A sharp extension of Halin's removable-edge theorem to matchings

Abstract

A subgraph $H$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-E(H)$ remains $k$-connected. Halin proved that every $k$-connected graph $G$ with $\delta (G)\ge k+1$ has a $k$-removable edge. We extend this result from a single edge to matchings of any prescribed size by showing that, for positive integers $k$ and $m$, every $k$-connected graph $G$ with $\delta (G)\ge \max \{k+1,2m-2\}$ contains a $k$-removable matching of size $m$, unless $G\cong K_{2m-1}$, or $(k,m)=(1,2)$ and $G$ is a cycle. This confirms a conjecture of Li, Zhou, Fujita, and Mao. The minimum degree bound is sharp, and both exceptions are unavoidable. Consequently, $\max \{k+1,2m-1\}$ is the sharp minimum degree threshold guaranteeing such a matching without exceptions. The proof combines a prescribed-set strengthening of Halin's removable-edge theorem with an extremal analysis of maximum $k$-removable matchings.