4-coverable snarks, perfect matching cover, and Isaacs product

Abstract

The perfect matching cover index $\tau \left(G\right)$ of a bridgeless cubic graph $G$ is the minimum number of perfect matchings in $G$ that cover all edges of $G$. It is conjectured by Berge that $\tau \left(G\right)\le 5$ for every bridgeless cubic graph. Esperet and Mazzuoccolo (2013) proved that deciding whether a cubic bridgeless graph $G$ satisfies $\tau \left(G\right)\le 4$ is NP-complete. Some major conjectures hold for bridgeless cubic graphs $G$ with $\tau \left(G\right)=4$, such as the cycle double cover conjecture and the Alon–Tarsi shortest cycle cover conjecture. In this paper, we study the perfect matching cover index of snarks obtained by the Isaacs products of two cubic graphs. Our results indicate that the Isaacs product of certain snarks does not increase the perfect matching cover index. As corollaries, we show that the perfect matching cover indices of some families of snarks with arbitrarily large oddness equal to 4.