# Competitive Coloring: A Structural Approach

2016

Li-Mei Lim

Aaron Williams

Eric Kramer

## Abstract

This thesis is an exploration of the graph-coloring game with a focus on structural graph theory, particularly on the relationship between local and global properties of graphs. A graph is simply a collection of dots with lines connecting some of the dots. The graphcoloring game is a two person game that can be played on any graph as follows. The two players, Alice and Bob, are given a graph and k colors with which to color the dots. Alice begins and then they alternate turns. In each turn, a player colors one uncolored dot in the graph without making any of the lines have both endpoints colored the same color. If at any point in the game, no such move is possible, Bob wins the game. Otherwise, if gameplay continues until all of the dots are colored, Alice wins. We are interested in how many colors Alice needs in order to win the game when Bob plays optimally. This number is called the game chromatic number of the graph. We study how the game chromatic number relates to other graph parameters in order to improve our understanding of what structures the game chromatic number detects. We are able to give partial answers about both the local and global structure of graphs for which the game chromatic number detects nothing more than the clique number. In addition, we show that a particular local structure bears most of the responsibility for making the game chromatic number and clique number distinct. Finally, we provide examples of graphs for which the game chromatic number depends only on the maximum number of lines connected to a single dot in the graph. Before presenting these results in their precise mathematical terms, we give a brief introduction to graph theory as a whole followed by a survey of previous results about the graph-coloring game.