . , n 1 . { 2 2 α,2 − } n (d) The super (n + 4, 3)-edge antimagic total labeling α3 of Pn: n α3 (vi) = i, for i = 1, 2, . . . , n, n α (vivi+1) = n + i, for i = 1, 2, . . . , n 1. 3 − The edge weights of Pn constitute the arithmetic progression of difference 3: n wαn (vivi+1) = n + 1 + 3i = C + 3i : i = 1, 2, . . . , n 1 . { 3 α,3 − } In this section we will use also (a, d)-edge antimagic vertex labelings of Pn for two n differences d = 1 and d = 2. These labelings β : V (Pn) 1, 2, . . . , n we define d → { } in the following way: 116 Chapter 8. Antimagic Labeling of Plane Graphs n n (e) The ( + 2, 1)-edge antimagic vertex labeling β of Pn: d 2 e 1 i+1 n 2 , for i 1 (mod 2), β1 (vi) = ≡ n + i , for i 0 (mod 2). d 2 e 2 ≡ The set of edge weights of Pn consists of the consecutive integers lnm n wβn (vivi+1) = + 1 + i = C + i : i = 1, 2, . . . , n 1 . { 1 2 β,1 − } n (f) The (3, 2)-edge antimagic vertex labeling β2 of Pn: n β2 (vi) = i, for i = 1, 2, . . . , n. The edge weights of Pn constitute the arithmetic progression of difference 2: n wβn (vivi+1) = 1 + 2i = C + 2i : i = 1, 2, . . . , n 1 . { 2 β,2 − } 8.1.2 Partitions with Determined Differences For construction of vertex and edge labelings of plane graphs we will use the parti- tions of a set of integers with determined differences. Let n, k, d and i be positive integers. We will consider the partition n of the Pk,d set 1, 2, . . . , kn into n, n 2, k-tuples such that the difference between the sum { } ≥ of the numbers in the (i + 1)th k-tuple and the sum of the numbers in the i-th k-tuple is always equal to the constant d, where i = 1, 2, . . . , n 1. Thus they form − an arithmetic sequence with the difference d. By the symbol k,d(i) we denote the P i-th k-tuple in the partition with the difference d, where i = 1, 2, . . . , n. Let P n (i) be the sum of the numbers in n (i). Evidently P n (i + 1) Pk,d Pk,d Pk,d − P n (i) = d. It is obvious that if there exists a partition of the set 1, 2, . . . , kn Pk,d { } with the difference d, there also exists a partition with the difference d. By the − notation n (i) c we mean that we add the constant c to every number in n (i). Pk,d ⊕ Pk,d If k = 1 then only the following partition of the set 1, 2, . . . , n is possible { } n (i) = i , for i = 1, 2, . . . , n. P1,1 { } 117 Chapter 8. Antimagic Labeling of Plane Graphs If k = 2 then we have several partitions of the set 1, 2,..., 2n . Let us define the { } partitions into 2-tuples in the following way: n (i) = i, 2n + 1 i , P2,0 { − } X n (i) = 2n + 1, for i = 1, 2, . . . , n; P2,0 n (i) = i, n + i , P2,2 { } X n (i) = n + 2i, for i = 1, 2, . . . , n; P2,2 n (i) = 2i 1, 2i , P2,4 { − } X n (i) = 4i 1, for i = 1, 2, . . . , n. P2,4 − Moreover, for 3 n 1 (mod 2) ≤ ≡ n+1 i−1 i−1 n 2 + 2 , n + 1 + 2 , for i 1 (mod 2); 2,1(i) = { } ≡ P i , n + n+1 + i , for i 0 (mod 2); { 2 2 2 } ≡ X n + 1 n (i) = n + + i, for i = 1, 2, . . . , n. P2,1 2 n n Note that we are able to obtain the partitions into 2-tuples 2,0(i) and 2,2(i) as P P n (i) n (i) n , where s, t = 1. We can use this idea to construct the other P1,s ∪ P1,t ⊕ ± partitions. More precisely, n (i) = n (i) n (i) ln , Pk,d Pl,s ∪ Pm,t ⊕ where k = l + m and d = s + l. For example, we are able to obtain n (i) from the partitions n (i), s = 1 P3,d P1,s ± and n (i), t = 0, 2, 4 and also t = 1 for n odd. It means, n exists for P2,t ± ± ± P3,d d = 1, 3, 5 and if n 1 (mod 2) also for d = 0, 2. Moreover, we are able to ± ± ± ≡ ± construct n in the following way P3,9 n (i) = 3(i 1) + 1, 3(i 1) + 2, 3(i 1) + 3 , P3,9 { − − − } X n (i) = 9i 3, for i = 1, 2, . . . , n. P3,9 − 118 Chapter 8. Antimagic Labeling of Plane Graphs Thus n exists for d = 1, 3, 5, 9 and if n 1 (mod 2) also for d = 0, 2. P3,d ± ± ± ± ≡ ± For the partition into 4-tuples we can use the following fact n (i) = n (i) n (i) ln , P4,d Pl,s ∪ Pm,t ⊕ where l = 3, m = 1 or l = 2, m = 2. Also n (i) = 4(i 1) + 1, 4(i 1) + 2, 4(i 1) + 3, 4(i 1) + 4 , P4,16 { − − − − } X n (i) = 16i 6, for i = 1, 2, . . . , n. P4,16 − Thus n exists for d = 0, 2, 4, 6, 8, 10, 16 and if n 1 (mod 2) also for P4,d ± ± ± ± ± ± ≡ d = 1, 3, 5. ± ± ± Let us note that each of the defined partition n has the property that Pk,d X n (i) = Cn + di, Pk,d k,d n where Ck,d is a constant depending on the parameters k and d. 8.1.3 d-Antimagic Labeling for Certain Families of Plane Graphs In this section, we shall use the edge antimagic labelings of paths Pn and the partitions of the set 1, 2, . . . , kn with determined differences described in the { } previous two sections to examine the existence of a super d-antimagic labeling for several families of plane graphs. The friendship graph Fn is a set of n triangles having a common central vertex, say v, and otherwise disjoint. The friendship graph Fn has 2n vertices of degree 2, say vi, ui for i = 1, 2, . . . , n, and 3n edges, say viv, uiv, viui for i = 1, 2, . . . , n. Let us define the 3-sided face fi, i = 1, 2, . . . , n, as the face bounded by the edges vvi, viui and uiv and let f be the external unbounded face. Theorem 8.1 The friendship graph Fn, n 2, has a super d-antimagic labeling ♦ ≥ of type (1, 1, 0) for d 1, 3, 5, 7, 9, 11, 13 . ∈ { } 119 Chapter 8. Antimagic Labeling of Plane Graphs Moreover, if n 1 (mod 2) then the graph Fn also admits a super d-antimagic ≡ labeling of type (1, 1, 0) for d 0, 2, 4, 6, 8, 10 . ∈ { } Proof. We define the bijection g1 : V (Fn) E(Fn) 1, 2,..., 5n + 1 as follows: ∪ → { } n g1(vi), g1(ui) = (i), i = 1, 2, . . . , n, { } P2,k g1(v) = 2n + 1, n g1(viv), g1(viui), g1(uiv) = (i) (2n + 1), i = 1, 2, . . . , n, { } P3,l ⊕ where k = 0, 2, 4 or for n 1 (mod 2) also k = 1, and l = 1, 3, 5, 9 or ± ± ≡ ± ± ± ± ± for n 1 (mod 2) also l = 0, 2. ≡ ± Observe that the vertices are labeled by the smallest possible numbers 1, 2,..., 2n+ 1. Moreover, for the weight of the face fi, i = 1, 2, . . . , n, we obtain wg1 (fi) = g1(vi) + g1(ui) + g1(v) + g1(viv) + g1(viui) + g1(uiv) X X = n (i) + (2n + 1) + ( n (i) (2n + 1)) P2,k P3,l ⊕ n n = C2,k + ki + (2n + 1) + C3,l + li + 3(2n + 1) n n = C2,k + C3,l + 4(2n + 1) + (k + l)i. As k = 0, 2, 4 or for n 1 (mod 2) also k = 1, and l = 1, 3, 5, 9 or for ± ± ≡ ± ± ± ± ± n 1 (mod 2) also l = 0, 2, we obtain that k + l 1, 3, 5, 7, 9, 11, 13 and for ≡ ± ∈ { } n odd we also get k + l 0, 2, 4, 6, 8, 10 . Thus under the labeling g1 the weights ∈ { } of the 3-sided faces form an arithmetic sequence with the desired difference, which completes the proof. For an illustration of type (1, 1, 0) of super 9-antimagic labeling for friendship graph F6, see Figure 8.1, with the labeling is defined by: n g1(vi), g1(ui) = i, 2n + 1 i = (i), i = 1, 2, . . . , n = 6, { } { − } P2,0 g1(v) = 2n + 1 = 13, g1(viv), g1(viui), g1(uiv) = 3(i 1) + 1, 3(i 1) + 2, 3(i 1) + 3 (2n + 1) { } { − − − } ⊕ = n (i) (2n + 1), i = 1, 2, . . . , n = 6. P3,9 ⊕ 120 Chapter 8. Antimagic Labeling of Plane Graphs 115 12 714 16 2 30 31 17 18 6 11 29 19 13 28 20 8 3 27 26 22 21 525 23 10 924 4 Figure 8.1: A super 9-antimagic labeling of type (1, 1, 0) for F6. If we replace every edge viui, i = 1, 2, . . . , n, of the friendship graph Fn by a path of length two with vertices vi, wi, ui, then we obtain a graph, say Bn, with the vertex set V (Bn) = vi, wi, ui, v : i = 1, 2, . . . , n and the edge set E(Bn) = { } viv, uiv, viwi, wiui : i = 1, 2, . . . , n . Let us define the 4-sided face fi, i = { } 1, 2, . . . , n, as the face bounded by the edges vvi, viwi, wiui and uiv and let f be the external unbounded face. Theorem 8.2 The graph Bn, n 2, has a super d-antimagic labeling of type ♦ ≥ (1, 1, 0) for d 1, 3, 5,..., 21, 25 . ∈ { } Moreover, if n 1 (mod 2) then the graph Bn also admits a super d-antimagic ≡ labeling of type (1, 1, 0) for d 0, 2, 4,..., 18 . ∈ { } Proof. We define the bijection g2 : V (Bn) E(Bn) 1, 2,..., 7n + 1 in the ∪ → { } following way: n g2(vi), g2(wi), g2(ui) = (i), i = 1, 2, . . . , n, { } P3,k g2(v) = 3n + 1, n g2(viv), g2(viwi), g2(wiui), g2(uiv) = (i) (3n + 1), i = 1, 2, . . . , n. { } P4,l ⊕ 121 Chapter 8. Antimagic Labeling of Plane Graphs Observe that the vertices are labeled by the numbers 1, 2,..., 3n + 1. Moreover, for the weight of the face fi, i = 1, 2, . . . , n, we have wg2 (fi) = g2(vi) + g2(wi) + g2(ui) + g2(v) + g2(viv) + g2(viwi) + g2(wiui) + g2(uiv) X X = n (i) + (3n + 1) + ( n (i) (3n + 1)) P3,k P4,l ⊕ n n = C3,k + ki + (3n + 1) + C4,l + li + 4(3n + 1) n n =C3,k + C4,l + 5(3n + 1) + (k + l)i, where k = 1, 3, 5, 9 or for n 1 (mod 2) also k = 0, 2, and l = 0, 2, 4, ± ± ± ± ≡ ± ± ± 6, 8, 10, 16 or for n 1 (mod 2) also l = 1, 3, 5. It means that g2 is ± ± ± ± ≡ ± ± ± a super d-antimagic labeling of type (1, 1, 0) of Bn, for d = 1, 3, 5,..., 21, 25 and if n 1 (mod 2) then d = 0, 2, 4,..., 18. ≡ A triangular snake En is a triangular cactus whose block-cutpoint graph is a path, that is, En is obtained from a path v1, v2, . . . , vn+1 by joining vi and vi+1 to a new vertex ui, for i = 1, 2, . . . , n. Let fi be the 3-sided face, i = 1, 2, . . . , n, bounded by the edges viui, uivi+1 and vivi+1. We denote the external unbounded face by the symbol f. Theorem 8.3 The graph En, n 2, has a super d-antimagic labeling of type ♦ ≥ (1, 1, 0) for d 0, 1, 2,..., 12 . ∈ { } Proof. Define the bijection g3 : V (En) E(En) 1, 2,..., 5n + 1 as follows: ∪ → { } n+1 g3(vi) = αk (vi), i = 1, 2, . . . , n + 1, n+1 g3(ui) = αk (vivi+1), i = 1, 2, . . . , n, n g3(viui), g3(uivi+1), g3(vi+1vi) = (i) (2n + 1), i = 1, 2, . . . , n. { } P3,l ⊕ 122 Chapter 8. Antimagic Labeling of Plane Graphs The labeling g3 assigns the numbers 1, 2,..., 2n + 1 to the vertices of the graph En. For the weight of the face fi, i = 1, 2, . . . , n, we have wg3 (fi) = g3(vi) + g3(ui) + g3(vi+1) + g3(viui) + g3(uivi+1) + g3(vi+1vi) X n = wαn+1 (vivi+1) + 3,l(i) (2n + 1) k P ⊕ n+1 n = Cα,k + ki + C3,l + li + 3(2n + 1) n+1 n = Cα,k + C3,l + 3(2n + 1) + (k + l)i, where k = 0, 1, 2, 3 and l = 1, 3, 5, 9, moreover for n 1 (mod 2) ± ± ± ± ± ± ± ≡ also l = 0, 2. Much as in the proof of the previous theorem we obtain that for ± d 0, 1, 2,..., 12 the bijection g3 is a super d-antimagic labeling of type (1, 1, 0) ∈ { } of the graph En. For an illustration, see Figure 8.2 of E5 with 10-antimagic labeling of type (1, 1, 0), n+1 g3(vi) = α1 (vi) = i, for i = 1, 2 . . . , n + 1 = 6, n+1 g3(ui) = α (vivi+1) = 2n i = 10 i, 1 − − g3(viui), g3(uivi+1), g3(vi+1vi) = 3(i 1) + 1, 3(i 1) + 2, 3(i 1) + 3 (2n + 1) { } { − − − } ⊕ = n (i) (2n + 1), for i = 1, 2, . . . , n + 1 = 6. P3,9 ⊕ 114 2 17 3 20 4 23 5 26 6 12 13 15 16 18 19 21 22 24 25 11 10 9 8 7 Figure 8.2: A super 10-antimagic labeling of type (1, 1, 0) for E5. If we replace every edge vivi+1, i = 1, 2, . . . , n, of the triangular snake En by a path of length two with vertices vi, wi, vi+1, then we obtain a graph, say Gn, with the vertex set V (Gn) = v1, v2, . . . , vn+1, u1, u2, . . . , un, w1, w2, . . . wn and the edge set { } E(Gn) = viui, uivi+1, viwi, wivi+1 : i = 1, 2, . . . , n . Let us define the 4-sided face { } fi, i = 1, 2, . . . , n, as the face bounded by the edges viui, uivi+1, viwi and wivi+1 and let f be the external unbounded face. 123 Chapter 8. Antimagic Labeling of Plane Graphs Theorem 8.4 The graph Gn, n 2, has a super d-antimagic labeling of type ♦ ≥ (1, 1, 0) for d 0, 1, 2,..., 22 . ∈ { } Proof. Define the bijection g4 : V (Gn) E(Gn) 1, 2,..., 7n+1 in the following ∪ → { } way: n+1 g4(vi) = βk (vi), i = 1, 2, . . . , n + 1, n g4(ui), g4(wi) = (i) (n + 1), i = 1, 2, . . . , n, { } P2,l ⊕ g4(viui), g4(uivi+1), g4(viwi), g4(wivi+1) { } = n (i) (3n + 1), i = 1, 2, . . . , n. P4,m ⊕ It is easy to verify that the labeling g4 assigns integers 1, 2,..., 3n+1 to the vertices. By direct computation we find that the weight of the face fi, i = 1, 2, . . . , n, admits the value wg4 (fi) = g4(vi) + g4(vi+1) + g4(ui) + g4(wi) + g4(viui) + g4(uivi+1) + g4(viwi) + g4(wivi+1) X n X n =wβn+1 (vivi+1) + 2,l(i) (n + 1) + 4,m(i) (3n + 1) k P ⊕ P ⊕ n+1 n n = Cβ,k + ki + C2,l + li + 2(n + 1) + C4,m + mi + 4(3n + 1) n+1 n n =Cβ,k + C2,l + C4,m + 14n + 6 + (k + l + m)i, where k = 1, 2, l = 0, 2, 4 and m = 0, 2, 4, 6, 8, 10, 16. After some ± ± ± ± ± ± ± ± ± ± manipulations we find that there exists a super d-antimagic labeling of the graph Gn for every difference d 0, 1, 2,..., 22 . ∈ { } Let the ladder Ln be a Cartesian product Ln Pn P2 of a path on n vertices with ' × a path on two vertices. Let V (Ln) = v1, v2, . . . , vn, u1, u2, . . . , un be the vertex { } set and E(Ln) = vivi+1, uiui+1 : i = 1, 2, . . . , n 1 viui : i = 1, 2, . . . , n be { − } ∪ { } the edge set of ladder. Let us define the 4-sided face fi, i = 1, 2, . . . , n, as the face bounded by the edges vivi+1, vi+1ui+1, uiui+1 and viui and let f be the external unbounded face. Theorem 8.5 The ladder Ln Pn P2, n 3, admits a super d-antimagic ♦ ' × ≥ labeling of type (1, 1, 0) for d 0, 1, 2,..., 10 . ∈ { } 124 Chapter 8. Antimagic Labeling of Plane Graphs Proof. Construct the bijective function g5 : V (Ln) E(Ln) 1, 2,..., 5n 2 ∪ → { − } as follows: n g5(vi) = βk (vi), i = 1, 2, . . . , n, n g5(ui) = βl (vi) + n, i = 1, 2, . . . , n, n g5(viui) = βm(vi) + 2n, i = 1, 2, . . . , n, n−1 g5(vivi+1), g5(uiui+1) = (i) (3n), i = 1, 2, . . . , n 1. { } P2,t ⊕ − It is a routine procedure to verify that the vertices are labeled by the smallest possi- ble numbers 1, 2,..., 2n. Moreover, for the weight of the face fi, i = 1, 2, . . . , n 1, − we obtain wg5 (fi) = g5(vi) + g5(vi+1) + g5(ui) + g5(ui+1) + g5(viui) + g5(vi+1ui+1) + g5(vivi+1) + g5(uiui+1) =w n (vivi+1) + (w n (vivi+1) + 2n) + (w n (vivi+1) + 4n) βk βl βm X + n−1(i) (3n) P2,t ⊕ n n n = Cβ,k + ki + Cβ,l + li + 2n + Cβ,m + mi + 4n n−1 + C2,t + ti + 6n n n n n−1 =Cβ,k + Cβ,l + Cβ,m + C2,t + 12n + (k + l + m + t)i. As k = 1, 2, l = 1, 2, m = 1, 2 and t = 0, 2, 4, we obtain that ± ± ± ± ± ± ± ± k + l + m + t 0, 1, 2, 3,..., 10 . This completes the proof. ∈ { } See Figure 8.3 for L4 = P4 P2 with 0-antimagic labeling of type (1, 1, 0) when × k = 1, l = 1, m = 1 and t = 1. − − Another variation of a ladder graph is specified as follows. A ladder Ln, n 2, is ≥ a graph obtained by completing the ladder Ln Pn P2 by n 1 edges such that ' × − V (Ln) = v1, v2, . . . , v2n is the vertex set and E(Ln) = v1v2, v2v3, . . . , v2n−1v2n, { } { v1v3, v2v4, . . . , v2n−2v2n is the edge set. } Theorem 8.6 The graph Ln, n 2, admits a super d-antimagic labeling of type ♦ ≥ (1, 1, 0) for d 0, 1, 2,..., 6 . ∈ { } 125 Chapter 8. Antimagic Labeling of Plane Graphs 1 14 3 13 2 15 4 12 1011 9 8 16 6 18 7 17 5 Figure 8.3: A super 0-antimagic labeling of type (1, 1, 0) for L4. Proof. Construct the bijective function g6 : V (Ln) E(Ln) 1, 2,..., 6n 1 ∪ → { − } in the following way: g6(vi) = i, i = 1, 2,..., 2n, 2n−1 g6(vivi+1) = β (vi) + 2n, i = 1, 2,..., 2n 1, k − 2n−2 g6(vivi+2) = (i) (4n 1), i = 1, 2,..., 2n 2. P1,l ⊕ − − Note that the labeling g6 assigns integers 1, 2,..., 2n to the vertices and a weight of the face fi, i = 1, 2,..., 2n 2, is − wg6 (fi) = g6(vi) + g6(vi+1) + g6(vi+2) + g6(vivi+1) + g6(vi+1vi+2) + g6(vivi+2) 2n−1 = i + (i + 1) + (i + 2) + w (vivi+1) + 4n βk X + 2n−2(i) (4n 1) P1,l ⊕ − = 3 + 3i + C2n−1 + ki + 4n + C2n−2 + li + (4n 1) β,k 1,l − 2n−1 2n−2 =Cβ,k + C1,l + 8n + 2 + (3 + k + l)i. Since k = 1, 2, and l = 1 we are able to show that 3 + k + l 0, 1, 2,..., 6 . ± ± ± ∈ { } This implies that the labeling g6 is a super d-antimagic labeling of type (1, 1, 0) for d 0, 1, 2,..., 6 of the graph Ln. ∈ { } If we replace every edge vivi+1 (respectively, every edge uiui+1), i = 1, 2, . . . , n 1, − of the ladder Ln Pn P2 by a path of length two with vertices vi, wi, vi+1 ' × 126 Chapter 8. Antimagic Labeling of Plane Graphs (respectively, ui, wn−1+i, ui+1), then we obtain a graph, say Hn, with the ver- tex set V (Hn) = v1, v2, . . . , vn, u1, u2, . . . , un, w1, w2, . . . , w2n−2 and the edge set { } E(Hn) = viwi, wivi+1, uiwn−1+i, wn−1+iui+1 : i = 1, 2, . . . , n 1 viui : i = { − } ∪ { 1, 2, . . . , n . } Let us define the 6-sided face fi, i = 1, 2, . . . , n 1, as the face bounded by the − edges viwi, wivi+1, vi+1ui+1, ui+1wn−1+i, wn−1+iui, uivi and let f be the external unbounded face. Theorem 8.7 The graph Hn, n 2, admits a super d-antimagic labeling of type ♦ ≥ (1, 1, 0) for d 0, 1, 2,..., 26 . ∈ { } Proof. We define the bijection g7 : V (Hn) E(Hn) 1, 2,..., 9n 6 in the ∪ → { − } following way: n g7(vi) = βk (vi), i = 1, 2, . . . , n, n g7(ui) = βl (vi) + n, i = 1, 2, . . . , n, n−1 g7(wi), g7(wn−1+i) = (i) (2n), i = 1, 2, . . . , n 1, { } P2,m ⊕ − n g7(viui) = β (vi) + 4n 2, i = 1, 2, . . . , n, t − g7(viwi), g7(wivi+1), g7(uiwn−1+i), g7(wn−1+iui+1) { } = n−1(i) (5n 2), i = 1, 2, . . . , n 1. P4,s ⊕ − − 127 Chapter 8. Antimagic Labeling of Plane Graphs It is easy to see that the vertices of Hn are labeled by the smallest possible integers 1, 2,..., 4n 2. For the weight of the face fi, i = 1, 2, . . . , n 1, we get − − wg7 (fi) = g7(vi) + g7(vi+1) + g7(ui) + g7(ui+1) + g7(wi) + g7(wn−1+i) + g7(viui) + g7(vi+1ui+1) + g7(viwi) + g7(wivi+1) + g7(uiwn−1+i) + g7(wn−1+iui+1) X n−1 =wβn (vivi+1) + wβn (vivi+1) + 2n + (i) (2n) k l P2,m ⊕ n X n−1 + w (vivi+1) + 8n 4 + (i) (5n 2) βt − P4,s ⊕ − n n n−1 = Cβ,k + ki + Cβ,l + li + 2n + C2,m + mi + 2(2n) + Cn + ti + 8n 4 + Cn−1 + si + 4(5n 2) β,t − 4,s − =Cn + Cn + Cn + Cn−1 + Cn−1 + 34n 12 + (k + l + m + t + s)i. β,k β,l β,t 2,m 4,s − Since k = 1, 2, l = 1, 2, m = 0, 2, 4, t = 1, 2 and s = 0, 2, 4, 6, ± ± ± ± ± ± ± ± ± ± ± 8, 10, 16, we get that k+l+m+t+s 0, 1, 2,..., 26 . Thus g7 is the required ± ± ± ∈ { } super d-antimagic labeling of type (1, 1, 0) of the graph Hn. This completes the proof. For an example, see Figure 8.4 of a super 20-antimagic labeling for H5, where k = 1, l = 1, m = 0, t = 2 and s = 16. 1 24 11 25 4 28 12 29 232 13 33 5 36 14 37 3 19 20 21 22 23 6 26 18 27 9 30 17 31 734 16 35 10 38 15 39 8 Figure 8.4: A super 20-antimagic labeling of type (1, 1, 0) for H5. 128 Chapter 8. Antimagic Labeling of Plane Graphs 8.2 Super d-Antimagic Labeling of Type (1, 1, 1) 8.2.1 Super 1-Antimagic Labeling of Type (1, 1, 1) In this section, we deal with (super) 1-antimagic labelings of type (1, 1, 1) for disjoint unions of graphs. By mG we denote the disjoint union of m copies of a graph G. The symbol zext is used to denote the unique external face in the plane graph. Define h∗ as (face) weights from h labeling. The main result is the following. Theorem 8.8 Let G(V,E,F ) be a plane graph. If there exists a (super) 1- ♦ antimagic labeling h of type (1, 1, 1) of G such that h(zext) = V (G) + E(G) + | | | | F (G) then, for every positive integer m, the graph mG also admits a (super) 1- | | antimagic labeling of type (1, 1, 1). Proof. Let G be a plane graph with p vertices, q edges and f + 1 faces. Let h be a (super) 1-antimagic labeling of type (1, 1, 1) of a graph G such that h(zext) = p + q + f + 1. Thus h : V (G) E(G) F (G) 1, 2, . . . , p + q + f + 1 ∪ ∪ −→ { } and for every positive integer s, the set of weights of all s-sided faces in G is ∗ s s h (z ), z F (G) = Ws(G) = as, as + 1, . . . , as + fs 1 , { ∈ } { − } where as is a positive integer and fs is the number of s-sided faces in G. By the symbol xi we denote an element (a vertex, an edge or a face) in the i- th copy of G, denoted by Gi, corresponding to the element x in G, that is, x ∈ V (G) E(G) F (G). ∪ ∪ We define the labeling g for vertices, edges and faces of mG in the following way. m[h(x) 1] + i if x V (G) F (G), g(xi) = − ∈ ∪ mh(x) + 1 i if x E(G), for i = 1, 2, . . . , m. − ∈ g(zext) =mp + mq + mf + 1, 129 Chapter 8. Antimagic Labeling of Plane Graphs Let t 1, 2, . . . , p + q + f . We consider two cases. ∈ { } Case 1: If the number t is assigned by the labeling h to some vertex or to some face of G then the corresponding vertices or faces in the copies in mG will receive the following labels under the labeling g m(t 1) + 1, m(t 1) + 2, . . . m(t 1) + i,... mt − − − in G1 in G2 . . . in Gi . . . in Gm Case 2: If the number t is assigned by the labeling h to some edge of G then the corresponding edges in the copies in mG will have labels mt, mt 1, . . . mt + 1 i,... m(t 1) + 1. − − − in G1 in G2 . . . in Gi . . . in Gm. It is easy to see that the vertex labels, edge labels and face labels in mG are not overlapping, and the maximum used label is mp + mq + mf + 1. Thus, g is a total labeling of G. Moreover, if the labeling h is super 1-antimagic, that is, the numbers 1, 2, . . . , p are used to label the vertices of G, then the numbers 1, 2, . . . , mp are used to label the vertices of mG under the labeling g. This means that g is super. To conclude the proof, we need to show that for every positive integer s, the face weights of all s-sided faces in mG form an arithmetic sequence with the difference 1. Let zs be an s-sided face in G. By x z we understand that the element x, a ∼ vertex or an edge, is incident with the face z. Define g∗ as face weights. For the s s face weight of the s-sided face zi corresponding to the face z in the i-th copy of G, i = 1, 2, . . . , m, we have 130 Chapter 8. Antimagic Labeling of Plane Graphs ∗ s X X s g (zi ) = g(vi) + g(ei) + g(zi ) s s vi∼zi ei∼zi X X = [m(h(v) 1) + i] + [mh(e) + 1 i] + [m(h(zs) 1) + i] v∼zs − e∼zs − − X X =[m h(v) ms + is] + [m h(e) + s is] + [mh(zs) m + i] v∼zs − e∼zs − − X X =m[ h(v) + h(e) + h(zs)] + s ms m + i v∼zs e∼zs − − =mh∗(zs) + s m(s + 1) + i = mh∗(zs) + const + i, − for i = 1, 2, . . . , m, where const = s m(s + 1). − As ∗ s s h (z ), z F (G) = Ws(G) = as, as + 1, . . . , as + fs 1 , { ∈ } { − } we have that the face weights of the s-sided faces in all the components are in G1: mas + const + 1, m(as + 1) + const + 1, . . . m(as + fs 1) + const + 1 − in G2: mas + const + 2, m(as + 1) + const + 2, . . . m(as + fs 1) + const + 2 . . . − . . . in Gi: mas + const + i, m(as + 1) + const + i, . . . m(as + fs 1) + const + i . . . − . . . in Gm: mas + const + m, m(as + 1) + const + m, . . . m(as + fs 1) + const + m. − It is clear from this array that the weights of the s-sided faces are distinct and consecutive: Ws(mG) = mas + const + 1, mas + const + 2, . . . , m(as + fs) + const . { } 131 Chapter 8. Antimagic Labeling of Plane Graphs This implies that mG has a (super) 1-antimagic labeling of type (1, 1, 1). For an example, based on 0-antimagic labeling of type (1, 1, 0) for the ladder L4 in Figure 8.3 and we can have 1-antimagic labeling of type (1, 1, 1) for the ladder L4 as shown in Figure 8.5. Then we have 1-antimagic labeling of type (1, 1, 1) for 2L4 in Figure 8.6. 22 1 14 3 13 2 15 4 1219 1020 11 21 9 8 16 6 18 7 17 5 Figure 8.5: A super 1-antimagic labeling of type (1, 1, 1) for L4. 1 28 5 26 3 30 7 2437 2039 22 41 18 15 32 11 36 13 34 9 43 2 27 6 25 4 29 8 2338 1940 21 42 17 16 31 12 35 14 33 10 Figure 8.6: A super 1-antimagic labeling of type (1, 1, 1) for 2L4. 132 Chapter 8. Antimagic Labeling of Plane Graphs We present one more example; when G with faces of two different sizes other than the exterior face has 1-antimagic labeling of type (1, 1, 1) as shown in Figure 8.7, then we obtain 1-antimagic labeling of type (1, 1, 1) for 2G in Figure 8.8. 30 3 22 5 21 4 23 6 11 12 1 15 2027 1828 19 29 17 16 2 14 13 10 24 8 26 9 25 7 Figure 8.7: A super 1-antimagic labeling of type (1, 1, 1) for G. 5 44 9 42 7 46 11 22 24 1 29 4053 3655 38 57 34 31 3 28 26 19 48 15 52 17 50 13 59 6 43 10 41 8 45 12 21 23 2 30 3954 3556 37 58 33 32 4 27 25 20 47 16 51 18 49 14 Figure 8.8: A super 1-antimagic labeling of type (1, 1, 1) for 2G. 133 Chapter 8. Antimagic Labeling of Plane Graphs 8.2.2 Super d-Antimagic Labeling of Type (1, 1, 1) In this section we will deal with the (super) d-antimagic labelings of type (1, 1, 1) for plane graphs containing faces of the same type. More precisely, we will study plane graphs with inner faces that are 3-sided and 4-sided. Theorem 8.9 Let G(V,E,F ) be a plane graph with 3-sided inner faces. Let h ♦ be a (super) d-antimagic labeling of type (1, 1, 1) of G such that h(zext) = V (G) + | | E(G) + F (G) . | | | | (i) If d = 1, 5, 7 then, for every positive integer m, the graph mG also admits a (super) d-antimagic labeling of type (1, 1, 1). (ii) If G is a tripartite graph and d = 3 then, for every positive integer m, the graph mG also admits a (super) d-antimagic labeling of type (1, 1, 1). (iii) If G is a tripartite graph and d = 0, 2, 4, 6 then, for every odd positive integer m, the graph mG also admits a (super) d-antimagic labeling of type (1, 1, 1). Proof. Let G be a plane graph with 3-sided inner faces with p vertices, q edges and f3 + 1 faces. Let h be a (super) d-antimagic labeling of type (1, 1, 1) of G such that h(zext) = p + q + f3 + 1. Thus h : V (G) E(G) F (G) 1, 2, . . . , p + q + f3 + 1 ∪ ∪ −→ { } and the set of weights of all 3-sided faces in G is W3(G) = a, a + d, . . . , a + (f3 1)d , { − } where a is a positive integer and f3 is the number of 3-sided faces in G. By the symbol xi we denote an element (a vertex, an edge or a face) in the i- th copy of G, denoted by Gi, corresponding to the element x in G, that is, x ∈ V (G) E(G) F (G). ∪ ∪ Case d = 1 For d = 1 the result follows immediately from Theorem 8.8. 134 Chapter 8. Antimagic Labeling of Plane Graphs Case d = 5 For d = 5 we define the labeling g for vertices, edges and faces of mG in the following way. m[h(x) 1] + i if x V (G) E(G), g(xi) = − ∈ ∪ mh(x) + 1 i if x F (G), − ∈ g(zext) =mp + mq + mf3 + 1, for i = 1, 2, . . . , m. As in the proof of the previous theorem, it is easy to prove that g is a bijection from the set of elements of mG to the set 1, 2, . . . , mp+mq+mf3 +1. 3 3 For the face weight of the 3-sided face zi corresponding to the face z in the i-th copy of G, i = 1, 2, . . . , m we have ∗ 3 X X 3 g (zi ) = g(vi) + g(ei) + g(zi ) 3 3 vi∼zi ei∼zi X X = [m(h(v) 1) + i] + [m(h(e) 1) + i] + [mh(z3) + 1 i] − − − v∼z3 e∼z3 X X =[m h(v) 3m + 3i] + [m h(e) 3m + 3i] + [mh(z3) + 1 i] − − − v∼z3 e∼z3 X X =m[ h(v) + h(e) + h(z3)] 6m + 1 + 5i − v∼z3 e∼z3 =mh∗(z3) 6m + 1 + 5i, − for i = 1, 2, . . . , m. As ∗ 3 3 h (z ), z F (G) = W3(G) = a, a + 5, . . . , a + (f3 1)5 , { ∈ } { − } we have that the face weights of the 3-sided faces in all the components are 135 Chapter 8. Antimagic Labeling of Plane Graphs in G1: ma 6m + 6, m(a + 5) 6m + 6, . . . m(a + (f3 1)5) 6m + 6 − − − − in G2: ma 6m + 11, m(a + 5) 6m + 11, . . . m(a + (f3 1)5) 6m + 11 . − . − . − − . . . in Gi: ma 6m + 1 + 5i, m(a + 5) 6m + 1 + 5i, . . . m(a + (f3 1)5) 6m + 1 + 5i . − . − . − − . . . in Gm: ma m + 1, m(a + 5) m + 1, . . . m(a + (f3 1)5) m + 1. − − − − Thus, the weights of all 3-sided faces are distinct and form an arithmetic sequence with the difference d = 5. This implies that mG has a (super) 5-antimagic labeling of type (1, 1, 1). Case d = 7 For d = 7 we define the labeling g for vertices, edges and faces of mG such that g(xi) =m[h(x) 1] + i if x V (G) E(G) F (G) − ∈ ∪ ∪ g(zext) =mp + mq + mf3 + 1, for i = 1, 2, . . . , m. Using methods similar to those of the previous case, we can show that g is a (super) 7-antimagic labeling of type (1, 1, 1) of mG. In the next part of the proof we will assume that the plane graph with 3-sided inner faces is tripartite. Let G be a tripartite graph with the partite sets V1,V2 and V3. Then E(G) = V1V2 V2V3 V1V3, where the juxtaposition of two partite sets ∪ ∪ denotes the set of edges between those two sets. Again, we will distinguish several cases according to the difference d. Case d = 3 and G is tripartite 136 Chapter 8. Antimagic Labeling of Plane Graphs In this case we define the total labeling g of mG in the following way m[h(x) 1] + i if x V2 V3 E(G), g(xi) = − ∈ ∪ ∪ mh(x) + 1 i if x V1 F (G), − ∈ ∪ g(zext) =mp + mq + mf3 + 1, for i = 1, 2, . . . , m. Case d = 0, G is tripartite and m is odd We define the total labeling g of mG in the following way. m[h(x) 1] + i if x V1 V1V2 V1V3, i = 1, 2, . . . , m, − ∈ ∪ ∪ mh(x) + 1 i if x V2V3 F (G), i = 1, 2, . . . , m, − m + 1 ∈ ∪ m−1 m[h(x) 1] + i + if x V2, i = 1, 2,..., 2 , g(xi) = − 2 ∈ m 1 m+1 m+3 m[h(x) 1] + i − if x V2, i = , , . . . , m, − − 2 ∈ 2 2 m−1 m[h(x) 1] + m + 1 2i if x V3, i = 1, 2,..., , − − ∈ 2 m+1 m+3 m[h(x) 1] + 2m + 1 2i if x V3, i = , , . . . , m, − − ∈ 2 2 g(zext) =mp + mq + mf3 + 1. Case d = 2, G is tripartite and m is odd We define the total labeling g of mG such that m[h(x) 1] + i if x V1 V1V2 V1V3, i = 1, 2, . . . , m, − ∈ ∪ ∪ mh(x) + 1 i if x V2V3 F (G), i = 1, 2, . . . , m, − ∈ ∪ i + 1 m[h(x) 1] + if x V2, i = 1, 3, . . . , m, − 2 ∈ g(xi) = m + i + 1 m[h(x) 1] + if x V2, i = 2, 4, . . . , m 1, − 2 ∈ − m + i m[h(x) 1] + if x V3, i = 1, 3, . . . , m, − 2 ∈ i m[h(x) 1] + if x V3, i = 2, 4, . . . , m 1, − 2 ∈ − g(zext) =mp + mq + mf3 + 1. 137 Chapter 8. Antimagic Labeling of Plane Graphs Case d = 4, G is tripartite and m is odd We define the total labeling g of mG in the following way. m[h(x) 1] + i if x V1 E(G) F (G), i = 1, 2, . . . , m, − m + 1 ∈ ∪ ∪ m−1 m[h(x) 1] + i + if x V2, i = 1, 2,..., 2 , − 2 ∈ m 1 m+1 m+3 g(xi) = m[h(x) 1] + i − if x V2, i = , , . . . , m, − − 2 ∈ 2 2 m−1 m[h(x) 1] + m + 1 2i if x V3, i = 1, 2,..., , − − ∈ 2 m+1 m+3 m[h(x) 1] + 2m + 1 2i if x V3, i = , , . . . , m, − − ∈ 2 2 g(zext) =mp + mq + mf3 + 1. Case d = 6, G is tripartite and m is odd We define the total labeling g of mG in the following way m[h(x) 1] + i if x V1 E(G) F (G), i = 1, 2, . . . , m, − ∈ ∪ ∪ i + 1 m[h(x) 1] + if x V2, i = 1, 3, . . . , m, − 2 ∈ m + i + 1 g(xi) = m[h(x) 1] + if x V2, i = 2, 4, . . . , m 1, − 2 ∈ − m + i m[h(x) 1] + if x V3, i = 1, 3, . . . , m, − 2 ∈ i m[h(x) 1] + if x V3, i = 2, 4, . . . , m 1, − 2 ∈ − g(zext) =mp + mq + mf3 + 1. It is not difficult to prove that the considered labelings satisfy the desired properties, thus they are (super) d-antimagic of type (1, 1, 1). For the plane graphs with all inner 4-sided faces we can prove Theorem 8.10 Let G(V,E,F ) be a plane graph with 4-sided inner faces. Let h ♦ be a (super) d-antimagic labeling of type (1, 1, 1) of G such that h(zext) = V (G) + | | E(G) + F (G) . If d = 1, 3, 5, 7, 9 then, for every positive integer m, the graph mG | | | | also admits a (super) d-antimagic labeling of type (1, 1, 1). 138 Chapter 8. Antimagic Labeling of Plane Graphs Proof. Let G be a plane graph with 4-sided inner faces. Let G have p vertices, q edges and f4 + 1 faces. Let h be a (super) d-antimagic labeling of type (1, 1, 1) of G such that h(zext) = p + q + f4 + 1. Thus h : V (G) E(G) F (G) 1, 2, . . . , p + q + f4 + 1 ∪ ∪ −→ { } and the set of weights of all 4-sided faces in G is W4(G) = a, a + d, . . . , a + (f4 1)d , { − } where a is a positive integer and f4 is the number of 4-sided faces in G. As all inner faces of G are 4-sided, G is a bipartite graph with the partite sets V1 and V2. By the symbol xi we denote an element (a vertex, an edge or a face) in the i-th copy of G, denoted by Gi, corresponding to the element x in G, that is, x V (G) E(G) F (G). ∈ ∪ ∪ For d = 1 the result follows from Theorem 8.8. To prove the result for differences d = 3, 5, 7, 9, it is sufficient to show that the following labelings have the desired properties. Case d = 3 The total labeling g of mG is defined as follows. m[h(x) 1] + i if x V1 E(G), g(xi) = − ∈ ∪ mh(x) + 1 i if x V2 F (G), − ∈ ∪ g(zext) =mp + mq + mf4 + 1, for i = 1, 2, . . . , m. Case d = 5 We define the total labeling g of mG in the following way. 139 Chapter 8. Antimagic Labeling of Plane Graphs m[h(x) 1] + i if x V1 E(G) F (G), g(xi) = − ∈ ∪ ∪ mh(x) + 1 i if x V2, − ∈ g(zext) =mp + mq + mf4 + 1, for i = 1, 2, . . . , m. Case d = 7 In this case the total labeling g of mG is defined such that m[h(x) 1] + i if x V (G) E(G), g(xi) = − ∈ ∪ mh(x) + 1 i if x F (G), − ∈ g(zext) =mp + mq + mf4 + 1, for i = 1, 2, . . . , m. Case d = 9 We define the total labeling g of mG such that g(xi) =m[h(x) 1] + i if x V (G) E(G) F (G), − ∈ ∪ ∪ g(zext) =mp + mq + mf4 + 1, for i = 1, 2, . . . , m. As an illustration for Case d = 5, we first construct the 4-antimagic labeling of type (1, 1, 0) for L4 by applying the construction in the proof of Theorem 8.5 when k = 1, l = 2, m = 1 and t = 0 and then we have easily the super 5-antimagic labeling of type (1, 1, 1) for L4 in Figure 8.9. 140 Chapter 8. Antimagic Labeling of Plane Graphs 22 1 13 3 14 2 15 4 919 1120 10 21 12 5 18 6 17 7 16 8 Figure 8.9: A super 5-antimagic labeling of type (1, 1, 1) for L4. See Figure 8.10, for 2L4 with a super 5-antimagic labeling of type (1, 1, 1). 1 25 6 27 3 29 8 1737 2139 19 41 23 10 35 11 33 14 31 15 43 2 26 5 28 4 30 7 1838 2240 20 42 24 9 36 12 34 13 32 16 Figure 8.10: A super 5-antimagic labeling of type (1, 1, 1) for 2L4. It is possible to generalize the results in the previous theorems for plane graphs containing only k-sided faces except for the external face, where k is a positive integer, k 3. ≥ 141 Chapter 8. Antimagic Labeling of Plane Graphs Theorem 8.11 Let k, d be positive integers, k 3. Let G(V,E,F ) be a plane ♦ ≥ graph with k-sided inner faces. Let h be a (super) d-antimagic labeling of type (1, 1, 1) of G such that h(zext) = V (G) + E(G) + F (G) . | | | | | | (i) If d = 1, 2k 1 then, for every positive integer m, the graph mG also admits ± a (super) d-antimagic labeling of type (1, 1, 1). (ii) Moreover, if k is even and d = k 1 then, for every positive integer m, the ± graph mG also admits a (super) d-antimagic labeling of type (1, 1, 1). Proof. Let G be a plane graph with k-sided inner faces. Let G have p vertices, q edges and fk + 1 faces. Let h be a (super) d-antimagic labeling of type (1, 1, 1) of G such that h(zext) = p + q + fk + 1. Thus h : V (G) E(G) F (G) 1, 2, . . . , p + q + fk + 1 ∪ ∪ −→ { } and the set of weights of all k-sided faces in G is Wk(G) = a, a + d, . . . , a + (fk 1)d , { − } where a is a positive integer and fk is the number of k-sided faces in G. By the symbol xi we denote an element (a vertex, an edge or a face) in the i- th copy of G, denoted by Gi, corresponding to the element x in G, that is, x ∈ V (G) E(G) F (G). ∪ ∪ For d = 1 the results follow from Theorem 8.8. We consider the following cases. Case d = 2k + 1 The total labeling g of mG is defined as follows g(xi) =m[h(x) 1] + i if x V (G) E(G) F (G), − ∈ ∪ ∪ g(zext) =mp + mq + mfk + 1, for i = 1, 2, . . . , m. 142 Chapter 8. Antimagic Labeling of Plane Graphs Case d = 2k 1 − We define the total labeling g of mG such that m[h(x) 1] + i if x V (G) E(G), g(xi) = − ∈ ∪ mh(x) + 1 i if x F (G), − ∈ g(zext) =mp + mq + mfk + 1, for i = 1, 2, . . . , m. If k is even, then G is a bipartite graph with the bipartite set V1 and V2. We define the labeling g in the following way. Case d = k 1 and k is even − m[h(x) 1] + i if x V1 E(G), g(xi) = − ∈ ∪ mh(x) + 1 i if x V2 F (G), − ∈ ∪ g(zext) =mp + mq + mfk + 1, for i = 1, 2, . . . , m. Case d = k + 1 and k is even m[h(x) 1] + i if x V1 E(G) F (G), g(xi) = − ∈ ∪ ∪ mh(x) + 1 i if x V2, − ∈ g(zext) =mp + mq + mfk + 1, for i = 1, 2, . . . , m. Using a technique similar to that used in the proof of Theorem 8.8, we are able to show that the foregoing labelings are (super) d-antimagic of type (1, 1, 1). We can formulate similar results also for plane graphs with two kinds of inner faces. Theorem 8.12 Let k be a positive integer, k 3. Let G(V,E,F ) be a plane ♦ ≥ graph with k-sided and (k+1)-sided inner faces. Let h be a (super) (2k+1)-antimagic 143 Chapter 8. Antimagic Labeling of Plane Graphs labeling of type (1, 1, 1) of G such that h(zext) = V (G) + E(G) + F (G) . Then, | | | | | | for every positive integer m, the graph mG also admits a (super) (2k +1)-antimagic labeling of type (1, 1, 1). Proof. Let G be a plane graph with k-sided and (k + 1)-sided inner faces. Let G have p vertices, q edges and fk +fk+1 +1 faces. Let h be a (super) (2k+1)-antimagic labeling of type (1, 1, 1) of G such that h(zext) = p + q + fk + fk+1 + 1. Thus h : V (G) E(G) F (G) 1, 2, . . . , p + q + fk + fk+1 + 1 ∪ ∪ −→ { } and the set of weights of all s-sided faces in G is Ws(G) = as, as + d, . . . , as + (fs 1)d , { − } where as is a positive integer, s = k, k + 1 and fs is the number of s-sided faces in G. By the symbol xi we denote an element (a vertex, an edge or a face) in the i- th copy of G, denoted by Gi, corresponding to the element x in G, that is, x ∈ V (G) E(G) F (G). By the symbols Fk(G), Fk+1(G) we denote the set of all ∪ ∪ k-sided, (k + 1)-sided inner faces in G, respectively. It is not difficult to prove that the labeling g of mG defined as m[h(x) 1] + i if x V E(G) Fk(G), g(xi) = − ∈ ∪ ∪ mh(x) + 1 i if x Fk+1(G), − ∈ g(zext) =mp + mq + m(fk + fk+1) + 1, for i = 1, 2, . . . , m, is a (super) (2k + 1)-antimagic labeling of type (1, 1, 1). 8.2.3 Some Classes of Plane Graphs In this section we will deal with some special classes of plane graphs. The friendship graph Fn is a set of n triangles having a common center vertex as defined in Section 8.1. It is known that 144 Chapter 8. Antimagic Labeling of Plane Graphs Proposition 8.1 [11] The friendship graph Fn, n 2, has a super d-antimagic ≥ labeling of type (1, 1, 1) for d 0, 2, 4,..., 20 . ∈ { } Moreover, if n 1 (mod 2) then the graph Fn also admits a super d-antimagic ≡ labeling of type (1, 1, 1) for d 1, 3, 5, 7, 9, 11, 15, 17 . ∈ { } A triangular snake is a connected graph whose blocks are the cycles C3 and its block- cutpoint graph is a path, see [71]. By En we denote the triangular snake embedded in the plane with the vertex set V (En) = v1, v2, . . . , vn+1, u1, u2, . . . , un and { } the edge set E(En) = vivi+1, viui, uivi+1 : i = 1, 2, . . . , n . The face set F (En) { } contains n 3-sided faces and one external infinite face. It is also known that Proposition 8.2 [11] The graph En, n 2, has a super d-antimagic labeling of ≥ type (1, 1, 1) for d 0, 1, 2,..., 19 . ∈ { } Immediately, from Theorem 8.9, we get Corollary 8.1 Let m, n, d, n 2, m 1 be nonnegative integers. Then for m ♦ ≥ ≥ odd, the graphs mFn and mEn admit a super d-antimagic labeling of type (1, 1, 1), for d = 0, 2, 4, 6. Moreover, if n 1 (mod 2) then mFn and mEn admit the super ≡ d-antimagic labeling of type (1, 1, 1), for d = 1, 3, 5, 7. Let Bn be a graph consisting of a set of n cycles C4 having a common center vertex. Consider the graph Bn embedded in the plane with the vertex set V (Bn) = vi, ui, wi, v : i = 1, 2, . . . , n and the edge set E(Bn) = viv, wiv, viui, uiwi : i = { } { 1, 2, . . . , n . The face set F (Bn) contains n 4-sided faces and one external infinite } face. It is known that Proposition 8.3 [11] The graph Bn, n 2, has a super d-antimagic labeling of ≥ type (1, 1, 1) for d 0, 2, 4, 6,..., 28, 30, 34 . ∈ { } Moreover, if n 1 (mod 2) then the graph Bn also admits a super d-antimagic ≡ labeling of type (1, 1, 1) for d 1, 3, 5,..., 27 . ∈ { } A quadrilateral snake is a connected graph whose blocks are the cycles C4. By Gn we denote the quadrilateral snake embedded in the plane with the vertex set V (Gn) = v1, v2, . . . , vn+1, u1, u2, . . . , un, w1, w2, . . . wn and the edge set E(Gn) = { } 145 Chapter 8. Antimagic Labeling of Plane Graphs viui, uivi+1, viwi, wivi+1 : i = 1, 2, . . . , n . The face set F (Gn) contains n 4-sided { } faces and one external infinite face. It is known that Proposition 8.4 [11] The graph Gn, n 2, has a super d-antimagic labeling of ≥ type (1, 1, 1) for d 0, 1, 2,..., 31 . ∈ { } Combining these results with Theorem 8.10, we obtain Corollary 8.2 Let m, n, d, 3 n 1 (mod 2), m 1 be nonnegative integers. ♦ ≤ ≡ ≥ Then the graph mBn admits a super d-antimagic labeling of type (1, 1, 1), for d = 1, 3, 5, 7, 9. Corollary 8.3 Let m, n, d, n 2, m 1 be nonnegative integers. Then the ♦ ≥ ≥ graph mGn admits super d-antimagic labeling of type (1, 1, 1), for d = 1, 3, 5, 7, 9. 8.3 Conclusion We have studied the super d-antimagic labelings of type (1, 1, 0) for seven families of plane graphs. We have shown that there exist super d-antimagic labelings of type (1, 1, 0) for these graphs for wide variety of differences d. We have also examined the super d-antimagic labelings of type (1, 1, 1) for disjoint unions of graphs. The results on super d-antimagic labelings of type (1, 1, 1) for disjoint unions of graphs with k-sided inner faces, k 3, for various values d, are provided. Finally, some ≥ special classes of plane graphs, for instance mFn, mEn, mBn, have been proved to be d-antimagic of type (1, 1, 1), for various values d. We conclude this chapter with the following open problems for super d-antimagic labelings of type (1, 1, 0). Open Problem 8.1 Find the upper bound for the feasible values of the difference d which make a super d-antimagic labelings of type (1, 1, 0) possible for the studied families of plane graphs. Open Problem 8.2 Find other feasible values of the difference d and the corre- sponding super d-antimagic labelings of type (1, 1, 0) for the studied families of plane graphs. 146 Chapter 9 Summary In this thesis, we considered three different types of graph labelings, namely, vertex antimagic edge labeling, vertex and edge antimagic total labeling, and d-antimagic labeling (face antimagic labeling of plane graphs). The first part of the thesis was devoted to the classical problem ‘vertex antimagic edge labelings of graphs’ that was introduced by Hartsfield and Ringel in 1990. We proved that vertex antimagic edge labelings exist for some particular families of graphs, for example, some classes of regular and non regular graphs. We also proved that for any degree sequence pertaining to a tree there exists a vertex antimagic labeling for a tree with that degree sequence; and if a tree has at least seven vertices, there exist two different vertex antimagic edge trees with that degree sequence. However, at the time of submission of this thesis even the weaker conjecture of Hartsfield and Ringel conjecture ‘Every tree different from K2 is vertex antimagic edge’ is still open. The second part of this dissertation contained total antimagic labeling and totally antimagic total, and d-antimagic labeling (in particular, super d-antimagic labelings of types (1, 1, 0) and (1, 1, 1)). Throughout the thesis we presented open problems which were summarized at the end of each chapter. For future research we particularly recommend the study of vertex antimagic edge labelings of regular graphs and trees, and totally antimagic total labelings of par- ticular families of graphs. The study of super d-antimagic labeling of types (1, 1, 0) and (1, 1, 1) for disconnected graphs is also an interesting area of research in graph labeling. 147 Chapter 9. Summary We conclude by presenting all the new open problems that have been presented in the chapters of this thesis. 9.1 Open Problems n Open Problem 9.1 Is it possible to find vertex antimagic labelings for all AG, n 2, and T n,n 3, where G is any regular graph? ≥ G ≥ In this thesis we have proved that for any degree sequence pertaining to a tree, a unicyclic and regular graphs there exists a vertex antimagic labeling for a graph corresponding to that degree sequence. More generally, we have the following prob- lem. Open Problem 9.2 For every degree sequence pertaining to a graph, there exists an antimagic labeling for a graph corresponding to that degree sequence. This is weaker than Hartsfield and Ringel Conjecture. Open Problem 9.3 Given any degree sequence pertaining to a unicyclic graph, is it possible to construct an antimagic unicyclic graph corresponding to that de- gree sequence that is different from the one obtained in the proof of Theorem 5.18? Specifically, can this graph be constructed using Construction 2 in the proof of The- orem 5.16? The problems are likely to be difficult. The following two problems may be more accessible. Open Problem 9.4 Find new classes of totally antimagic total graphs. Open Problem 9.5 Find some necessary and/or sufficient conditions for a graph to be totally antimagic total. Open Problem 9.6 Characterize totally antimagic total graphs. 148 Chapter 9. Summary Investigating classes of graphs that are simultaneously magic in some way and antimagic in another way is a new interesting research direction. For example, it might be interesting to investigate the next two problems. 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