The eternal game chromatic number of a graph
Game coloring is a 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 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. The goal of this paper is to introduce this problem, consider several variations of this game, show its behavior on some elementary classes of graphs, and make some conjectures.
Journal of Combinatorial Mathematics and Combinatorial Computing
Klostermeyer, W.F., Mendoza, H. (2021) The eternal game chromatic number of a graph. Journal of Combinatorial Mathematics and Combinatorial Computing, 116, 13-25.