Perfect matchings in highly cyclically connected regular graphs

Abstract

A leaf matching operation on a graph consists of removing a vertex of degree~$1$ together with its neighbour from the graph. For $k\geq 0$, let $G$ be a $d$-regular cyclically $(d-1+2k)$-edge-connected graph of even order. We prove that for any given set $X$ of $d-1+k$ edges, there is no $1$-factor of $G$ avoiding $X$ if and only if either an isolated vertex can be obtained by a series of leaf matching operations in $G-X$, or $G-X$ has an independent set that contains more than half of the vertices of~$G$. To demonstrate how to check the conditions of the theorem we prove several statements on $2$-factors of cubic graphs. For $k\ge 3$, we prove that given a cubic cyclically $(4k-5)$-edge-connected graph $G$ and three paths of length $k$ such that the distance of any two of them is at least $8k-17$, there is a $2$-factor of $G$ that contains one of the paths . We provide a similar statement for two paths when $k=3$ and $k=4$. As a corollary we show that given a vertex $v$ in a cyclically $7$-edge-connected cubic graph, there is a $2$-factor such that $v$ is in a circuit of length greater than $7$.