Monochromatic cycle covers in random graphs

Abstract

A classic result of Erdos, Gyarfas and Pyber states that for every coloring of the edges of K-n with r colors, there is a cover of its vertex set by at most f(r)=O(r2logr) vertex-disjoint monochromatic cycles. In particular, the minimum number of such covering cycles does not depend on the size of K-n but only on the number of colors. We initiate the study of this phenomenon in the case where K-n is replaced by the random graph G(n,p). Given a fixed integer r and p=p(n)>= n-1/r+epsilon, we show that with high probability the random graph G similar to G(n,p) has the property that for every r-coloring of the edges of G, there is a collection of f '(r)=O(r8logr) monochromatic cycles covering all the vertices of G. Our bound on p is close to optimal in the following sense: if pMUCH LESS-THAN(logn/n)1/r, then with high probability there are colorings of G similar to G(n,p) such that the number of monochromatic cycles needed to cover all vertices of G grows with n