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 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 graph 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$e$称为对某一定向$o$可删除的，若$o$在$G-e$上的限制构成强定向。2021年，Hörsch与Szigeti针对$3$-边连通图提出了一种新参数——Frank数，该参数对$k$-边连通性进行了细化。Frank数定义为使$G$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。他们证明每个$3$-边连通图的Frank数至多为$7$，且当这些图同时为$3$-边可着色图时该参数至多为$3$。本文强化了上述结果：我们证明每个$3$-边连通图的Frank数至多为$4$，而每个$3$-边连通且$3$-边可着色图的Frank数为$2$。后者同时证实了Barát与Blázsik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，并利用这些条件通过算法计算表明：在不超过$36$个顶点的循环$4$-边连通三次图中，仅Petersen图的Frank数大于$2$。