The Lov\'asz-Cherkassky theorem in countable graphs

TL;DR

This paper extends the Lovász-Cherkassky theorem from finite to countable graphs, providing a structural generalization that leverages infinite graph theory and Menger's theorem.

Contribution

It formulates and proves a generalization of the Lovász-Cherkassky theorem for countable graphs, expanding its applicability to infinite graph structures.

Findings

The theorem is successfully extended to countable graphs.

The proof utilizes the infinite Menger's theorem and Nash-Williams' characterization.

The generalization maintains the theorem's core properties in an infinite setting.

Abstract

Lov\'asz 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 Lov\'asz-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 Erd\H{o}s-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.