Equitable coloring of random graphs

An 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+ǫ < p < 0.99 for some 0 < ǫ, 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−ǫ) < p < 0.99 for some 0 < ǫ, then almost surely χ(G(n, p)) ≤ χ=(G(n, p)) ≤ (2 + o(1))χ(G(n, p)) holds, and if log 1+ǫ n n < p < 0.99 for some 0 < ǫ, then almost surely we have χ(G(n, p)) ≤ χ=(G(n, p)) = Oǫ(χ(G(n, p))).