Menger’s theorem for infinite graphs

Abstract

We prove that Menger’s theorem is valid for infinite graphs, in the following strong version: let A and B be two sets of vertices in a possibly infinite digraph. Then there exist a set \(\mathcal{P}\) of disjoint AB paths, and a set S of vertices separating A from B, such that S consists of a choice of precisely one vertex from each path in \(\mathcal{P}\). This settles an old conjecture of Erdős.

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Correspondence to Ron Aharoni.

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Aharoni, R., Berger, E. Menger’s theorem for infinite graphs. Invent. math. 176, 1–62 (2009). https://doi.org/10.1007/s00222-008-0157-3

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Keywords

  • Bipartite Graph
  • Disjoint Path
  • Maximal Wave
  • Terminal Vertex
  • Countable Case