In an orientation O of a graph G, an edge e is deletable if O − e is strongly connected. For a 3-edge-connected graph G, Hörsch and Szigeti defined the Frank number as the minimum k for which G admits k orientations such that every edge e of G is deletable in at least one of the k orientations. They conjectured the Frank number is at most 3 for every 3-edge-connected graph G. They proved the Petersen graph has Frank number 3, but this was the only example with this property. We show an infinite class of graphs having Frank number 3. Hörsch and Szigeti showed every 3-edge-colorable 3-edge-connected graph has Frank number at most 3. It is tempting to consider non-3-edge-colorable graphs as candidates for having Frank number greater than 2. Snarks are some-times a good source of finding critical examples or counterexamples. One might suspect various snarks should have Frank number 3. However, we prove several candidate infinite classes of snarks have Frank number 2. This holds also for the generalized Petersen Graphs GP (2s + 1, s). We formulate numerous conjectures inspired by our experience. © The author(s).
János et al. (Mon,) studied this question.