Optimal covers with Hamilton cycles in random graphs

Abstract

A packing of a graph G with Hamilton cycles is a set of edge-disjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in G _n,p a.a.s. has size ⌊δ(G _n,p )/2⌋. Glebov, Krivelevich and Szabó recently initiated research on the ‘dual’ problem, where one asks for a set of Hamilton cycles covering all edges of G. Our main result states that for $\tfrac{{log^{117} n}} {n} \leqslant p \leqslant 1 - n^{ - 1/8}$, a.a.s. the edges of G _n,p can be covered by ⌈Δ (G _n,p )/2⌉ Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szabó, which holds for p ≥ n ^−1+ɛ. Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.