Coloring triangle-free graphs and network games

A proper vertex coloring of a graph is an assignment of colors to all vertices such that adjacent vertices have distinct colors. The chromatic number [chi](G) of a graph G is the minimum number of colors required for a proper vertex coloring. In this dissertation, we give some background on graph coloring and applications of the probabilistic method to graph coloring problems. We then give three results about graph coloring. * Let G be a triangle-free graph with maximum degree [Delta](G). We show that the chromatic number [chi](G) is less than 67(1 + o(1))[Delta;] log [Delta]. This number is best possible up to a constant factor for triangle-free graphs. * We give a randomized algorithm that properly colors the vertices of a triangle- free graph G on n vertices using O([Delta](G)/ log [Delta](G)) colors. The algorithm takes O(n [Delta]² log [Delta] (G)) time and succeeds with high probability, provided [Delta](G) is greater than log¹⁺[epsilon]) n for a positive constant [epsilon]. We analyze a network(graph) coloring game. In each round of the game, each player, as a node in a network G, randomly chooses one of the available colors that is different from all colors played by its neighbors in the previous round. We show that the coloring game converges to its Nash equilibrium if the number of colors is at least [Delta](G) + 2. Examples are given for which convergence does not happen with [Delta](G) + 1 colors. We also show that with probability at least 1 - [delta], the number of rounds required is O(log(n/[delta]))