The Lovász-Cherkassky theorem in countable graphs

Abstract

Lovász and Cherkassky discovered in the 1970s independently that if $G$ is a finite graph with a given set $T$ of terminal vertices such that $G$ is inner Eulerian, then the maximal number of edge-disjoint paths connecting distinct vertices in $T$ is ${\sum }_{t\in T}\lambda (t,T-t)$ where $\lambda$ is the local edge-connectivity function. The optimality of a system of edge-disjoint $T$-paths in the Lovász-Cherkassky theorem is witnessed by the existence of certain cuts by Menger's theorem. The infinite generalisation of Menger's theorem by Aharoni and Berger (earlier known as the Erdős-Menger Conjecture) together with the characterization of infinite Eulerian graphs due to Nash-Williams makes it possible to generalise the theorem for infinite graphs in a structural way. The aim of this paper is to formulate this generalisation and prove it for countable graphs.