Removable matchings in $2$-connected graphs

Abstract

A matching $M$ of a $2$-connected graph $G$ is removable if $G-M$ is $2$-connected, extending Halin's classical notion of a removable edge. We prove that for every integer $d\ge 5$, every $2$-connected graph $G$ with $δ(G)\ge d$ and $|V(G)|\ge 2d$ has a removable $d$-matching. This is best possible, since the complete bipartite graph $K_{d,\,n-d}$ on $n\,(\ge 2d)$ vertices has no $(d+1)$-matching. Consequently, our result gives a complete answer, for every $d\ge 5$, to a question of Li, Zhou, Fujita, and Mao on the maximum size of a removable matching guaranteed by the minimum degree. The previously best known bound, due to Li, Zhou, Fujita, and Mao and to Chu, Kim, and Park, guaranteed a removable $(d-2)$-matching under the same assumptions. Our proof is based on an analysis of \emph{minimal non-removable matchings}, matchings that are not removable although all of their proper submatchings are.