The strong Nash-Williams orientation theorem for rayless 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.