On Frank's conjecture on k-connected orientations

TL;DR

This paper disproves Frank's conjecture that weakly 2k-connected graphs always have k-vertex-connected orientations and proves that deciding such orientations is NP-complete for k≥3.

Contribution

The paper provides a counterexample to Frank's conjecture and establishes NP-completeness for the orientation decision problem when k≥3.

Findings

Disproved Frank's conjecture for k≥3

Proved NP-completeness of the orientation problem for k≥3

Identified limitations of connectivity orientation algorithms

Abstract

We disprove a conjecture of Frank stating that each weakly 2k-connected has a k-vertex-connected orientation. For k at least 3, we also prove that the problem of deciding whether a graph has a k-vertex-connected orientation is NP-complete.