A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every
coloring of the edges of Kn 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 Kn but only on the number of colors. We initiate the study of this
phenomena in the case where Kn 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(r8logr) monochromatic cycles covering all the
vertices of G. Our bound on p is close to optimal in the following sense:
if p≪(logn/n)1/r, 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.Comment: 24 pages, 1 figure (minor changes, added figure