Cycles and matchings in randomly perturbed digraphs and hypergraphs

TL;DR

This paper demonstrates that modest random perturbations to dense hypergraphs and digraphs typically induce complex spanning structures like Hamilton cycles and perfect matchings, with results proven to be tight.

Contribution

It provides new theoretical results showing how random perturbations guarantee the emergence of spanning substructures in various dense discrete graphs.

Findings

Adding linear random edges ensures perfect matchings and Hamilton cycles in hypergraphs.

Random perturbations make dense digraphs pancyclic.

Perturbing tournaments yields multiple edge-disjoint Hamilton cycles.

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 Szemer\'edi'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.