Optimal covers with Hamilton cycles in random graphs

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 \lfloor delta(G_n,p) /2 \rfloor. Glebov, Krivelevich and Szab\'o 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 G_n,p can be covered by \lceil Delta(G_n,p)/2 \rceil Hamilton cycles. This is clearly optimal and improves an approximate result of Glebov, Krivelevich and Szab\'o, which holds for p > n^{-1+\eps}. Our proof is based on a result of Knox, K\"uhn and Osthus on packing Hamilton cycles in pseudorandom graphs.Comment: final version of paper (to appear in Combinatorica

Completing Partial Packings of Bipartite Graphs

Given a bipartite graph $H$ and an integer $n$, let $f(n;H)$ be the smallest integer such that, any set of edge disjoint copies of $H$ on $n$ vertices, can be extended to an $H$-design on at most $n+f(n;H)$ vertices. We establish tight bounds for the growth of $f(n;H)$ as $n \rightarrow \infty$. In particular, we prove the conjecture of F\"uredi and Lehel \cite{FuLe} that $f(n;H) = o(n)$. This settles a long-standing open problem