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

Understand

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 $n-1$.