Connectivity of orientations of 3-edge-connected graphs

TL;DR

This paper investigates the Frank number of 3-edge-connected graphs, establishing upper bounds, specific examples like the Petersen graph, and exploring computational complexity and related conjectures.

Contribution

It introduces the Frank number for 3-edge-connected graphs, proves an upper bound of 7, and connects the concept to the Berge-Fulkerson conjecture and NP-completeness.

Findings

Frank number of any 3-edge-connected graph is at most 7.

The Petersen graph has a Frank number of 3.

Deciding if all edges in a subset can be deletable in one orientation is NP-complete.

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 {\it 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 $f(G)\leq 7$ 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.