On Sum Coloring of Graphs

On Sum Coloring of Graphs Mohammadreza Salavatipour Master of Science Graduate Department of Computer Science University of Toronto 2000 The sum coloring problem asks to nd a vertex coloring of a given graph G, using natural numbers, such that the total sum of the colors of vertices is minimized amongst all proper vertex colorings of G. This minimum total sum is the chromatic sum of the graph, (G), and a coloring which achieves this total sum is called an optimum coloring. There are some graphs for which the optimum coloring needs more colors than indicated by the chromatic number. The minimum number of colors needed in any optimum coloring of a graph is called the strength of the graph, which we denote by s(G). Trivially (G) s(G). In this thesis we present various results about the sum coloring problem. We prove the NP-Hardness of nding the vertex strength for graphs with = 6 and also give some logarithmic upper bounds for the vertex strength of graphs with small chromatic number. We also prove that the sum coloring problem is NP-complete for split graphs. Polynomial time algorithms are presented for the sum coloring of k-split graphs, P4-reducible graphs, chain bipartite graphs, and cobipartite graphs. We can extend the idea of sum coloring to edge coloring and de ne the edge chromatic sum and the edge strength of a graph similarly. We prove that the edge sum coloring and the edge strength problems are both NP-complete for cubic graphs. Also we give a polynomial time algorithm to solve the edge sum coloring problem on trees, and show that using the Monadic Second Order Logic we can solve this problem on partial k-trees with bounded degree in linear time. ii Acknowledgements I do not see the completion of my M.Sc program as a personal achievement; there are many people who have generously helped me along the way. I would like to thank the people who have supported and encouraged me. First and foremost, I would like to thank my advisor, Derek Corneil. He is an excellent advisor and a great researcher. His deep experience, sharpness, and insight in di erent areas of graph theory provided me a great source of inspiration. He has helped me with his great suggestions, good questions, and new ideas whenever I was distracted. This thesis would not have been possible without his supervision. I also thank Mike Molloy, the second reader of this thesis, for his helpful comments. From the very beginning years of my study, I have received great support from my parents. After being far from my family, and from my wife for a while, I realized that I love them more than I could imagine it before. I don't know how to thank my family, and in particular my mother for her unconditional love. And about my wife, I have no words to thank her who is the primary reason for my happy life. For many useful discussions, I would like to thank my fellow graduate students Mohammad Mahdian, Kiumars Kaveh, and Babak Farzad. I can not forget my primary teachers in university, who directed me to this area of science, Dr. Ebad Mahmoodian and Dr. Mohammad Ghodsi. Finally, I thank the members of the department who made the department a great place to work and study. I also thank the University of Toronto for nancial assistance. iii .Ý÷‘“Âúõ ¤¢‘õ ,ôÃþÃä ÂÆÞû‚“ ÝþÀ֗ 1 iv