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 $\chi_g(G)$ 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.) $c_1\chi(G_{n,p})\leq \chi_g(G_{n,p})\leq c_2\chi(G_{n,p})$ 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. $\chi_g(G_{n,3})=4$.

Keywords

  1. game chromatic number
  2. random graphs
  3. sparse

MSC codes

  1. 05C80
  2. 05C57

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 768 - 790
ISSN (online): 1095-7146

History

Submitted: 10 January 2012
Accepted: 10 December 2012
Published online: 18 April 2013

Keywords

  1. game chromatic number
  2. random graphs
  3. sparse

MSC codes

  1. 05C80
  2. 05C57

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