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The chromatic number of random graphs. (English) Zbl 0666.05033
Let $$G_ p$$ denote a random graph with vertex set $$\{$$ 1,2,...,n$$\}$$, in which the edges are chosen independently and with probability $$p=p(n)$$, $$0<p<1$$. The main aim of the paper is the study of the clique number $$cl(G_ p)$$ and chromatic number $$\chi (G_ p)$$ of $$G_ p$$ for p constant. The notation and terminology used are standard. It is proved that for a fixed probability p, $$0<p<1$$, almost every random graph $$G_{n,p}$$ has chromatic number $(+o(1))\log (1/(1-p))\frac{n}{\log n}.$ This result improves the estimation of $$\chi (G_ p)$$ presented in D. W. Matula [ibid. 7, 275-284 (1987; Zbl 0648.05049)] and is best possible. Further, some results concerning independent sets and the chromatic number of graphs $$G_ p$$ for varying probabilities are sketched.
Reviewer: U.Baumann

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C80 Random graphs (graph-theoretic aspects)
##### Keywords:
random graph; clique number; chromatic number; independent sets
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##### References:
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