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Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  14 March 2016

MICHAEL KRIVELEVICH ,
MATTHEW KWAN  and
BENNY SUDAKOV
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 6997801, Israel (e-mail: krivelev@post.tau.ac.il)
MATTHEW KWAN
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland (e-mail: matthew.kwan@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland (e-mail: matthew.kwan@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
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Abstract

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We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to a dense k-uniform hypergraph ensures the (asymptotically almost sure) existence of a perfect matching or a loose Hamilton cycle. The proof involves an interesting application of Szemerédi's Regularity Lemma, which might be independently useful. We next prove that digraphs with certain strong expansion properties are pancyclic, and use this to show that adding a linear number of random edges typically makes a dense digraph pancyclic. Finally, we prove that perturbing a certain (minimum-degree-dependent) number of random edges in a tournament typically ensures the existence of multiple edge-disjoint Hamilton cycles. All our results are tight.

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C35: Extremal problems

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 6 , November 2016 , pp. 909 - 927
Copyright
Copyright © Cambridge University Press 2016 

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