Skip to main content
SpringerLink
Log in

Proof of Toft’s Conjecture: Every Graph Containing No Fully Odd K 4 Is 3-Colorable

Extended Abstract

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1449))

Included in the following conference series:

  • 359 Accesses

Abstract

The graph 3-coloring problem arises in connection with certain scheduling and partition problems. As is well known, this problem is NP-complete and therefore intractable in general unless NP = P. The present paper is devoted to the 3-coloring problem on a large class of graphs, namely, graphs containing no fully odd K 4, where a fully odd K 4 is a subdivision of K 4 such that each of the six edges of the K 4 is subdivided into a path of odd length. In 1974, Toft conjectured that every graph containing no fully odd K 4 can be vertex-colored with three colors. The purpose of this paper is to prove Toft’s conjecture.

This work was supported in part by the Air Force Office of Scientific Research under grant F49620-93-1-0041 awarded to Rutgers Cent

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. K. Appel and W. Haken, Every Planar Map Is Four Colorable, A.M.S. Contemp. Math.98 (1989).

    Google Scholar 

  2. P. Catlin, Hajós Graph-Coloring Conjecture: Variations and Counterexamples, J. Combin. Theory Ser. B26 (1979), 268–274.

    Article  MathSciNet  Google Scholar 

  3. V. Chvátal, On Certain Polytopes Associated with Graphs, J. Combin. Theory Ser. B18 (1975), 138–154.

    Article  MathSciNet  Google Scholar 

  4. G. A. Dirac, A Property of 4-Chromatic Graphs and Some Remarks on Critical Graphs, J. London Math. Soc.27 (1952), 85–92.

    Article  MathSciNet  Google Scholar 

  5. H. Hadwiger, Über eine Klassifikation der Streckenkomplexe, Vierteljahrsch. Naturforsch. Ges. Zürich88 (1943), 133–142.

    MATH  Google Scholar 

  6. G. Hajós, Über eine Konstruktion nicht n-färbbarer Graphen, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe10 (1961), 116–117.

    Google Scholar 

  7. T. R. Jensen and F. B. Shepherd, Note on a Conjecture of Toft, Combinatorica15 (1995), 373–377.

    Article  MathSciNet  Google Scholar 

  8. U. Krusenstjerna-Hafstrøm and B. Toft, Special Subdivisions of K 4 and 4-Chromatic Graphs, Monatsh. Math.89 (1980), 101–110.

    Article  MathSciNet  Google Scholar 

  9. N. Robertson, D. Sanders, P. Seymour and R. Thomas, The Four-Colour Theorem, J. Combin. Theory Ser. B70 (1997), 2–44.

    Article  MathSciNet  Google Scholar 

  10. N. Robertson, P. Seymour and R. Thomas, Hadwiger’s Conjecture for K 6-Free Graphs, Combinatorica13 (1993), 279–361.

    Article  MathSciNet  Google Scholar 

  11. E. C. Sewell and L. E. Trotter, Stability Critical Graphs and Even Subdivisions of K 4, J. Combin. Theory Ser. B59 (1993), 74–84.

    Article  MathSciNet  Google Scholar 

  12. C. Thomassen and B. Toft, Non-separating Induced Cycles in Graphs, J. Combin. Theory Ser. B31 (1981), 199–224.

    Article  MathSciNet  Google Scholar 

  13. B. Toft, “Problem 10”, in: Recent Advances in Graph Theory, Proc. of the Symposium held in Prague, June 1974 (M. Fiedler, ed.), Academia Praha, 1975, pp. 543–544.

    Google Scholar 

  14. K. Wagner, Über eine Eigenschaft der ebene Komplexe, Math. Ann.114 (1937), 570–590.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Hong Kong, Hong Kong

    Wenan Zang

Authors
  1. Wenan Zang
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, 11529, Nankang, Taipei, Taiwan

    Wen-Lian Hsu

  2. Department of Computer Science, Yale University, 51 Prospect Street, New Haven, CT, 06520, USA

    Ming-Yang Kao

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Zang, W. (1998). Proof of Toft’s Conjecture: Every Graph Containing No Fully Odd K 4 Is 3-Colorable. In: Hsu, WL., Kao, MY. (eds) Computing and Combinatorics. COCOON 1998. Lecture Notes in Computer Science, vol 1449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68535-9_30

Download citation

  • DOI: https://doi.org/10.1007/3-540-68535-9_30

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64824-6

  • Online ISBN: 978-3-540-68535-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation