On the minimal number of edges in color-critical graphs


A graphG isk-critical if it has chromatic numberk, but every proper subgraph of it is (k−1)-colorable. This paper is devoted to investigating the following question: for givenk andn, what is the minimal number of edges in ak-critical graph onn vertices, with possibly some additional restrictions imposed? Our main result is that for everyk≥4 andn>k this number is at least\(\left( {\frac{{k - 1}}{2} + \frac{{k - 3}}{{2(k^2 - 2k - 1)}}} \right)n\), thus improving a result of Gallai from 1963. We discuss also the upper bounds on the minimal number of edges ink-critical graphs and provide some constructions of sparsek-critical graphs. A few applications of the results to Ramsey-type problems and problems about random graphs are described.

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Alon andJ. H. Spencer:The probabilistic method, Wiley, New York, 1992.

    Google Scholar 

  2. [2]

    B. Bollobás:Random graphs, Academic Press, New York, 1985.

    Google Scholar 

  3. [3]

    B. Bollobás andH. R. Hind: Graphs without large triangle-free subgraphs,Discrete Math. 87 (1991), 119–131.

    Google Scholar 

  4. [4]

    R. L. Brooks: On colouring the nodes of a network,Proc. Cambridge Phil. Soc.,37 (1941), 194–197.

    Google Scholar 

  5. [5]

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

    Google Scholar 

  6. [6]

    P. Erdős: Some new applications of probability methods to combinatorial analysis and graph theory,Proc. 5th S.E. Conf. in Combinatorics, Graph Theory and Computing, (1974), 39–51.

  7. [7]

    P. Erdős andC. A. Rogers: The construction of certain graphs,Canad. J. Math.,14 (1962), 702–707.

    Google Scholar 

  8. [8]

    P. Erdős andP. Tetali: Representations of integers as the sum ofk terms,Random Struct. Alg.,1 (1990), 245–261.

    Google Scholar 

  9. [9]

    M. Fekete: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koefficienten,Math. Z.,17 (1923), 228–249.

    Google Scholar 

  10. [10]

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

    Google Scholar 

  11. [11]

    T. R. Jensen andB. Toft:Graph coloring problems, Wiley, New York, 1995.

    Google Scholar 

  12. [12]

    H. A. Kierstead, E. Szemerédi andW. T. Trotter: On coloring graphs with locally small chromatic number,Combinatorica,4 (1984), 183–185.

    Google Scholar 

  13. [13]

    M. Krivelevich:K S-free graphs without largeK r-free subgraphs,Combinat., Probab. Comput.,3 (1994), 349–354.

    Google Scholar 

  14. [14]

    M. Krivelevich: Bounding Ramsey numbers through large deviation inequalities,Random Struct. Alg.,7 (1995), 145–155.

    Google Scholar 

  15. [15]

    N. Linial andYu. Rabinovich: Local and global clique numbers,J. Combin. Theory, Ser. B.,61 (1994), 5–15.

    Google Scholar 

  16. [16]

    T. Luczak: A note on the sharp concentration of the chromatic number of random graphs,Combinatorica,11 (1991), 295–297.

    Google Scholar 

  17. [17]

    V. D. Milman andG. Schechtman:Asymptotic theory of finite dimensional normed spaces, Lecture Notes in Mathematics 1200, Springer Verlag, Berlin and New York, 1986.

    Google Scholar 

  18. [18]

    H. Sachs andM. Stiebitz: Colour-critical graphs with vertices of low valency,Annals of Discrete Math.,41 (1989), 371–396.

    Google Scholar 

  19. [19]

    H. Sachs andM. Stiebitz: On constructive, methods in the theory of colour-critical graphs,Discrete Math.,74 (1989), 201–226.

    Google Scholar 

  20. [20]

    E. Shamir andJ. Spencer: Sharp concentration of the chromatic number of random graphsG n,p ,Combinatorica,7 (1987), 124–129.

    Google Scholar 

  21. [21]

    M. Stiebitz: Proof of a conjecture of T. Gallai concernign connectivity properties of colour-critical graphs,Combinatorica,2 (1982), 315–323.

    Google Scholar 

  22. [22]

    B. Toft: Color-critical graphs, and hypergraphs,J. Combin. Theory, Ser. B,16 (1974), 145–161.

    Google Scholar 

  23. [23]

    D. A. Youngs: 4-Chromatic projective graphs,J Graph Theory,21 (1996), 219–227.

    Google Scholar 

Download references

Author information



Additional information

This research forms part of a Ph.D. thesis written by the author under the supervision of Professor Noga Alon. Research supported in part by a Charles Clore Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Krivelevich, M. On the minimal number of edges in color-critical graphs. Combinatorica 17, 401–426 (1997). https://doi.org/10.1007/BF01215921

Download citation

Mathematics Subject Classification (1991)

  • 05 C 15
  • 05 C 35
  • 05 C 80