Distance restricted matching extensions in regular non-bipartite graphs

TL;DR

This paper extends a bipartite graph matching theorem to non-bipartite graphs, providing conditions under which a matching can be extended to a perfect matching in regular graphs with certain connectivity and ear structure properties.

Contribution

It introduces non-bipartite analogues of a recent bipartite matching theorem, involving conditions on edge-disjoint odd ears and cyclic edge cuts for extending matchings.

Findings

Conditions for extending matchings in non-bipartite graphs are established.

Results apply to graphs with specific connectivity and ear structure properties.

Theorems hold for various degrees and distance conditions, including cases with no distance restrictions.

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 $M$.