Connectivity of orientations of 3-edge-connected graphs

Abstract

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(G) 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(G)≤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.