The eternal game chromatic number of random graphs

Abstract

The eternal vertex colouring game, recently introduced byKlostermeyer and Mendoza (2018), is a version of the vertex colouring game, a well studied graph game. The game is played by two players, Alice and Bob on a graph G with a set of colours {1,…,k}. The game consists of rounds, such that in each round, every vertex is coloured exactly once. In the first round, players alternate turns with Alice playing first. During their turn, each player first picks yet uncoloured vertex and then colours it by any colour so that the resulting partial colouring of the graph is proper (if at least one such colour exists). During all further rounds, players keep choosing vertices alternately. After choosing a vertex, the player assigns a colour to the vertex which is distinct from its current colour such that the resulting colouring is still proper. Each vertex retains its colour between rounds until it is recoloured. Bob wins if at any point the chosen vertex does not have a legal recolouring, while Alice wins if the game is continued indefinitely. The eternal game chromatic number χg∞(G) is the smallest number k such that Alice has a winning strategy.In this note, we study the eternal game chromatic number of random graphs. We show that with high probability χg∞(Gn,p)=(p2+o(1))n for odd n, and also for even n when p=1k for some k∈N. The upper bound applies for even n and any other value of p as well, but we conjecture in this case this upper bound is not sharp. Finally, we answer a question posed by Klostermeyer and Mendoza.