The Nash-Williams orientation theorem for graphs with countably many ends

Abstract

Nash-Williams proved in 1960 that a finite graph admits a $k$-arc-connected orientation if and only if it is $2k$-edge-connected, and conjectured that the same result should hold for all infinite graphs, too.

Progress on Nash-Williams’s problem was made by C. Thomassen, who proved in 2016 that all $8k$-edge-connected infinite graphs admit a $k$-arc connected orientation, and by the first author, who recently showed that edge-connectivity of $4k$ suffices for locally-finite, 1-ended graphs.

In the present article, we establish the optimal bound $2k$ in Nash-Williams’s conjecture for all locally finite graphs with countably many ends.