On the maximum fraction of edges covered by $t$ perfect matchings in a cubic bridgeless graph

Abstract

A conjecture of Berge and Fulkerson (1971) states that every cubic bridgeless graph contains 6 perfect matchings covering each edge precisely twice, which easily implies that every cubic bridgeless graph has three perfect matchings with empty intersection (this weaker statement was conjectured by Fan and Raspaud in 1994). Let $m_t$ be the supremum of all reals $\alpha \le 1$ such that for every cubic bridgeless graph $G$, there exist $t$ perfect matchings of $G$ covering a fraction of at least $\alpha$ of the edges of $G$. It is known that the Berge–Fulkerson conjecture is equivalent to the statement that $m_5=1$, and implies that $m_4=\frac{14}{15}$ and $m_3=\frac{4}{5}$. In the first part of this paper, we show that $m_4=\frac{14}{15}$ implies $m_3=\frac{4}{5}$, and $m_3=\frac{4}{5}$ implies the Fan–Raspaud conjecture, strengthening a recent result of Tang, Zhang, and Zhu. In the second part of the paper, we prove that for any $2\le t\le 4$ and for any real $\tau$ lying in some appropriate interval, deciding whether a fraction of more than (resp. at least) $\tau$ of the edges of a given cubic bridgeless graph can be covered by $t$ perfect matching is an NP-complete problem. This resolves a conjecture of Tang, Zhang, and Zhu.