Monochromatic cycle covers in random graphs

A classic result of Erdős, Gyárfás 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(r² log r) 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+𝜀}, we show that with high probability the random graph G ~ G(n, p) has the property that for every r-coloring of the edges of G, there is a collection of f^′(r) = O(r⁸ log r) monochromatic cycles covering all the vertices of G. Our bound on p is close to optimal in the following sense: if (Formula presented.), then with high probability there are colorings of G ~ G(n, p) such that the number of monochromatic cycles needed to cover all vertices of G grows with n.