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
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
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 -