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. Inspired by a problem of Frank, 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 the parameter is at most $3$. Here we strengthen both results by showing that every $3$-edge-connected graph has Frank number at most $4$ and that every graph which is $3$-edge-connected and $3$-edge-colourable has Frank number $2$. The latter also confirms a conjecture by Bar\'at and Bl\'azsik. 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 $G$ 的一条边 $e$ 被称为对于某个定向 $o$ 是可删除的，若 $o$ 在 $G-e$ 上的限制是一个强定向。受弗兰克问题的启发，2021 年 Hörsch 和 Szigeti 为 $3$ 边连通图提出了一个新参数，称为弗兰克数，该参数细化了 $k$ 边连通性。弗兰克数定义为 $G$ 的最小定向数，使得 $G$ 的每条边在至少一个定向中是可删除的。他们证明了每个 $3$ 边连通图的弗兰克数至多为 $7$，并且当这些图也是 $3$ 边可着色时，该参数至多为 $3$。本文通过证明每个 $3$ 边连通图的弗兰克数至多为 $4$，以及每个 $3$ 边连通且 $3$ 边可着色的图的弗兰克数为 $2$，改进了这两个结果。后者也证实了 Barát 和 Blázsik 的一个猜想。此外，我们证明了三次图具有弗兰克数 $2$ 的两个充分条件，并利用它们设计了一个算法，通过计算表明在顶点数不超过 $36$ 的循环 $4$ 边连通三次图中，只有彼得森图的弗兰克数大于 $2$。