Edge-connectivity between (edge-)ends of infinite graphs

TL;DR

This paper extends classical finite graph connectivity theorems to infinite graphs by utilizing the concept of edge-ends, providing new formulations and generalizations of Menger's and Lovász-Cherkassky theorems.

Contribution

It introduces an edge version of Menger's Theorem and generalizes the Lovász-Cherkassky Theorem for infinite graphs with edge-ends.

Findings

Established an edge version of Menger's Theorem for infinite graphs.

Generalized Lovász-Cherkassky Theorem to infinite graphs with edge-ends.

Connected edge-connectivity properties with topological and combinatorial approaches.

Abstract

In infinite graph theory, the notion of ends, first introduced by Freudenthal and Jung for locally finite graphs, plays an important role when generalizing statements from finite graphs to infinite ones. Nash-Willian's Tree-Packing Theorem and MacLane's Planarity Criteria are examples of results that allow a topological approach, in which ends might be considered as endpoints of rays. In fact, there are extensive works in the literature showing that classical theorems of (vertex-)connectivity for finite graphs can be discussed regarding ends, in a more general context. However, aiming to generalize results of edge-connectivity, this paper recalls the definition of edge-ends in infinite graphs due to Hahn, Laviolette and \v{S}ir\'a\v{n}. In terms of that object, we state an edge version of Menger's Theorem (following a previous work of Polat) and generalize the Lov\'asz-Cherkassky Theorem for infinite graphs with edge-ends (inspired by a paper of Jacobs, Jo\'o, Knappe, Kurkofka and Melcher).