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 \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.
Keywords
Cite
@article{arxiv.1203.3868,
title = {Optimal covers with Hamilton cycles in random graphs},
author = {Dan Hefetz and Daniela Kühn and John Lapinskas and Deryk Osthus},
journal= {arXiv preprint arXiv:1203.3868},
year = {2013}
}
Comments
final version of paper (to appear in Combinatorica)