Equitable coloring of random graphs

Abstract

AbstractAn equitable coloring of a graph is a proper vertex coloring such that the sizes of any two color classes differ by at most one. The least positive integer k for which there exists an equitable coloring of a graph G with k colors is said to be the equitable chromatic number of G and is denoted by χ=(G). The least positive integer k such that for any k′ ≥ k there exists an equitable coloring of a graph G with k′ colors is said to be the equitable chromatic threshold of G and is denoted by χ=*(G). In this paper, we investigate the asymptotic behavior of these coloring parameters in the probability space G(n,p) of random graphs. We prove that if n−1/5+ϵ < p < 0.99 for some 0 < ϵ, then almost surely χ(G(n,p)) ≤ χ=(G(n,p)) = (1 + o(1))χ(G(n,p)) holds (where χ(G(n,p)) is the ordinary chromatic number of G(n,p)). We also show that there exists a constant C such that if C/n < p < 0.99, then almost surely χ(G(n,p)) ≤ χ=(G(n,p)) ≤ (2 + o(1))χ(G(n,p)). Concerning the equitable chromatic threshold, we prove that if n−(1−ϵ) < p < 0.99 for some 0 < ϵ, then almost surely χ(G(n,p)) ≤ χ=* (G(n,p)) ≤ (2 + o(1))χ(G(n,p)) holds, and if ${{\log^{1+\epsilon}n}\over {n}}$ < p < 0.99 for some 0 < ϵ, then almost surely we have χ(G(n,p)) ≤ χ=*(G(n,p)) = Oϵ(χ(G(n,p))). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009