Introduction to Competitive Graph Coloring

Abstract

We consider a collection of coloring games played on finite graphs. These games all involve two players, Alice and Bob, alternating coloring the vertices (or edges or both) from a finite set of colors. At each step, the players must use legal colors, where the definition of legal depends on which variation of the game is being considered. The first player, Alice, wants to ensure that all relevant elements (vertices and/or edges) are colored. The second player, Bob, wants to force a situation in which there is an uncolored element that cannot be legally colored. For each version of the game the parameter of interest finds the least number of colors that must be available so that Alice has a winning strategy for the game. We examine this game for a number of classes of graphs, including trees, forests, outerplanar graphs, and planar graphs. We also note a number of unexpected interesting properties these parameters have and highlight how rich these games can be to explore for undergraduate researchers and their mentors.