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

TL;DR

This paper proves Nash-Williams's conjecture for all locally finite graphs with countably many ends, establishing that $2k$-edge-connectivity guarantees a $k$-arc-connected orientation, extending previous partial results.

Contribution

It establishes the optimal $2k$ bound for Nash-Williams's orientation theorem in all locally finite graphs with countably many ends, confirming the conjecture in this broad class.

Findings

Proves the $2k$-edge-connectivity condition suffices for orientation.

Extends Nash-Williams's theorem to graphs with countably many ends.

Provides a complete solution for locally finite 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.