## Abstract

Given a graph G and an integer k, two players take turns coloring the vertices of G one by one using k colors so that neighboring vertices get different colors. The first player wins iff at the end of the game all the vertices of G are colored. The game chromatic number χg(G) is the minimum k for which the first player has a winning strategy. The paper [T. Bohman, A. M. Frieze, and B. Sudakov, Random Structures Algorithms, 32 (2008), pp. 223-235] began the analysis of the asymptotic behavior of this parameter for a random graph G_{n,p}. This paper provides some further analysis for graphs with constant average degree, i.e., np = O(1), and for random regular graphs. We show that with high probability (w.h.p.) c_{1}χ(G_{n,p}) ≤ χg(G_{n,p}) ≤ c_{2}χ(Gn,p) for some absolute constants 1 < c_{1} < c_{2}. We also prove that if G _{n,3} denotes a random n-vertex cubic graph, then w.h.p. χg(G _{n,3}) = 4.

Original language | English |
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Pages (from-to) | 768-790 |

Number of pages | 23 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 27 |

Issue number | 2 |

DOIs | |

State | Published - 2013 |

Externally published | Yes |

## Keywords

- Game chromatic number
- Random graphs
- Sparse