The Eternal Game Chromatic Number of a Graph

TL;DR

This paper introduces the concept of the eternal game chromatic number, a dynamic graph coloring measure where players alternate coloring and re-coloring vertices to maintain proper coloring indefinitely.

Contribution

It defines the eternal game chromatic number, explores its properties, and analyzes its behavior across various elementary graph classes, introducing new variations of the game.

Findings

Eternal game chromatic number varies across graph classes

Player strategies depend on the number of colors available

New variations of the game exhibit distinct behaviors

Abstract

Game coloring is a well-studied two-player game in which each player properly colors one vertex of a graph at a time until all the vertices are colored. An `eternal' version of game coloring is introduced in this paper in which the vertices are colored and re-colored from a color set over a sequence of rounds. In a given round, each vertex is colored, or re-colored, once, so that a proper coloring is maintained. Player 1 wants to maintain a proper coloring forever, while player 2 wants to force the coloring process to fail. The eternal game chromatic number of a graph $G$ is defined to be the minimum number of colors needed in the color set so that player 1 can always win the game on $G$. We consider several variations of this new game and show its behavior on some elementary classes of graphs.