Critical graphs without triangles: An optimum density construction

Abstract

We construct dense, triangle-free, chromatic-critical graphs of chromatic number k for all k ≥ 4. For k ≥ 6 our constructions have \(> \left( {\tfrac{1} {4} - \varepsilon } \right)n^2\) edges, which is asymptotically best possible by Turán’s theorem. We also demonstrate (nonconstructively) the existence of dense k-critical graphs avoiding all odd cycles of length ≤ for any and any k≥4, again with a best possible density of \(> \left( {\tfrac{1} {4} - \varepsilon } \right)n^2\) edges for k ≥ 6. The families of graphs without triangles or of given odd-girth are thus rare examples where we know the correct maximal density of k-critical members (k ≥ 6).

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References

  1. [1]

    S. Brandt: On the structure of dense triangle-free graphs, Combinatorica 20 (1999), 237–245.

    Google Scholar 

  2. [2]

    S. Brandt and S. Thomassé: Dense triangle-free graphs are 4-colorable: A solution to the Erdos-Simonovits problem, http://www.lirmm.fr/~thomasse/liste/vega11.pdf.

  3. [3]

    M. El-Zahar and N. Sauer: The chromatic number of the product of two 4-chromatic graphs is 4, Combinatorica, 5 (1985), 121–126.

    Article  MATH  MathSciNet  Google Scholar 

  4. [4]

    P. Erdős, D. J. Kleitman and B. L. Rothschild: Asymptotic enumeration of K n-free graphs, in: Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo II, pages 19–27. Atti dei Convegni Lincei, No. 17. Accad. Naz. Lincei, Rome, 1976.

    Google Scholar 

  5. [5]

    P. Erdős and M. Simonovits: On a valence problem in extremal graph theory, Discrete Mathematics, 5 (1973), 323–334.

    Article  MathSciNet  Google Scholar 

  6. [6]

    P. Erdős and A. H. Stone: On the structure of linear graphs, Bulletin of the American Mathematical Society, 52 (1946), 1087–1091.

    Article  MathSciNet  Google Scholar 

  7. [7]

    A. Gyárfás: Personal communication, 2005.

  8. [8]

    A. Gyárfás, T. Jensen and M. Stiebitz: On Graphs With Strongly Independent Color-Classes, Journal of Graph Theory 46 (2004), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  9. [9]

    R. Häggkvist: Odd cycles of specified length in nonbipartite graphs, in: Graph Theory, pages 89–99. Annals of Discrete Mathematics, Vol. 13. North-Holland, Amsterdam-New York, 1982.

    Google Scholar 

  10. [10]

    T. R. Jensen: Dense critical and vertex-critical graphs, Discrete Mathematics, 258 (2002), 63–84.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    G. Jin: Triangle-free four-chromatic graphs, Discrete Mathematics, 145 (1995), 151–170.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    A. V. Kostochka and M. Stiebnitz: On the number of edges in colour-critical graphs and hypergraphs, Combinatorica 20 (2000), 521–530.

    Article  MATH  MathSciNet  Google Scholar 

  13. [13]

    I. Kříž: A hypergraph-free construction of highly chromatic graphs without short cycles, Combinatorica 9 (1989), 227–229.

    Article  MATH  MathSciNet  Google Scholar 

  14. [14]

    L. Lovász: On chromatic number of graphs and set-systems, Acta Math. Hungar. 19 (1968), 59–67.

    MATH  Google Scholar 

  15. [15]

    L. Lovász: Self-dual polytopes and the chromatic number of distance graphs on the sphere, Acta Scientiarum Mathematicarum 45 (1983), 317–323.

    MATH  Google Scholar 

  16. [16]

    J. Mycielski: Sur le coloriage des graphes, Colloq. Math., 3 (1955), 161–162.

    MATH  MathSciNet  Google Scholar 

  17. [17]

    A. Schrijver: Vertex-critical subgraphs of kneser graphs, Nieuw Arch. Wiskd. III. Ser. 26 (1978), 454–461.

    MATH  MathSciNet  Google Scholar 

  18. [18]

    M. Stiebitz: Beiträge zur Theorie der färbungskritschen Graphen, PhD thesis, Technical University Ilmenaur, 1985.

    Google Scholar 

  19. [19]

    C. Tardif: Fractional chromatic numbers of cones over graphs, Journal of Graph Theory 38 (2001), 87–94.

    Article  MATH  MathSciNet  Google Scholar 

  20. [20]

    C. Thomassen: On the chromatic number of triangle-free graphs of large minimum degree, Combinatorica 22 (2002), 591–596.

    Article  MATH  MathSciNet  Google Scholar 

  21. [21]

    C. Thomassen: On the chromatic number of pentagon-free graphs of large minimum degree, Combinatorica 27 (2007), 241–243.

    Article  MATH  MathSciNet  Google Scholar 

  22. [22]

    B. Toft: On the maximal number of edges of critical k-chromatic graphs, Studia Sci. Math. Hungar. 5 (1970), 461–470.

    MathSciNet  Google Scholar 

  23. [23]

    A. Zykov: On some properties of linear complexes (in Russian), Matem. Sbornik 24 (1949), 163–187.

    MathSciNet  Google Scholar 

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Correspondence to Wesley Pegden.

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Pegden, W. Critical graphs without triangles: An optimum density construction. Combinatorica 33, 495–512 (2013). https://doi.org/10.1007/s00493-013-2440-1

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Mathematics Subject Classification (2010)

  • 05C15
  • 05C35