Skip to main content
Log in

Ore’s conjecture for k=4 and Grötzsch’s Theorem

Combinatorica Aims and scope Submit manuscript

Abstract

A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k−1)-colorable. Let f k (n) denote the minimum number of edges in an n-vertex k-critical graph. In a very recent paper, we gave a lower bound, f k (n)≥(k, n), that is sharp for every n≡1 (mod k−1). It is also sharp for k=4 and every n≥6. In this note, we present a simple proof of the bound for k=4. It implies the case k=4 of two conjectures: Gallai in 1963 conjectured that if n≡1 (mod k−1) then \(f_k (n)\tfrac{{(k + 1)(k - 2)n - k(k - 3)}} {{2(k - 1)}}\), and Ore in 1967 conjectured that for every k≥4 and \(n \geqslant k + 2,f_k (n + k - 1) = f(n) + \tfrac{{k - 1}} {2}(k - \tfrac{2} {{k - 1}})\). We also show that our result implies a simple short proof of Grötzsch’s Theorem that every triangle-free planar graph is 3-colorable.

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

Access this article

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. O. V. Borodin, A. V. Kostochka, B. Lidicky and M. Yancey: Short proofs of coloring theorems on planar graphs, European J. of Combinatorics 36 (2014), 314–321.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. A. Dirac: A theorem of R. L. Brooks and a conjecture of H. Hadwiger, Proc. London Math. Soc. 7 (1957), 161–195.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Farzad, M. Molloy: On the edge-density of 4-critical graphs, Combinatorica 29 (2009), 665–689.

    Article  MATH  MathSciNet  Google Scholar 

  4. T. Gallai: Kritische Graphen I, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963), 165–192.

    MATH  MathSciNet  Google Scholar 

  5. T. Gallai: Kritische Graphen II, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963), 373–395.

    MathSciNet  Google Scholar 

  6. H. Götzsch: Zur Theorie der diskreten Gebilde. VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-Nat. Reihe 8 (1958/1959), 109–120 (in German).

    MathSciNet  Google Scholar 

  7. T. R. Jensen and B. Toft: Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1995.

    MATH  Google Scholar 

  8. T. R. Jensen and B. Toft: 25 pretty graph colouring problems, Discrete Math. 229 (2001), 167–169.

    Article  MATH  MathSciNet  Google Scholar 

  9. A. V. Kostochka and M. Yancey: Ore’s Conjecture is almost true, submitted.

  10. O. Ore: The Four Color Problem, Academic Press, New York, 1967.

    MATH  Google Scholar 

  11. C. Thomassen: A short list color proof of Göotzsch’s theorem, J. Combin. Theory Ser. B 88 (2003), 189–192.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Alexandr Kostochka.

Additional information

Research of this author is supported in part by NSF grant DMS-0965587 and by grants 12-01-00448 and 12-01-00631 of the Russian Foundation for Basic Research.

Research of this author is partially supported by the Arnold O. Beckman Research Award of the University of Illinois at Urbana-Champaign and from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students.”

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kostochka, A., Yancey, M. Ore’s conjecture for k=4 and Grötzsch’s Theorem. Combinatorica 34, 323–329 (2014). https://doi.org/10.1007/s00493-014-3020-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-014-3020-x

Keywords

Navigation