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：若图$G$的某个定向$o$限制在$G-e$上为强定向，则称边$e$对该定向$o$是可删除的。受Frank问题的启发，Hörsch与Szigeti于2021年针对3-边连通图提出了一个称为弗兰克数的新参数，该参数细化了k-边连通性。弗兰克数定义为：使$G$的每条边在至少一个定向中可删除的$G$的最小定向数目。他们证明了每个3-边连通图的弗兰克数至多为7，且当这类图同时是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，以及每个同时满足3-边连通与3-边可着色的图其弗兰克数为2，从而强化了上述两个结果。后者也证实了Barát与Blázsik的一个猜想。此外，我们证明了立方图具有弗兰克数2的两个充分条件，并利用这些条件通过算法计算表明：在36个顶点以内，彼得森图是唯一一个弗兰克数大于2的循环4-边连通立方图。