On the connectivity of infinite graphs and 2-complexes
An analogue of the classical Menger-Whitney theorem equating the connectivity of a locally finite graph with its separation number is proved for locally finite 2-complexes, as conjectured by E. Y. Woon, who proved the finite case [Isr. J. Math. 52, 177-192 (1985; Zbl 0593.05046)]. Let \(n\) be a positive integer and let \(P\) denote a locally finite 2-complex with the following properties: (i) every vertex and edge are contained in a triangle, (ii) every two edges are joined by an alternating sequence of edges and triangles in which consecutive terms are incident, (iii) each triangle is incident with at most one edge that is incident with no other triangle, and (iv) all other edges are incident with at least \(n\) triangles. Let \(A\) and \(B\) each denote a set of vertices of \(P\) or a set of ends of \(P\). Theorem: There exist \(n\) internally disjoint paths (or rays or double rays) joining \(A\) and \(B\) in \(P\) if and only if \(P\) contains no set \(J\) of edges with \(| J| <n\) such that \(A\) and \(B\) are in different components of \(P-J\).
scientific article; zbMATH DE number 1321835
scientific article