  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.
