The Erdös-Menger conjecture for source/sink sets with disjoint closures

Authors:
Reinhard Diestel
Affiliations:
Mathematisches Seminar der Universität Hamburg, Bundesstraße 55, D-20146 Hamburg, Germany
Venue:
Journal of Combinatorial Theory Series B
Year:
2005

Citing 4
Cited 0

Menger's theorem for countable graphs

Journal of Combinatorial Theory Series A
The countable Erdös-Menger conjecture with ends

Journal of Combinatorial Theory Series B
Graph-theoretical versus topological ends of graphs

Journal of Combinatorial Theory Series B
The Cycle Space of an Infinite Graph

Combinatorics, Probability and Computing

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Abstract

Erdös conjectured that, given an infinite graph G and vertex sets A, B ⊆ V(G), there exist a set P of disjoint A-B paths in G and an A-B separator X 'on' P, in the sense that X consists of a choice of one vertex from each path in P. We prove the conjecture for vertex sets A and B that have disjoint closures in the usual topology on graphs with ends. The result can be extended by allowing A, B and X to contain ends as well as vertices.