Matchings and Independent Sets of a Fixed Size in Regular Graphs

TL;DR

This paper uses entropy methods to asymptotically bound the number of fixed-size matchings and independent sets in regular graphs, providing evidence for existing conjectures.

Contribution

It introduces entropy-based bounds that match conjectured asymptotic counts for fixed-size matchings and independent sets in regular graphs.

Findings

Bounds agree with counts in disjoint unions of complete bipartite graphs

Provides asymptotic evidence for conjectures of Friedland et al. and Kahn

Bounds derived from partition function estimates

Abstract

We use an entropy based method to study two graph maximization problems. We upper bound the number of matchings of fixed size $\ell$ in a $d$-regular graph on $N$ vertices. For $\frac{2\ell}{N}$ bounded away from 0 and 1, the logarithm of the bound we obtain agrees in its leading term with the logarithm of the number of matchings of size $\ell$ in the graph consisting of $\frac{N}{2d}$ disjoint copies of the complete bipartite graph $K_{d,d}$. This provides asymptotic evidence for a conjecture of S. Friedland {\it et al.}. We also obtain an analogous result for independent sets of a fixed size in regular graphs, giving asymptotic evidence for a conjecture of J. Kahn. Our bounds on the number of matchings and independent sets of a fixed size are derived from bounds on the partition function (or generating polynomial) for matchings and independent sets.