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

Michael Anastos; David Fabian; Alp Müyesser; Tibor Szabó

paper

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

arXiv:2212.03100

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 $a_{1}, a_{2}, a_{3} \in N$ 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.