  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 $7$.
