Eulerian orientations and vertex-connectivity

Abstract

It is well-known that every Eulerian orientation of an Eulerian 2k-edge-connected undirected graph is k-arc-connected. A long-standing goal in the area has been to obtain analogous results for vertex-connectivity. Levit, Chandran and Cheriyan recently proved in Levit et al. (2018) that every Eulerian orientation of a hypercube of dimension 2k is k-vertex-connected. Here we provide an elementary proof for this result.We also show other families of 2k-regular graphs for which every Eulerian orientation is k-vertex-connected, namely the even regular complete bipartite graphs, the incidence graphs of projective planes of odd order, the line graphs of regular complete bipartite graphs and the line graphs of complete graphs.Furthermore, we provide a simple graph counterexample for a conjecture of Frank attempting to characterize graphs admitting at least one k-vertex-connected orientation.