A mixed version of Menger's theorem

Abstract

An(a, b)-n-fan means a union ofn internally disjoint paths. Menger's theorem states that a graphG has an(a, b)-n-fan if and only ifG isn-connected betweena andb. We show thatG contains λ edge-disjoint(a, b)-n-fans if and only if for anyk withk≤0≤min{n−1, |V(G)|−2} and for any subsetX ofV(G)-{a, b} with cardinalityk, G-X is λ(n-k)-edge-connected betweena andb.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Bollobás: “Extremal Graph Theory”, Academic Press New York, 1978.

    Google Scholar 

  2. [2]

    G. Chartrand, andL. Lesniak: “Graphs & Digraphs” second edition. Wadsworth, Belmont, California, 1986.

    Google Scholar 

  3. [3]

    G. A. Dirac: In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterleitungen,Math. Nachr. 22 (1960), 61–85.

    Google Scholar 

  4. [4]

    R. L. Ford, andD. R. Fulkerson: “Flows in Networks”, Princeton University Press, New Jersey, 1962.

    Google Scholar 

  5. [5]

    D. König: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre.Math. Ann. 77 (1916), 453–465.

    Google Scholar 

  6. [6]

    K. Menger: Zur allgemeinen Kurventheorie.Fund. Math. 10 (1927), 96–115.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Egawa, Y., Kaneko, A. & Matsumoto, M. A mixed version of Menger's theorem. Combinatorica 11, 71–74 (1991). https://doi.org/10.1007/BF01375475

Download citation

AMS subject classification (1980)

  • 05 C 40