Results and open problems in matchings in regular graphs

TL;DR

This survey explores bounds on the number of k-matchings in regular graphs, discussing known results, conjectures, and open problems across bipartite, non-bipartite, and infinite graphs, highlighting recent advances like the Lovász-Plummer conjecture.

Contribution

It provides a comprehensive overview of current bounds, conjectures, and open problems related to matchings in regular graphs, including recent proofs and continuous analogs.

Findings

Upper matching conjecture holds for perfect matchings

Lovász-Plummer conjecture proved for cubic bridgeless graphs

Discussion of minimum haffnians and infinite regular graphs

Abstract

This survey paper deals with upper and lower bounds on the number of $k$-matchings in regular graphs on $N$ vertices. For the upper bounds we recall the upper matching conjecture which is known to hold for perfect matchings. For the lower bounds we first survey the known results for bipartite graphs, and their continuous versions as the van der Waerden and Tverberg permanent conjectures and its variants. We then discuss non-bipartite graphs. Little is known beyond the recent proof of the Lov\'asz-Plummer conjecture on the exponential growth of perfect matchings in cubic bridgeless graphs. We discuss the problem of the minimum of haffnians on the convex set of matrices, whose extreme points are the adjacency matrices of subgraphs of the complete graph corresponding to perfect matchings. We also consider infinite regular graphs. The analog of $k$-matching is the $p$-monomer entropy, where $p\in [0,1]$ is the density of the number of matchings.