Majority coloring game

Abstract

A majority coloring of a graph is a coloring of its vertices such that for each vertex v, at least half of the neighbors of v have different color than v. Let μ(G) denote the least number of colors needed for a majority coloring of a graph G. It is well known and easy to prove that μ(G)≤2 for every graph G. We consider a game-theoretic variant of this parameter, the majority game chromatic numberμg(G), defined via a two person coloring game in which one of the players tries to produce a majority coloring of the whole graph, while the other tends to prevent it. We prove that μg(G) is in general unbounded. On the other hand, it is not hard to see that μg(G) is not greater than colg(G), the game coloring number of G. We improve this trivial bound for some classes of graphs. In particular, we prove that μg(T)≤3 for every complete binary tree T. This may suggest that μg(G) is bounded for graphs with bounded coloring number col(G). However, we prove that, contrary to this intuition, μg(G) is unbounded for graphs with col(G)=3.