Majority coloring games
A vertex coloring of graph satisfies the majority rule, if for each vertex $v$ at most half of its neighbors receive the same color as $v$. A coloring which satisfies the majority rule is called majority coloring. The problem of such colorings was introduced in [1,5] and continued with different variants in [2,4]. We consider its game version. For given graph $G$ and color set $C$ two players, Alice and Bob, in alternating turns color vertices with respect to the majority rule. Alice wins when every vertex becomes colored, while goal for Bob is to create a vertex which cannot be colored with any color belonging to the set $C$ without breaking the majority rule. We show that if the color set $C$ is finite, there exists a graph $G$ on which Bob has winning strategy. Number of colors that Alice needs to have to win the game on graph $G$ is clearly bounded by game coloring number of $G$. We improve that bound for some classes of graphs.
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