Cycles and matchings in randomly perturbed digraphs and hypergraphs
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 kuniform 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 (minimumdegreedependent) number of random edges in a tournament typically ensures the existence of multiple edgedisjoint Hamilton cycles. All our results are tight.
 Publication:

arXiv eprints
 Pub Date:
 January 2015
 arXiv:
 arXiv:1501.04816
 Bibcode:
 2015arXiv150104816K
 Keywords:

 Mathematics  Combinatorics
 EPrint:
 17 pages, 2 figures. Addressed referee's comments, streamlined proof of Lemma 6