Antifactors in bipartite multigraphs

TL;DR

This paper extends a bipartite graph antifactors result to multigraphs with prime power degrees, linking perfect matchings to algebraic conditions, and proposes a conjecture on perfect matchings distribution.

Contribution

It generalizes a known bipartite graph antifactors theorem to multigraphs under prime power degree conditions and introduces a conjecture on perfect matchings distribution.

Findings

Extension of the antifactors result to multigraphs with prime power degrees.

Necessary condition on the number of perfect matchings for multigraphs.

Conjecture on the distribution of perfect matchings modulo q in random bipartite q-regular graphs.

Abstract

Let $G$ be a $q$-regular bipartite graph with bipartition $(U,V)$. It was proved by Lu, Wang, and Yan in 2020 that $G$ has a spanning subgraph $H$ such that each vertex of $U$ has degree 1 in $H$, and each vertex of $V$ has degree distinct from 1 in $H$. We extend the result to multigraphs, under the condition that $q$ is a prime power and the number of perfect matchings of $G$ is not divisible by $q$. The condition on the number of perfect matchings is necessary for multigraphs. We conclude with a conjecture on the limiting distribution of the number of perfect matchings modulo $q$ in a random bipartite $q$-regular graph.