On covering cubic graphs with three perfect matchings

Abstract

For a bridgeless cubic graph $G$, $m_3(G)$ is the ratio of the maximum number of edges of $G$ covered by the union of $3$ perfect matchings to $|E(G)|$. We prove that for any $r\in [4/5,1)$, there exist infinitely many cubic graphs $G$ such that $m_3(G)=r$. For any $r\in [9/10,1)$, there exist infinitely many cyclically $4$-connected cubic graphs $G$ with $m_3(G)=r$.