The strong Nash-Williams orientation theorem for rayless graphs

TL;DR

This paper extends Nash-Williams' strong orientation theorem to rayless graphs, showing that all such infinite graphs can be oriented to preserve a lower bound on directed paths between vertices.

Contribution

It proves that rayless graphs, a class of infinite graphs, admit orientations satisfying Nash-Williams' path-counting property, partially confirming the conjecture for infinite graphs.

Findings

Rayless graphs have orientations with the Nash-Williams property.

The result extends the theorem from finite to a class of infinite graphs.

Provides a partial answer to Nash-Williams' conjecture for infinite graphs.

Abstract

In 1960, Nash-Williams proved his strong orientation theorem that every finite graph has an orientation in which the number of directed paths between any two vertices is at least half the number of undirected paths between them (rounded down). Nash-Williams conjectured that it is possible to find such orientations for infinite graphs as well. We provide a partial answer by proving that all rayless graphs have such an orientation.