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


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.

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  1. [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. [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. [3]

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

    Article  MATH  MathSciNet  Google Scholar 

  4. [4]

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

    MATH  MathSciNet  Google Scholar 

  5. [5]

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

    MathSciNet  Google Scholar 

  6. [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. [7]

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

    Google Scholar 

  8. [8]

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

    Article  MATH  MathSciNet  Google Scholar 

  9. [9]

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

  10. [10]

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

    Google Scholar 

  11. [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 

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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.”

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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

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  • Planar Graph
  • Simple Proof
  • Chromatic Number
  • Proper Coloring
  • Graph Coloring Problem