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Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

Published online by Cambridge University Press:  14 March 2016

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)

Abstract

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.

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Copyright
Copyright © Cambridge University Press 2016 

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References

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