Journal Article10.1017/S0963548300001073
Menger's Theorem for a Countable Source Set
Ron Aharoni,Reinhard Diestel +1 more
TL;DR: This paper proves that Menger's theorem is true for graphs that contain a set of disjoint paths to B from all but countably many vertices of A when A is countable.
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Abstract: Paul Erdős has conjectured that Menger's theorem extends to infinite graphs in the following way: whenever A, B are two sets of vertices in an infinite graph, there exist a set of disjoint A−B paths and an A−B separator in this graph such that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths to B from all but countably many vertices of A. In particular, the conjecture is true when A is countable.
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Citations
Menger’s theorem for infinite graphs
Ron Aharoni,Eli Berger +1 more
TL;DR: 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, and then there exist a set of disjoint A-B paths, and a set S of nodes separating A from B, such that S consists of a choice of precisely one vertex from each path as mentioned in this paper.
Infinite circuits in locally finite graphs
Henning Bruhn
- 04 Jul 2005
TL;DR: In this article, it is shown that the cycle space of Diestel and Kuhn can be extended to locally finite graphs, where the circles are precisely the homeomorphic images of the unit circle in the Freudenthal compactification of the locally finite graph.
The countable Erdös-Menger conjecture with ends
TL;DR: It is proved, for countable graphs G, the extension of this conjecture in which A, B and X are allowed to contain ends as well as vertices, and where the closure of A avoids B and vice versa.
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A Few Remarks on a Conjecture of Erdős on the Infinite Version of Menger’s Theorem
Ron Aharoni
- 01 Jan 2013
TL;DR: The lemma to which Erdős’ conjecture on the extension of Menger’s theorem to infinite graphs can be reduced is proved in two special cases: graphs with countable out-degrees, and graphs with no unending paths.
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References
König's Duality Theorem for Infinite Bipartite Graphs
TL;DR: In this paper, a couplage n and un recouvrement p consiste in un choix de 1 sommet a partir d'une arete dans n.
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Cycles and rays
Geňa Hahn,Gert Sabidussi,Robert E. Woodrow,Rays: Basic Structures in Finite,Infinite Graphs +4 more
- 01 Jan 1990
TL;DR: Hilton and Rodger as mentioned in this paper considered the problem of infinite versions of Menger and Gallai-Milgram Theorems for Ordered Sets and Graphs and proposed a solution to it.
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Menger's theorem for countable graphs
TL;DR: The countable case of a conjecture of Erdos is settled in this paper, where the choice of precisely one vertex from each path in a disjoint path set is determined.
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