Optimal covers with Hamilton cycles in random graphs
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by
Dan Hefetz and Daniela Kühn and John Lapinskas and Deryk Osthus
2013
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 delta(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 log^117n / n
< p < 1-n^-1/8, a.a.s. the edges of G_n,p can be covered by
Delta(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+. Our proof is based on a result of Knox, Kühn and Osthus on
packing Hamilton cycles in pseudorandom graphs.
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