Optimal covers with Hamilton cycles in random graphs

A packing of a graph G with Hamilton cycles is a set of edgedisjoint Hamilton cycles in G. Such packings have been studied intensively and recent results imply that a largest packing of Hamilton cycles in Gn,p a.a.s. has size bδ(Gn,p)/2c. 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 log 117 n n ≤ p ≤ 1− n−1/8, a.a.s. the edges of Gn,p can be covered by d∆(Gn,p)/2e Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szabó, which holds for p ≥ n−1+ε. Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.