Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets

TL;DR

This paper investigates matchings in multigraphs formed by three perfect matchings, proving existence results under certain conditions and proposing conjectures that extend classical combinatorial conjectures to multiset settings.

Contribution

It establishes new existence theorems for matchings with prescribed intersections in multigraphs and introduces conjectures generalizing the Ryser-Brualdi-Stein conjecture for multisets.

Findings

Existence of a matching with specified intersections in multigraphs with three perfect matchings.

Bound of n-2 is tight for general multigraphs.

Conjecture that bipartite case holds with bound n-1, supported by a construction.

Abstract

We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general. We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.