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On two Hamilton cycle problems in random graphs

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Abstract

We study two problems related to the existence of Hamilton cycles in random graphs. The first question relates to the number of edge disjoint Hamilton cycles that the random graph G n,p contains. δ(G)/2 is an upper bound and we show that if p ≤ (1 + o(1)) ln n/n then this upper bound is tight whp. The second question relates to how many edges can be adversarially removed from G n,p without destroying Hamiltonicity. We show that if p ≥ K ln n/n then there exists a constant α > 0 such that whp G − H is Hamiltonian for all choices of H as an n-vertex graph with maximum degree Δ(H) ≤ αK ln n.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Alan Frieze

  2. Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Michael Krivelevich

Authors
  1. Alan Frieze
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  2. Michael Krivelevich
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Corresponding author

Correspondence to Alan Frieze.

Additional information

Research supported in part by NSF grant CCR-0200945.

Research supported in part by USA-Israel BSF Grant 2002-133 and by grant 526/05 from the Israel Science Foundation.

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Frieze, A., Krivelevich, M. On two Hamilton cycle problems in random graphs. Isr. J. Math. 166, 221–234 (2008). https://doi.org/10.1007/s11856-008-1028-8

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  • DOI: https://doi.org/10.1007/s11856-008-1028-8

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