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 -