Inventiones mathematicae

, Volume 176, Issue 1, pp 1–62 | Cite as

Menger’s theorem for infinite graphs

  • Ron AharoniEmail author
  • Eli Berger


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 \(\mathcal{P}\) of disjoint AB 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 \(\mathcal{P}\). This settles an old conjecture of Erdős.


Bipartite Graph Disjoint Path Maximal Wave Terminal Vertex Countable Case 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aharoni, R.: Menger’s theorem for graphs containing no infinite paths. Eur. J. Comb. 4, 201–204 (1983)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Aharoni, R.: König’s duality theorem for infinite bipartite graphs. J. Lond. Math. Soc. 29, 1–12 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aharoni, R.: Menger’s theorem for countable graphs. J. Comb. Theory, Ser. B 43, 303–313 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Aharoni, R.: Matchings in graphs of size ℵ1. J. Comb. Theory, Ser. B 36, 113–117 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Aharoni, R.: Matchings in infinite graphs. J. Comb. Theory, Ser. B 44, 87–125 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Aharoni, R.: Linkability in countable-like webs. In: Hahn, G., Sabidussi, G., Woodrow, R.E. (eds.) Cycles and Rays: Proceedings of the NATO Advanced Research Workshop on “Cycles and Rays: Basic Structures in Finite and Infinite Graphs”, held in Montreal, Canada, May 3–9, 1987, pp. 1–8. Springer (1987)Google Scholar
  7. 7.
    Aharoni, R.: Infinite matching theory. Discrete Math. 95, 5–22 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Aharoni, R.: A few remarks on a conjecture of Erdős on the infinite version of Menger’s theorem. In: Graham, R.L., Nesteril, J. (eds.) The Mathematics of Paul Erdős, pp. 394–408. Springer, Berlin Heidelberg (1997)Google Scholar
  9. 9.
    Aharoni, R., Diestel, R.: Menger’s theorem for countable source sets. Comb. Probab. Comput. 3, 145–156 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Aharoni, R., Korman, V.: Greene–Kleitman’s theorem for infinite posets. Order 9, 245–253 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Aharoni, R., Nash-Williams, C.S.J., Shelah, S.: A general criterion for the existence of transversals. Proc. Lond. Math. Soc. 47, 43–68 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ahlswede, R., Khachatrian, L.H.: A counterexample to Aharoni’s strongly maximal matching conjecture. Discrete Math. 149, 289 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Damerell, M.R., Milner, E.C.: Necessary and sufficient conditions for transversals of countable set systems. J. Comb. Theory, Ser. A 17, 350–374 (1974)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Diestel, R.: Graph Theory, 1st edn. Springer (1997)Google Scholar
  15. 15.
    Fodor, G.: Eine Bemerkung zur Theorie der regressive Funktionen. Acta Sci. Math. 17, 139–142 (1956)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Gallai, T.: Ein neuer Beweis eines Mengerschen Satzes. J. Lond. Math. Soc. 13, 188–192 (1938)CrossRefGoogle Scholar
  17. 17.
    Hall, P.: On representatives of subsets. J. Lond. Math. Soc. 10, 26–30 (1935)zbMATHCrossRefGoogle Scholar
  18. 18.
    König, D.: Graphs and matrices. Mat. Fiz. Lapok 38, 116–119 (1931) (Hungarian)zbMATHGoogle Scholar
  19. 19.
    König, D.: Theorie der endlichen und unendlichen Graphen. Akademischen Verlagsgesellschaft, Leipzig (1936) (Reprinted: Chelsea, New York, 1950)Google Scholar
  20. 20.
    Lovász, L., Plummer, M.D.: Matching Theory. Ann. Math., vol. 29. North Holland (1991)Google Scholar
  21. 21.
    McDiarmid, C.: On separated separating sets and Menger’s theorem. Congr. Numerantium 15, 455–459 (1976)MathSciNetGoogle Scholar
  22. 22.
    Menger, K.: Zur allgemeinen Kurventhoerie. Fundam. Math. 10, 96–115 (1927)zbMATHGoogle Scholar
  23. 23.
    Nash-Williams, C.S.J.A.: Infinite graphs – a survey. J. Comb. Theory 3, 286–301 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Nash-Williams, C.S.J.A.: Which infinite set-systems have transversals? – A possible approach. In: Combinatorics (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972), pp. 237–253. Inst. Math. Appl., Southend-on-Sea (1972)Google Scholar
  25. 25.
    Nash-Williams, C.S.J.A.: Marriage in denumerable societies. J. Comb. Theory, Ser. A 19, 335–366 (1975)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Nash-Williams, C.S.J.A.: Another criterion for marriage in denumerable societies. In: Bollobás, B. (ed.) Advances in Graph Theory. Ann. Discrete Math., vol. 3, pp. 165–179. North-Holland, Amsterdam (1978)CrossRefGoogle Scholar
  27. 27.
    Oellrich, H., Steffens, K.: On Dilworth’s decomposition theorem. Discrete Math. 15, 301–304 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Podewski, K.-P., Steffens, K.: Injective choice functions for countable families. J. Comb. Theory, Ser. B 21, 40–46 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Podewski, K.-P., Steffens, K.: Über Translationen und der Satz von Menger in unendlichen Graphen. Acta Math. Hung. 30, 69–84 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Fiedler, M.: Theory of graphs and its applications. In: Proceedings of the Symposium held in Smolenice, June 1963. Czechoslovak Academy of Sciences, Prague (1964)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsTechnionHaifaIsrael
  2. 2.Department of Mathematics, Faculty of Science and Science EducationHaifa UniversityHaifaIsrael

Personalised recommendations