, Volume 38, Issue 4, pp 887–934 | Cite as

A Brooks-Type Result for Sparse Critical Graphs

  • Alexandr Kostochka
  • Matthew Yancey


A graph G is k-critical if it has chromatic number k, but every proper subgraph of G is (k-1)-colorable. Let fk(n) denote the minimum number of edges in an n-vertex k-critical graph. Recently the authors gave a lower bound, \({f_k}\left( n \right) \geqslant \left\lceil {\frac{{\left( {k + 1} \right)\left( {k - 2} \right)\left| {V\left( G \right)} \right| - k\left[ {k - 3} \right]}}{{2\left( {k - 1} \right)}}} \right\rceil \), that solves a conjecture by Gallai from 1963 and is sharp for every n≡1 (mod k-1). It is also sharp for k=4 and every n≤6. In this paper we refine the result by describing all n-vertex k-critical graphs G with \(\left| {E\left( G \right)} \right| \geqslant \frac{{\left( {k + 1} \right)\left( {k - 2} \right)\left| {V\left( G \right)} \right| - k\left( {k - 3} \right)}}{{2\left( {k - 1} \right)}}\). In particular, this result implies exact values of f5(n) for n≤7.

Mathematics Subject Classification (2000)

05C15 05C35 


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  1. [1]
    V. Aksenov: Private communication (1976).Google Scholar
  2. [2]
    O. V. Borodin: Colorings of plane graphs: A survey, Discrete Math. 313 (2013), 517–533.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    O. V. Borodin, Z. Dvorák, A. V. Kostochka, B. Lidický and M. Yancey: Planar 4-critical graphs with four triangles, European J. of Combinatorics 41 (2014), 138–151.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    O. V. Borodin and A. V. Kostochka: Defective 2-colorings of sparse graphs, J. Combin. Theory (B) 104 (2014), 72–80.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    O. V. Borodin, A. V. Kostochka, B. Lidický and M. Yancey: Short proofs of coloring theorems on planar graphs, European J. of Combinatorics 36 (2014), 314–321.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    O. V. Borodin, A. V. Kostochka and M. Yancey: On 1-improper 2-coloring of sparse graphs, Discrete Mathematics 313 (2013), no. 22, 2638–2649.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    G. A. Dirac: Note on the colouring of graphs, Math. Z. 54 (1951), 347–353.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    G. A. Dirac: A property of 4-chromatic graphs and some remarks on critical graphs, J. London Math. Soc. 27 (1952), 85–92.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    G. A. Dirac: Some theorems on abstract graphs, Proc. London Math. Soc. 2 (1952), 69–81.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    G. A. Dirac: The structure of k-chromatic graphs, Fund. Math. 40 (1953), 42–55.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    G. A. Dirac: Map colour theorems related to the Heawood colour formula, J. London Math. Soc. 31 (1956), 460–471.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    G. A. Dirac: A theorem of R. L. Brooks and a conjecture of H. Hadwiger, Proc. London Math. Soc. 7 (1957), 161–195.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    G. A. Dirac: On the structure of 5- and 6-chromatic abstract graphs, J. Reine Angew. Math. 214–215 (1964), 43–52.MathSciNetzbMATHGoogle Scholar
  14. [14]
    G. A. Dirac: The number of edges in critical graphs, J. Reine Angew. Math. 268/269 (1974), 150–164.MathSciNetzbMATHGoogle Scholar
  15. [15]
    B. Farzad and M. Molloy: On the edge-density of 4-critical graphs, Combinatorica 29 (2009), 665–689.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    T. Gallai: Kritische Graphen I, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963), 165–192.MathSciNetzbMATHGoogle Scholar
  17. [17]
    T. Gallai: Kritische Graphen II, Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963), 373–395.MathSciNetzbMATHGoogle Scholar
  18. [18]
    H. Grötzsch: 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).MathSciNetGoogle Scholar
  19. [19]
    G. Hajós: Über eine Konstruktion nicht-n-färbbarer Graphen, Wiss. Z. Martin-Luther-Unive. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 116–117.Google Scholar
  20. [20]
    S. L. Hakimi: On realizability of a set of integers as degrees of the vertices of a linear graph. I., J. Soc. Indust. Appl. Math. 10 (1962), 496–506.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    T. R. Jensen and B. Toft: Graph Coloring Problems, Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, New York, 1995.zbMATHGoogle Scholar
  22. [22]
    T. R. Jensen and B. Toft: 25 pretty graph colouring problems, Discrete Math. 229 (2001), 167–169.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    H. Kierstead and L. Rabern: Personal communication.Google Scholar
  24. [24]
    A. V. Kostochka and M. Stiebitz: Excess in colour-critical graphs, in: Graph Theory and Combinatorial Biology, Balatonlelle (Hungary), 1996, Bolyai Society, Mathematical Studies 7, Budapest, 1999, 87–99.Google Scholar
  25. [25]
    A. V. Kostochka and M. Stiebitz: A new lower bound on the number of edges in colour-critical graphs and hypergraphs, J. Comb. Theory, Series B. 87 (2003), 374–402.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    A. V. Kostochka and M. Yancey: Ore's Conjecture on color-critical graphs is almost true, J. Comb. Theory, Series B. 109 (2014), 73–101.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    A. V. Kostochka and M. Yancey: Ore's Conjecture for k=4 and Grötzsch Theorem, Combinatorica 34 (2014), 323–329.MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    M. Krivelevich: On the minimal number of edges in color-critical graphs, Combinatorica 17 (1997), 401–426.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    O. Ore: The Four Color Problem, Academic Press, New York, 1967.zbMATHGoogle Scholar
  30. [30]
    L. Postle: Personal communication.Google Scholar
  31. [31]
    R. Steinberg: The state of the three color problem, Quo Vadis, Graph Theory?, Ann. Discrete Math. 55 (1993), 211–248.MathSciNetzbMATHGoogle Scholar
  32. [32]
    B. Toft: Color-critical graphs and hypergraphs, J. Combin. Theory 16 (1974), 145–161.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    B. Toft: Personal communication.Google Scholar
  34. [34]
    Zs Tuza: Graph coloring, in: Handbook of graph theory (J. L. Gross and J. Yellen Eds.), CRC Press, Boca Raton, FL, 2004. xiv+1167.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of Illinois at Urbana-ChampaignUrbanaUSA
  2. 2.Sobolev Institute of MathematicsNovosibirskRussia
  3. 3.Department of MathematicsUniversity of IllinoisUrbanaUSA

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