An edge $e$ of a graph $G$ is called deletable for some orientation $o$ if the restriction of $o$ to $G-e$ is a strong orientation. In 2021, H\"orsch and Szigeti proposed a new parameter for $3$-edge-connected graphs, called the Frank number, which refines $k$-edge-connectivity. The Frank number is defined as the minimum number of orientations of $G$ for which every edge of $G$ is deletable in at least one of them. They showed that every $3$-edge-connected graph has Frank number at most $7$ and that in case these graphs are also $3$-edge-colourable graphs the parameter is at most $3$. Here we strengthen the latter result by showing that such graphs have Frank number $2$, which also confirms a conjecture by Bar\'at and Bl\'aszik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number $2$ and use them in an algorithm to computationally show that the Petersen graph is the only cyclically $4$-edge-connected cubic graph up to $36$ vertices having Frank number greater than $2$.

翻译：图$G$的边$e$称为关于某定向$o$可删除，如果$o$限制在$G-e$上是一个强定向。2021年，H\"orsch和Szigeti提出了三边连通图的一个新参数，即Frank数，该参数细化了$k$边连通性。Frank数定义为使得图$G$的每条边至少在其中一种定向下可删除所需的最小定向数目。他们证明了每个三边连通图的Frank数至多为$7$，并且当这些图也是三边可着色图时，该参数至多为$3$。本文改进了后一结果，证明此类图的Frank数为$2$，这也证实了Bar\'at和Bl\'aszik的一个猜想。此外，我们给出了三次图具有Frank数$2$的两个充分条件，并将其用于算法中，通过计算表明在顶点数不超过$36$的循环四边连通三次图中，只有Petersen图的Frank数大于$2$。