We attempt to generalize a theorem of Nash-Williams stating that a graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. In a strongly connected digraph we call an arc deletable if its deletion leaves a strongly connected digraph. Given a 3-edge-connected graph $G$, we define its Frank number $f\left(G\right)$ to be the minimum number $k$ such that there exist $k$ orientations of $G$ with the property that every edge becomes a deletable arc in at least one of these orientations. We are interested in finding a good upper bound for the Frank number. We prove that $f\left(G\right)\le 7$ for every 3-edge-connected graph. On the other hand, we show that a Frank number of 3 is attained by the Petersen graph. Further, we prove better upper bounds for more restricted classes of graphs and establish a connection to the Berge–Fulkerson conjecture. We also show that deciding whether all edges of a given subset can become deletable in one orientation is NP-complete.

我们尝试推广Nash-Williams定理，指出一个图具有一个 $ķ$-arc-connected方向，当且仅当它是 $2ķ$-边缘连接。在强连通有向图我们所说的弧删除的，如果将其删除叶强连通有向图。给定一个三边连接图$G$，我们定义其弗兰克数 $F（G）$ 成为最小数量 $ķ$ 这样就存在 $ķ$ 的方向 $G$具有以下特性：每个边缘至少在这些方向之一上成为可删除的弧。我们有兴趣为弗兰克数找到一个好的上限。我们证明$F（G）\le 7$对于每个3边连接图。另一方面，我们证明彼得森图获得的弗兰克数为3。此外，我们证明了图的更严格限制的更好上限，并建立了与Berge-Fulkerson猜想的联系。我们还表明，确定给定子集的所有边缘是否可以在一个方向上删除是NP完全的。