6 research outputs found
Rainbow matchings in Dirac bipartite graphs
This is the peer reviewed version of the following article: Coulson, M, Perarnau, G. Rainbow matchings in Dirac bipartite graphs. Random Struct Alg. 2019; 55: 271– 289., which has been published in final form at https://doi.org/10.1002/rsa.20835. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived VersionsWe show the existence of rainbow perfect matchings in µn-bounded edge colorings of Dirac bipartite graphs, for a sufficiently small µ¿>¿0. As an application of our results, we obtain several results on the existence of rainbow k-factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree
Counting oriented trees in digraphs with large minimum semidegree
Let be an oriented tree on vertices with maximum degree at most
. If is a digraph on vertices with minimum
semidegree , then contains as a
spanning tree, as recently shown by Kathapurkar and Montgomery (in fact, they
only require maximum degree ). This generalizes the corresponding
result by Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. We investigate the
natural question how many copies of the digraph contains. Our main
result states that every such contains at least
copies of , which is optimal. This implies
the analogous result in the undirected case.Comment: 24 page