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 ℵ1. We prove the lemma (and hence the ℵ1 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.
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© 1997 Springer-Verlag Berlin Heidelberg
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Aharoni, R. (1997). A Few Remarks on a Conjecture of Erdős on the Infinite Version of Menger’s Theorem. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_34
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DOI: https://doi.org/10.1007/978-3-642-60406-5_34
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