Abstract
Suppose that two players take turns coloring the vertices of a given graph G with k colors. In each move the current player colors a vertex such that neighboring vertices get different colors. The first player wins this game if and only if at the end, all vertices are colored. The game chromatic number χg(G) is defined as the smallest k for which the first player has a winning strategy.Recently, Bohman, Frieze and Sudakov [Random Structures and Algorithms 2008] analysed the game chromatic number of random graphs and obtained lower and upper bounds of the same order of magnitude. In this paper we improve existing results and show that with high probability, the game chromatic number χg(Gn,p) of dense random graphs with p ≥ e-o(log n) is asymptotically twice as large as the ordinary chromatic number χ(Gn,p).
Highlights
Let k be a set of colors and let G be a graph with initially uncolored vertices
Suppose that two players take turns coloring the vertices of a given graph G with k colors
Maker always wins the game if the number of colors is larger than the maximum degree of G, because no vertex can the electronic journal of combinatorics 21(4) (2014), #P4.47 run out of colors
Summary
Let k be a set of colors and let G be a graph with initially uncolored vertices. Maker and Breaker play the following game: They alternately take turns and color one vertex per move such that the coloring remains proper, i.e., two neighbors never receive the same color. The game chromatic number χg(G) is defined as the smallest k for which Maker has a winning strategy, no matter how Breaker plays. Maker always wins the game if the number of colors is larger than the maximum degree of G, because no vertex can the electronic journal of combinatorics 21(4) (2014), #P4.47 run out of colors. We study the game chromatic number of the Erdős–Rényi random graph model Gn,p = (V, E). We improve the upper bound for dense random graphs. This result holds for constant values of p. Together with the first statement of Theorem 1 it implies the asymptotic value of the game chromatic number for dense random graphs: Corollary 3.
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