RAINBOW CONNECTION NUMBER OF GRAPH POWER AND GRAPH PRODUCTS A Thesis Submitted For the Degree of Master of Science (Engineering) in the Faculty of Engineering by Arunselvan. R Computer Science and Automation Indian Institute of Science BANGALORE { 560 012 November 2011 i c Arunselvan. R November 2011 All rights reserved TO My Parents Acknowledgements First of all, I would like to thank my advisor Dr. L. Sunil Chandran for all the help and support. I sincerely thank you for guiding me through my formative years in research. I would like to thank my friends from the theory lab - Subramanya, Deepak, Manu, Jasine, Rogers, Abhijit, Abhijin and others; who were a part and parcel of my life in the institute. Learning had never been so much fun before. I was able to lead a balanced life thanks to you folks. I remember discussions on both Cantor functions and Bach's partitas with the same fondness. I deeply appreciate all of your love and encouragement. I would also like to thank all the non-teaching staff members of our department due to whose hard work we have a smooth academic life. I would like to thank Dr. Y. Narahari for those wonderful words of wisdom and encouragement. Finally, I would like to thank Dr. Ashok Rao who has been a constant support throughout. i Abstract The minimum number of colors required to color the edges of a graph so that any two distinct vertices are connected by at least one path in which no two edges are colored the same is called its rainbow connection number. This graph parameter was introduced by Chartrand et al. in 2008. The problem has garnered considerable interest and several variants of the initial version have since been introduced. The rainbow connection number of a connected graph G is denoted by rc(G). It can be shown that the rainbow connection number of a tree on n vertices is n − 1. Hence jGj − 1 is an upper bound for rc(G) of any non-trivial graph G. For all non-trivial, bridge-less and connected graphs G, Basavaraju et al. showed that rc(G) can be upper-bounded by a quadratic function of its radius. In addition they also proved the tightness of the bound. It is clear that we cannot hope to get an upper-bound better than jGj − 1 in the case of graphs with bridges. An immediate and natural question is the following: Are there classes of bridge-less graphs whose rainbow connection numbers are linear functions of their radii? This question is of particular interest since the diameter is a trivial lower bound for rc(G). We answer in affirmative to the above question. In particular we studied three (graph) product operations (Cartesian, Lexicographic and Strong) and the graph powering operation. We were able to show that the rainbow connection number of the graph resulting from any of the above graph operations is upper-bounded by 2r(G) + c, where r(G) is radius of the resultant graph and c 2 f0; 1; 2g. ii Contents Acknowledgements i Abstract ii 1 Introduction 1 1.1 Rainbow Coloring: An Overview . 1 1.2 Graph Operations and Rainbow Coloring . 15 1.3 Our Results . 18 2 Preliminaries and Definitions 19 3 Graph Operations: An Introduction 22 3.1 The k − th Powering of a Graph H ..................... 22 3.2 The Cartesian Product of Two Graphs, G0 and H0 . 23 3.3 The Lexicographic Product of Two Graphs G0 and H . 25 3.4 The Strong Product of Two Graphs G0 and H0 . 26 4 Rainbow Connection Number of Graph Power and Graph Products 28 4.1 Rainbow Connection Number of the k-th Power of a Graph H . 28 4.2 Rainbow Connection Number of the Cartesian Product of Two Graphs G0 and H0 ..................................... 31 4.3 Rainbow Connection Number of the Lexicographic Product of Two Non- Trivial Graphs G0 and H ........................... 34 4.4 Rainbow Connection Number of the Strong Product of Two Non-Trivial, Connected Graphs G0 and H0 ............................... 41 Bibliography 50 iii Chapter 1 Introduction 1.1 Rainbow Coloring: An Overview Let G be a non-trivial, connected graph and let c : E(G) ! f1; 2; : : : ; kg be an edge coloring of G such that adjacent edges may be given the same color. We say that G is rainbow colored with respect to the edge coloring c, if every pair of vertices are connected by a rainbow path.A rainbow path with respect to an edge coloring is a path in G such that no two edges of the path are colored using the same color. The least number of colors required to rainbow color G is called the rainbow connection number of G and is denoted by rc(G). We familiarize ourselves with the concept by working with a few simple and small graphs. Let G be a non-trivial, complete graph on n vertices i.e. G = Kn. Let f be an edge coloring of G that colors every edge of G using the same color. Given any two vertices u; v 2 V (G), the edge (u; v) 2 E(G) is a trivial rainbow path between them. Thus we can say that f is a valid rainbow coloring of G. Since f uses just one color to edge-color G(= Kn), we have that rc(Kn) = 1 for any n ≥ 2. As our next example we consider a path, Pn, on n vertices. Let f be an edge coloring of Pn using < n − 1 number of colors. Observe that the end vertices of Pn cannot be rainbow connected with respect to f. Hence f is not a valid rainbow coloring of Pn. Thus we can say that any 1 Chapter 1. Introduction 2 valid rainbow coloring of Pn must use at least n − 1 colors i.e. rc(Pn) ≥ n − 1. But if we color all the edges of a graph distinctly then it is a trivial rainbow coloring of the graph. Hence rc(Pn) = n − 1. An immediate generalization of a path is a tree. Is the rainbow connection number of a tree also n − 1? where n is the number of vertices in the tree. We return to this question later in this section. As a final example consider G = K1;n, a star graph on n + 1 vertices and n edges. It can be verified that rc(K1;n) = n. Let us compare the rainbow connection numbers of the graphs considered hitherto to their chromatic indices (edge chromatic numbers). We know that rc(Kn≥2) = 1 while 0 0 χ (Kn≥2) ≥ n − 1. Given a path, Pn, we know that rc(Pn) = n − 1 while χ (Pn) = 2. In 0 the case of a star graph, K1;n, we have that χ (K1;n) = rc(K1;n) = n. In general we may conclude that the rainbow connection number and the chromatic index of a graph need not be related. The aim of this section is to present an overview of the evolution of the problem and the associated concepts. We also discuss a few interesting variants of the problem. Since the results are numerous, we do not list them all. A more exhaustive survey has been done by Li et al. [3]. There the results are discussed in considerable depth. For an introduction to rainbow connection number the reader is referred to the book by Chartrand and Zhang [4]. Rainbow coloring of was first introduced by Chartrand et al. in 2008 [5]. In the same paper they introduced a variant called the strong rainbow connection number of a graph. Let G be a non-trivial, connected graph and let f be an edge coloring of G. We say that G is strongly rainbow colored with respect to the edge coloring f if every pair of vertices, u and v, are connected by a rainbow path of length distG(u; v) (distance between u and v in G). The minimum number of colors required to strongly rainbow color a graph is called the strong rainbow connection number and is denoted by src(G). Since any strong rainbow coloring of G is also a valid rainbow coloring of it, the following relation holds: rc(G) ≤ src(G). We know that 9 u; v 2 V (G) such that distG(u; v) = diam(G). Hence Chapter 1. Introduction 3 the diameter is a trivial lower bound for the rainbow connection number. We thus have that diam(G) ≤ rc(G) ≤ src(G). Let G be any non-trivial, connected graph. Let G0 be a connected, spanning subgraph of G. Let f be a rainbow coloring of G0. Any edge e 2 E(G)nE(G0) is arbitrarily colored, only using colors used by f. This coloring is a rainbow coloring of G, hence we have that 0 rc(G) ≤ rc(G ). Given a tree T and a rainbow coloring fT of T , we claim that fT uses jT j − 1 number of colors. The proof of the claim involves arguing that if lesser number of colors are used then there exist a pair of vertices that are not rainbow connected. A simple implication of the claim is that rc(G) ≤ jGj − 1. Suppose all the edges of G are colored distinctly then G is trivially both rainbow colored and strongly rainbow colored. Hence we have the following trivial upper bound, rc(G) ≤ src(G) ≤ jE(G)j. It may be noted that the equality holds only when G is a tree. We now mention a few results by Chartrand et al.