Asymptotics of the Upper Matching Conjecture
Liviu Ilinca, Jeff Kahn

TL;DR
This paper establishes near-optimal upper bounds on the number of matchings of a given size in bipartite and regular graphs, advancing the understanding of the Upper Matching Conjecture.
Contribution
It provides the best known upper bounds for matchings in regular graphs, approaching the conjectured limits up to a small error term.
Findings
Bounds are tight up to an exponential error factor diminishing with degree d.
Progress represents the most significant advancement on the Upper Matching Conjecture to date.
Results apply to both bipartite and general graphs with specified degrees.
Abstract
We give upper bounds for the number of matchings of size in (i) bipartite graphs with specified degrees (), and (ii) general graphs with all degrees specified. In particular, for -regular, -vertex graphs, our bound is best possible up to an error factor of the form , where as . This represents the best progress to date on the "Upper Matching Conjecture" of Friedland, Krop, Lundow and Markstr\"om. Some further possibilities are also suggested.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Graph theory and applications · Advanced Graph Theory Research
