Skip to main content
SpringerLink
Log in

Elementary Graphs with Respect to f-Parity Factors

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

This note concerns the f-parity subgraph problem, i.e., we are given an undirected graph G and a positive integer value function \({f : V(G) \rightarrow \mathbb{N}}\), and our goal is to find a spanning subgraph F of G with deg F ≤ f and minimizing the number of vertices x with \({\deg_F(x) \not\equiv f(x) \, {\rm mod} \, {2}}\) . First we prove a Gallai–Edmonds type structure theorem and some other known results on the f-parity subgraph problem, using an easy reduction to the matching problem. Then we use this reduction to investigate barriers and elementary graphs with respect to f-parity factors, where an elementary graph is a graph such that the union of f-parity factors form a connected spanning subgraph.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amahashi A.: On factors with all degrees odd. Graphs Combin. 1, 111–114 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  2. Cornuéjols G.: General factors of graphs. J. Combin. Theory Ser. B 45, 185–198 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cui Y., Kano M.: Some results on odd factors of graphs. J. Graph Theory 12, 327–333 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Edmonds J.: Paths, trees, and flowers. Can. J. Math. 17, 449–467 (1965)

    MATH  MathSciNet  Google Scholar 

  5. Gallai T.: Kritische Graphen II. A Magyar Tud. Akad. Mat. Kut. Int. Közl. 8, 135–139 (1963)

    MATH  MathSciNet  Google Scholar 

  6. Gallai T.: Maximale Systeme unabhängiger Kanten. A Magyar Tud. Akad. Mat. Kut. Int. Közl. 9, 401–413 (1964)

    MATH  MathSciNet  Google Scholar 

  7. Kano M., Katona G.Y.: Odd subgraphs and matchings. Discrete Math. 250, 265–272 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kano M., Katona G.Y.: Structure theorem and algorithm on (1, f)-odd subgraphs. Discrete Math. 307, 1404–1417 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  9. Király, Z.: The calculus of barriers (to appear)

  10. Lovász, L.: The factorization of graphs, Combinatorial Structures and their Applications. In: Proc. Calgary Internat. Conf., Calgary, Alta., 1969, pp. 243–246 (1970)

  11. Lovász L.: On the structure of factorizable graphs. Acta Math. Acad. Sci. Hungar. 23, 179–195 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lovász L.: The factorization of graphs, II. Acta Math. Acad. Sci. Hungar. 23, 223–246 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Lovász L., Plummer M.D.: Matching Theory. North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam (1986)

    Google Scholar 

  14. Szabó, J.: A note on the degree prescribed factor problem, EGRES Technical Reports 2004-19. http://www.cs.elte.hu/egres

  15. Szabó, J.: Graph packings and the degree prescribed subgraph problem. PhD thesis, Eötvös University, Budapest (2006)

Download references

Author information

Authors and Affiliations

  1. Department of Computer and Information Sciences, Ibaraki University, Hitachi, Ibaraki, 316-8511, Japan

    Mikio Kano

  2. Department of Computer and Information Sciences, Technical University of Budapest, P.O. Box 91, Budapest, 1521, Hungary

    Gyula Y. Katona

  3. Egerváry Research Group (EGRES), Department of Operations Research, Eötvös University, Pázmány P. s. 1/C, Budapest, 1117, Hungary

    Jácint Szabó

Authors
  1. Mikio Kano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gyula Y. Katona
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jácint Szabó
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikio Kano.

Additional information

M. Kano’s research is supported by Grant-in-Aid for Scientific Research of Japan, G.Y. Katona’s research is partially supported by the Hungarian National Research Fund and by the National Office for Research and Technology (Grant Number OTKA 67651 and 78439), J. Szabó’s research is supported by OTKA grants T037547, K60802, TS 049788 and by European MCRTN Adonet, Contract Grant No. 504438.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kano, M., Katona, G.Y. & Szabó, J. Elementary Graphs with Respect to f-Parity Factors. Graphs and Combinatorics 25, 717–726 (2009). https://doi.org/10.1007/s00373-010-0878-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-010-0878-0

Keywords

Search

Navigation