TY - JOUR

T1 - Cycles and Matchings in Randomly Perturbed Digraphs and Hypergraphs

AU - Krivelevich, Michael

AU - Kwan, Matthew

AU - Sudakov, Benny

N1 - Publisher Copyright:
© 2016 Cambridge University Press.

PY - 2016/11/1

Y1 - 2016/11/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84961203440&partnerID=8YFLogxK

U2 - 10.1017/S0963548316000079

DO - 10.1017/S0963548316000079

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AN - SCOPUS:84961203440

SN - 0963-5483

VL - 25

SP - 909

EP - 927

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 6

ER -