Skip to main content
SpringerLink
Log in

Approximate Min-max Relations for Odd Cycles in Planar Graphs

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3509))

  • 1161 Accesses

Abstract

We study the ratio between the minimum size of an odd cycle vertex transversal and the maximum size of a collection of vertex-disjoint odd cycles in a planar graph. We show that this ratio is at most 10. For the corresponding edge version of this problem, Král and Voss [7] recently proved that this ratio is at most 2; we also give a short proof of their result.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Appel, K., Haken, W.: A proof of the four color theorem. Discrete Mathematics 16, 179–180 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  2. Diestel, R.: Graph Theory, 2nd edn. Graduate Texts in Mathematics, vol. 173. Springer, New-York (2000)

    Google Scholar 

  3. Dilworth, R.P.: A decomposition theorem for partially ordered sets. Annals of Mathematics 51, 161–166 (1950)

    Article  MathSciNet  Google Scholar 

  4. Frank, A.: A survey on T-joins, T-cuts, and conservative weightings. In Combinatorics, Paul Erdös is eighty, 2, Keszthely (1993), vol. 2 of Bolyai Society Mathematical Studies, pp. 213–252. János Bolyai Mathematical Society, Budapest (1996)

    Google Scholar 

  5. Goemans, M.X., Williamson, D.P.: Primal-dual approximation algorithms for feedback problems in planar graphs. Combinatorica 18, 37–59 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hadlock, F.: Finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing 4, 221–225 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  7. Král, D., Voss, H.: Edge-disjoint odd cycles in planar graphs. Journal of Combinatorial Theory, Series B 90, 107–120 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Lovász, L.: 2-matchings and 2-covers of hypergraphs. Acta Mathematica Academiae Scientiarum Hungaricae 26, 433–444 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  9. Rautenbach, D., Reed, B.: The Erdős-Pósa property for odd cycles in highly connected graphs. Combinatorica 21, 267–278 (2001); Paul Erdős and his mathematics (Budapest, 1999)

    Article  MathSciNet  Google Scholar 

  10. Reed, B.: On mangoes and blueberries. Combinatorica 19, 267–296 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Robertson, N., Sanders, D., Seymour, P., Thomas, R.: The four colour theorem. Journal of Combinatorial Theory, Series B 70, 2–44 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Algorithms and Combinatorics, vol. 24. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  13. Seymour, P.: On odd cuts and plane multicommodity flows. Proceedings of the London Mathematical Society (Third Series) 42, 178–192 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  14. Seymour, P.: Packing directed circuits fractionally. Combinatorica 15, 281–288 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  15. Thomassen, C.: On the presence of disjoint subgraphs of a specified type. Journal of Graph Theory 12, 101–111 (1988)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. GERAD, Montreal, Quebec, Canada

    Samuel Fiorini

  2. Université Libre de Bruxelles, Brussels, Belgium

    Samuel Fiorini

  3. McGill University, Montreal, Quebec, Canada

    Nadia Hardy, Bruce Reed & Adrian Vetta

Authors
  1. Samuel Fiorini
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nadia Hardy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bruce Reed
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Adrian Vetta
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität zu Köln,  

    Michael Jünger

  2. Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Volker Kaibel

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Fiorini, S., Hardy, N., Reed, B., Vetta, A. (2005). Approximate Min-max Relations for Odd Cycles in Planar Graphs. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_4

Download citation

  • DOI: https://doi.org/10.1007/11496915_4

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26199-5

  • Online ISBN: 978-3-540-32102-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation