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Perfect matching index vs. circular flow number of a cubic graph

The perfect matching index of a cubic graph $G$, denoted by $π(G)$, is the smallest number of perfect matchings that cover all the edges of $G$. According to the Berge-Fulkerson conjecture, $π(G)\le5$ for every bridgeless cubic graph~$G$. The class of graphs with $π\ge 5$ is of particular interest as many conjectures and open problems, including the famous cycle double cover conjecture, can be reduced to it. Although nontrivial examples of such graphs are very difficult to find, a few infinite families are known, all with circular flow number $Φ_c(G)=5$. It has been therefore suggested [Electron. J. Combin. 23 (2016), $\#$P3.54] that $π(G)\ge 5$ might imply $Φ_c(G)\ge 5$. In this article we dispel these hopes and present a family of cyclically $4$-edge-connected cubic graphs of girth at least $5$ (snarks) with $π\ge 5$ and $Φ_c\le 4+\frac23$.