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A Few Remarks on a Conjecture of Erdős on the Infinite Version of Menger’s Theorem

In: The Mathematics of Paul Erdős II

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  • Ron Aharoni

    (Technion, Department of Mathematics)

Abstract

Summary We discuss a few issues concerning Erdős’ conjecture on the extension of Menger’s theorem to infinite graphs. A key role is given to a lemma to which the conjecture can probably be reduced. The paper is intended to be expository, so rather than claim completeness of proofs, we chose to prove the reduction only for graphs of size ℵ 1. We prove the lemma (and hence the ℵ 1 case of the conjecture) in two special cases: graphs with countable out-degrees, and graphs with no unending paths. We also present new versions of the proofs of the (already known) cases of countable graphs and graphs with no infinite paths. A main tool used is a transformation converting the graph into a bipartite graph.

Suggested Citation

  • Ron Aharoni, 2013. "A Few Remarks on a Conjecture of Erdős on the Infinite Version of Menger’s Theorem," Springer Books, in: Ronald L. Graham & Jaroslav Nešetřil & Steve Butler (ed.), The Mathematics of Paul Erdős II, edition 2, pages 335-352, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-7254-4_21
    DOI: 10.1007/978-1-4614-7254-4_21
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