Edge‐Connectivity Between Edge‐Ends of Infinite Graphs

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‐Williams' Tree‐Packing Theorem and MacLane's Planarity Criteria are examples of results that allow a topological approach, in which ends may be considered as endpoints of rays. In fact, there are extensive studies in the literature showing that classical (vertex‐)connectivity theorems 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 Širáň. In terms of that object, we state an edge version of Menger's Theorem (following a previous work of Polat) and generalize the Lovász‐Cherkassky Theorem for infinite graphs with edge‐ends (inspired by a recent paper of Jacobs, Joó, Knappe, Kurkofka and Melcher).