Perfect matching cover indices of generalized rotation snarks

Let Γ be a 3-regular bridgeless graph and $\tau \left(\Gamma \right)$ be the perfect matching cover index of Γ. It is conjectured by Berge that $\tau \left(\Gamma \right)\le 5$. Esperet and Mazzuoccolo (2013) [4] proved that deciding whether Γ satisfies $\tau \left(\Gamma \right)\le 4$ is NP-complete. Máčajová and Škoviera (2021) [13] gave a family $R=\{R_k:k\ge 4\}$ of rotation snarks. We construct a family $\mathrm{FR}=\{F{R}_{2n+1}:n\ge 1\}$ of generalized rotation snarks. In this paper, we show that each $F{R}_{2n+1}$ satisfies $\tau \left(F{R}_{2n+1}\right)=4$. As a corollary, each $R_k$ satisfies $\tau \left(R_k\right)=4$.