TY - JOUR

T1 - On the game chromatic number of sparse random graphs

AU - Frieze, Alan

AU - Haber, Simcha

AU - Lavrov, Mikhail

PY - 2013

Y1 - 2013

N2 - 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 Gn,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.) c1χ(Gn,p) ≤ χg(Gn,p) ≤ c2χ(Gn,p) for some absolute constants 1 < c1 < c2. We also prove that if G n,3 denotes a random n-vertex cubic graph, then w.h.p. χg(G n,3) = 4.

AB - 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 Gn,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.) c1χ(Gn,p) ≤ χg(Gn,p) ≤ c2χ(Gn,p) for some absolute constants 1 < c1 < c2. We also prove that if G n,3 denotes a random n-vertex cubic graph, then w.h.p. χg(G n,3) = 4.

KW - Game chromatic number

KW - Random graphs

KW - Sparse

UR - http://www.scopus.com/inward/record.url?scp=84880401057&partnerID=8YFLogxK

U2 - 10.1137/120861953

DO - 10.1137/120861953

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AN - SCOPUS:84880401057

SN - 0895-4801

VL - 27

SP - 768

EP - 790

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

IS - 2

ER -