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 pn −1+ɛ. Our proof is based on a result of Knox, Kühn and Osthus on packing Hamilton cycles in pseudorandom graphs.

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Correspondence to Dan Hefetz.

Additional information

The research leading to these results was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement n. 258345 (D. Kühn).

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Hefetz, D., Kühn, D., Lapinskas, J. et al. Optimal covers with Hamilton cycles in random graphs. Combinatorica 34, 573–596 (2014). https://doi.org/10.1007/s00493-014-2956-z

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Mathematics Subject Classification (2000)

  • 05C45
  • 05C70
  • 05C80