Sharp lower bounds for the number of maximum matchings in bipartite multigraphs

TL;DR

This paper establishes sharp lower bounds on the number of maximum matchings in bipartite multigraphs under various degree and size conditions, refining classical results and providing constructions for extremal cases.

Contribution

It introduces new lower bounds for the count of maximum matchings in bipartite multigraphs considering degree and size constraints, extending and refining prior bounds.

Findings

Derived exact lower bounds for bipartite multigraphs with degree constraints.

Extended classical bounds by Hall to more general multigraph settings.

Provided constructions demonstrating the sharpness of the bounds.

Abstract

We study the minimum number of maximum matchings in a bipartite multigraph G with parts $X$ and $Y$ under various conditions, refining the well-known lower bound due to M. Hall. When $|X|=n$, every vertex in $X$ has degree at least $k$, and every vertex in $X$ has at least $r$ distinct neighbors, the minimum is $r!(k-r+1)$ when $n\ge r$ and is $[r+n(k-r)]\prod_{i=1}^{n-1}(r-i)$ when $n<r$. When every vertex has at least two neighbors and $|Y|-|X|=t\ge 0$, the minimum is $[(n-1)t+2+b](t+1)$, where $b=|E(G)|-2(n+t)$. We also determine the minimum number of maximum matchings in several other situations. We provide a variety of sharpness constructions.