Distance restricted matching extensions in regular non-bipartite graphs

Abstract

Let $m$ and $r$ be integers with $m \ge r \ge 3$ and let $G$ be an $r$-regular graph of even order. Let $M$ be a matching in $G$ of size $m$ such that each pair of edges in $M$ is at distance at least $3$. In 2023, Aldred et al. proved that if $G$ is cyclically $(mr-r+1)$-edge-connected and $G$ is bipartite, then there exists a perfect matching of $G$ containing $M$. In this paper, we present non-bipartite analogues of Aldred et al.'s theorem. An odd ear of $U \subseteq V(G)$ is a path of odd length whose ends lie in $U$ but whose internal vertices do not, or a cycle of odd length having exactly one vertex in $U$. Our first result shows that if $G$ is cyclically $(mr - m +1)$-edge-connected and there exist $mr - \left\lceil \frac{r}{2} \right\rceil + 1$ edge-disjoint odd ears of $V(M)$, then $M$ can be extended to a perfect matching of $G$. We further show that if $G$ contains $mr-r+1$ edge-disjoint odd ears of $V(M)$ and no cyclic edge cut in $G$ of size less than $(2m-1)(r-1)$ separates an odd cycle from another cycle, then $M$ can still be extended to a perfect matching. The second result extends Aldred et al.'s theorem to non-bipartite graphs in the case $r \ge 4$, and in the case when $r = 3$ and each pair of edges in $M$ is at distance at least $5$. It is also shown that the above results hold when $m \le r - 1$, without assuming the distance condition on $M$.