The Game Chromatic Number of Dense Random Graphs

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 the 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 is asymptotically twice as large as the ordinary chromatic number χ(Gn,p).

Introduction to Random Graphs

“…Bohman, Frieze and Sudakov [123] studied χ g for dense random graphs and proved that for such graphs, χ g is within a constant factor of the chromatic number. Keusch and Steger [501] proved that this factor is asymptotically equal to two. Frieze, Haber and Lavrov [346] extended the results of [123] to sparse random graphs.…”

On Monotonicity in Maker-Breaker Graph Colouring Games

“…Despite the difficulty of computing χ g (G) in general, the game chromatic number of various classes of graph has received significant attention. In particular, the class of planar graphs and various subclasses thereof have been studied extensively [10,15,21,19,22,18,20], and there has also been work on random graphs in various regimes [3,14,11].…”

The game chromatic number of a random hypergraph

“…The papers by Bohman, Frieze and Sudakov [6], Frieze, Haber and Lavrov [11] and by Keusch and Steger [16] discuss the game chromatic number of random graphs. In this paper we discuss the game chromatic number of random hypergraphs.…”

“…Here A, B color the vertices of H consecutively and a coloring is proper if there is no e ∈ E such that all vertices in e have the same color. This problem has hardly been studied, even in a deterministic setting, but we feel it is of interest to extend the results of [6], [11] and [16] to this setting.…”