A Short Proof of a Theorem of Kano and Yu on Factors in Regular Graphs

Abstract

In this note we present a short proof of the following result, which is a slight extension of a nice 2005 theorem by Kano and Yu. Let $e$ be an edge of an $r$-regular graph $G$. If $G$ has a 1-factor containing $e$ and a 1-factor avoiding $e$, then $G$ has a $k$-factor containing $e$ and a $k$-factor avoiding $e$ for every $k\in\{1,2,\ldots,r-1\}$.