$k$-Factor-Covered Regular Graphs

Abstract

A graph $G$ is called $f$-factor-covered if every edge of $G$ is contained in some $f$-factor. $G$ is called $f$-factor-deleted if $G$ – $e$ contains an $f$-factor for every edge $e$. Babler proved that every $r$-regular, $(r–1)$-edge-connected graph of even order has a $1$-factor. In the present article, we prove that every $2r$-regular graph of odd order is both $2m$-factor-covered and $2m$-factor-deleted for all integers $m$, $1\le m\le r–1$, and every $r$-regular, $(r–1)$-edge-connected graph of even order is both $m$-factor-covered and $m$-factor-deleted for all integers $m$, $1\le m\le \lfloor \frac{r}{2}\rfloor$.