The Chromatic Number of Random Regular Graphs

Given any integer d ≥ 3, let k be the smallest integer such that d < 2k log k. We prove that with high probability the chromatic number of a random d-regular graph is k, k+1, or k+2.

[1]  Florent Krzakala,et al.  Threshold values, stability analysis and high-q asymptotics for the coloring problem on random graphs , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[2]  Cristopher Moore,et al.  On the 2-Colorability of Random Hypergraphs , 2002, RANDOM.

[3]  Tomasz Luczak The chromatic number of random graphs , 1991, Comb..

[4]  Béla Bollobás,et al.  Random Graphs , 1985 .

[5]  A. Naor,et al.  The two possible values of the chromatic number of a random graph , 2005 .

[6]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[7]  Cristopher Moore,et al.  The Asymptotic Order of the k-SAT Threshold , 2002 .

[8]  N. Wormald,et al.  Models of the , 2010 .

[9]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[10]  Alan M. Frieze,et al.  On the independence and chromatic numbers of random regular graphs , 1992, J. Comb. Theory, Ser. B.

[11]  Cristopher Moore,et al.  The asymptotic order of the random k-SAT threshold , 2002, The 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002. Proceedings..

[12]  S. Ross A random graph , 1981 .

[13]  C. Moore,et al.  Almost All Graphs of Degree 4 are 3-colorable , 2001 .

[14]  Tomasz Luczak A note on the sharp concentration of the chromatic number of random graphs , 1991, Comb..