Menger’s Theorem for Infinite Graphs

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 P of disjoint A–B 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 P. This settles an old conjecture of Erdős. 1. History of the problem In 1931 Dénes König [17] proved a min-max duality theorem on bipartite graphs: Theorem 1.1. In any finite bipartite graph, the maximal size of a matching equals the minimal size of a cover of the edges by vertices. Here a matching in a graph is a set of disjoint edges, and a cover (of the edges by vertices) is a set of vertices meeting all edges. This theorem was the culmination of a long development, starting with a paper of Frobenius in 1912. For details on the intriguing history of this theorem, see [19]. Four years after the publication of König’s paper Phillip Hall [16] proved a result which he named “the marriage theorem”. To formulate it, we need the following notation: given a set A of vertices in a graph, we denote by N(A) the set of its neighbors. Theorem 1.2. In a finite bipartite graph with sides M and W there exists a marriage of M (that is, a matching meeting all vertices of M) if and only if |N(A)| ≥ |A| for every subset A of M . The two theorems are closely related, in the sense that they are easily derivable from each other. In fact, König’s theorem is somewhat stronger, in that the derivation of Hall’s theorem from it is more straightforward than vice versa. At the time of publication of König’s theorem, a theorem generalizing it considerably was already known. Definition 1.3. Let X,Y be two sets of vertices in a digraph D. A set S of vertices is called X–Y -separating if every X–Y -path meets S, namely if the deletion of S severs all X–Y -paths. Note that, in particular, S must contain X ∩ Y . The research of the first author was supported by grant no. 780-04 of the Israel Science Foundation, by GIF grant no. 2006311, by the Technion’s research promotion fund, and by the