Matchings of cycles and paths in directed graphs

Abstract

In this paper we present a Berge-Tutte-type theorem for a matching problem in directed graphs. This extends the maximum matching problem in undirected graphs, the maximum even factor problem in weakly symmetric directed graphs proposed by W. H. Cunningham and J. F. Geelen in [6], and a packing problem for cycles and edges in undirected graphs. We show an Edmonds-Gallai-type structural description of a canonical set attaining the minimum in the formula. We also give a generalization of the matching matroid to this concept.

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

References

  1. [1]

    G. Cornuéjols and D. Hartvigsen: An extension of matching theory, Journal of Combinatorial Theory Ser. B 40 (1986), 285–296.

    Article  MATH  Google Scholar 

  2. [2]

    G. Cornuéjols, D. Hartvigsen and W. Pulleyblank: Packing subgraphs in a graph, Op. Res. Letters, (1982), 139–143.

  3. [3]

    G. Cornuéjols and W. Pulleyblank: A matching problem with side conditions, Discrete Mathematics 29 (1980), 135–159.

    Article  MathSciNet  MATH  Google Scholar 

  4. [4]

    G. Cornuéjols and W. Pulleyblank: Critical graphs, matchings and tours or a hierarchy of the travelling salesman problem, Combinatorica 3(1) (1983), 35–52.

    Article  MathSciNet  MATH  Google Scholar 

  5. [5]

    W. H. Cunningham: Matching, Matroids and Extensions, Math. Program. Ser. B 91(3) (2002), 515–542.

    Article  MathSciNet  MATH  Google Scholar 

  6. [6]

    W. H. Cunningham and J. F. Geelen: The Optimal Path-Matching Problem, Combinatorica 17(3) (1997), 315–336.

    Article  MathSciNet  MATH  Google Scholar 

  7. [7]

    J. Edmonds: Paths, trees, and flowers, Canadian Journal of Mathematics 17 (1965), 449–467.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    S. Felsner: Orthogonal structures in directed graphs, Journal of Combinatorial Theory Ser. B 57(2) (1993), 309–321.

    Article  MathSciNet  MATH  Google Scholar 

  9. [9]

    A. Frank and L. Szegő: A Note on the Path-Matching Formula, J. of Graph Theory 41(2) (2002), 110–119.

    Article  MATH  Google Scholar 

  10. [10]

    T. Gallai: Maximale Systeme unabhängiger Kanten, A Magyar Tudományos Akadémia Matematika Kutatóintézetének Közleményei 9 (1964), 401–413.

    MathSciNet  MATH  Google Scholar 

  11. [11]

    J. Geelen: An algebraic approach to matching problems, manuscript.

  12. [12]

    T. Király and M. Makai: On polyhedra related to even factors, Proceedings of 10th International IPCO Conference, D. Bienstock, G. Nemhauser (eds.), Lecture Notes in Computer Science, Springer, 2004, 416–430.

  13. [13]

    Z. Király and J. Szabó: Generalized induced factor problems, Egres Technical Report, 2002.

  14. [14]

    M. Loebl and S. Poljak: Efficient Subgraph Packing, Journal of Combinatorial Theory Ser. B 59 (1993), 106–121.

    Article  MathSciNet  MATH  Google Scholar 

  15. [15]

    L. Lovász: On determinants, matchings and random algorithms, in Fundamentals of Computational Theory (L. Budach, ed.), Akademie-Verlag, Berlin, 1979, 565–574.

    Google Scholar 

  16. [16]

    L. Lovász and M. D. Plummer: Matching Theory, Akadémiai Kiadó, Budapest, 1986.

    Google Scholar 

  17. [17]

    Gy. Pap and L. Szegő: On the Maximum Even Factor in Weakly Symmetric Graphs, Journal of Combinatorial Theory Ser. B 91(2) (2004), 201–213.

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Gyula Pap.

Additional information

Research supported by the Hungarian National Foundation for Scientific Research Grant, OTKA T037547 and by European MCRTN Adonet, Contract Grant No. 504438.

The author is supported by the Egerváry Research Group of the Hungarian Academy of Sciences.

The author is a member of the Egerváry Research Group (EGRES).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pap, G., Szegő, L. Matchings of cycles and paths in directed graphs. Combinatorica 27, 383–398 (2007). https://doi.org/10.1007/s00493-007-2131-x

Download citation

Mathematics Subject Classification (2000)

  • 05C70