Extending Matchings to $2$-Factors

Abstract

Let $G$ be a finite $4$-regular cyclically $2k$-edge-connected simple graph for some integer $k\ge 1$. Let $E(k)$ be a set of $k$ independent edges in $G$ and $(E_1,E_2)$ be a partition of $E(k)$. We consider when there exists a $2$-factor in $G$ which excludes all edges of $E_1$, and includes all the edges of $E_2$. A complete characterization is provided.