2019
DOI: 10.1002/rsa.20835
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Rainbow matchings in Dirac bipartite graphs

Abstract: We 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.

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Cited by 16 publications
(23 citation statements)
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“…This topic has received considerable attention recently, with probabilistic tools and techniques from extremal graph theory allowing for major progress on longstanding problems. In this context, natural (rainbow) structures to seek include matchings, Hamilton cycles, spanning trees and triangle factors (see e.g., ). It is easy to see that results on edge‐colored Kn also imply results on patterns in symmetric n×n arrays.…”
Section: Introduction and Our Resultsmentioning
confidence: 99%
“…This topic has received considerable attention recently, with probabilistic tools and techniques from extremal graph theory allowing for major progress on longstanding problems. In this context, natural (rainbow) structures to seek include matchings, Hamilton cycles, spanning trees and triangle factors (see e.g., ). It is easy to see that results on edge‐colored Kn also imply results on patterns in symmetric n×n arrays.…”
Section: Introduction and Our Resultsmentioning
confidence: 99%
“…Our result and those of [4,8] suggest that for any Dirac-type problem, the rainbow problem for bounded colourings should have asymptotically the same degree threshold as the problem with no colours. In particular, it may be interesting to establish this for Hamilton cycles in hypergraphs (i.e.…”
Section: Discussionmentioning
confidence: 50%
“…For example, Erdős and Spencer [7] showed the existence of a rainbow perfect matching in any edge-colouring of K n,n that is (n − 1)/16-bounded, meaning that are at most (n−1)/16 edges of any given colour. Coulson and Perarnau [4] recently obtained a Dirac-type version of this result, showing that any o(n)-bounded edge-colouring of a subgraph of K n,n with minimum degree at least n/2 has a rainbow perfect matching. One could consider this a 'local resilience' version (as in [27]) of the Erdős-Spencer theorem.…”
mentioning
confidence: 99%
“…The central question is under which conditions on G and its edge coloring a rainbow copy of H is guaranteed. Here, H is usually a spanning subgraph such as a perfect matching , Hamilton cycle , spanning tree , or a general bounded degree graph . Closely related questions concern properly colored subgraphs and rainbow decompositions, which we shall discuss briefly in Section 1.3.…”
Section: Introductionmentioning
confidence: 99%