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.Comment: 17 pages, 2 figures. Addressed referee's comments, streamlined proof
of Lemma
