SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1287-1297, January 2021.
The perfect matching index of a cubic graph $G$, denoted by $\pi(G)$, is the smallest number of perfect matchings that cover all the edges of $G$. According to the Berge--Fulkerson conjecture, $\pi(G)\le5$ for every bridgeless cubic graph $G$. The class of graphs with $\pi\ge 5$ 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 $\Phi_c(G)=5$. It has been therefore suggested [Abreu et al., Electron. J. Combin., 23 (2016), P3.54] that $\pi(G)\ge 5$ might imply $\Phi_c(G)\ge 5$. In this article we dispel these hopes and present a family of cyclically 4-edge-connected cubic graphs of girth at least 5 with $\pi\ge 5$ and $\Phi_c\le 4+\frac23$.

中文翻译：

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三次图的完美匹配指数与循环流数

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1287-1297 页，2021 年 1 月。
三次图$G$的完美匹配指数，用$\pi(G)$表示，是覆盖$G$所有边的最小完美匹配数。根据 Berge--Fulkerson 猜想，$\pi(G)\le5$ 对于每个无桥三次图 $G$。具有 $\pi\ge 5$ 的图类特别有趣，因为许多猜想和开放问题，包括著名的循环双覆盖猜想，都可以归结为它。尽管此类图的非平凡示例很难找到，但已知有几个无限族，所有族都具有循环流数 $\Phi_c(G)=5$。因此，有人建议 [Abreu 等人，Electron. J. Combin., 23 (2016), P3.54] $\pi(G)\ge 5$ 可能意味着 $\Phi_c(G)\ge 5$。