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Packing, counting and covering Hamilton cycles in random directed graphs

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Abstract

A Hamilton cycle in a digraph is a cycle that passes through all the vertices, where all the arcs are oriented in the same direction. The problem of finding Hamilton cycles in directed graphs is well studied and is known to be hard. One of the main reasons for this is that there is no general tool for finding Hamilton cycles in directed graphs comparable to the so-called Posá ‘rotation-extension’ technique for the undirected analogue. Let D(n, p) denote the random digraph on vertex set [n], obtained by adding each directed edge independently with probability p. Here we present a general and a very simple method, using known results, to attack problems of packing and counting Hamilton cycles in random directed graphs, for every edge-probability p > logC(n)/n. Our results are asymptotically optimal with respect to all parameters and apply equally well to the undirected case.

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Authors and Affiliations

  1. Department of Mathematics, Yale University, New Haven, CT, 06520, USA

    Asaf Ferber

  2. Department of Mathematics, MIT, Cambridge, MA, 02139-4307, USA

    Asaf Ferber

  3. School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel

    Gal Kronenberg & Eoin Long

Authors
  1. Asaf Ferber
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  2. Gal Kronenberg
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  3. Eoin Long
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Correspondence to Gal Kronenberg.

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Ferber, A., Kronenberg, G. & Long, E. Packing, counting and covering Hamilton cycles in random directed graphs. Isr. J. Math. 220, 57–87 (2017). https://doi.org/10.1007/s11856-017-1518-7

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  • DOI: https://doi.org/10.1007/s11856-017-1518-7

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