TY - JOUR

T1 - Random regular graphs of non-constant degree

T2 - Concentration of the chromatic number

AU - Ben-Shimon, Sonny

AU - Krivelevich, Michael

N1 - Funding Information:
We would like to thank Alan Frieze for bringing Theorem 3.2 to our attention, Nick Wormald and Xavier Pérez-Giménez for providing us several of the references below and the anonymous referees for their helpful comments and corrections. Second author’s research was supported in part by a USA–Israel BSF grant, by a grant from the Israel Science Foundation, and by Pazy Memorial Award.

PY - 2009/6/28

Y1 - 2009/6/28

N2 - In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model Gn, d for d = o (n1 / 5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G (n, p) with p = frac(d, n). Our proof is largely based on ideas of Alon and Krivelevich who proved this two-point concentration result for G (n, p) for p = n- δ where δ > 1 / 2. The main tool used to derive such a result is a careful analysis of the distribution of edges in Gn, d, relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model.

AB - In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model Gn, d for d = o (n1 / 5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G (n, p) with p = frac(d, n). Our proof is largely based on ideas of Alon and Krivelevich who proved this two-point concentration result for G (n, p) for p = n- δ where δ > 1 / 2. The main tool used to derive such a result is a careful analysis of the distribution of edges in Gn, d, relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model.

KW - Chromatic number concentration

KW - Edge distribution

KW - Random regular graphs

UR - http://www.scopus.com/inward/record.url?scp=67349096845&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2008.12.014

DO - 10.1016/j.disc.2008.12.014

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AN - SCOPUS:67349096845

SN - 0012-365X

VL - 309

SP - 4149

EP - 4161

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 12

ER -