Towards Nash-Williams Orientation Conjecture for Infinite Graphs

Abstract

In 1960 Nash-Williams proved that an edge-connectivity of 2k is sufficient for a finite graph to have a k-arc-connected orientation. He then conjectured that the same is true for infinite graphs. In 2016, Thomassen, using his own results on the auxiliary lifting graph, proved that 8k-edge-connected infinite graphs admit a $k$-arc connected orientation. Here we improve this result for the class of $1$-ended locally-finite graphs and show that an edge-connectivity of 4k is enough in that case. Crucial to this improvement are results presented in a separate paper, by the same author of this paper, on the key concept of the lifting graph, extending results by Ok, Richter, and Thomassen.