On a class of degenerate extremal graph problems


Given a class ℒ of (so called “forbidden”) graphs, ex (n, ℒ) denotes the maximum number of edges a graphG n of ordern can have without containing subgraphs from ℒ. If ℒ contains bipartite graphs, then ex (n, ℒ)=O(n 2−c) for somec>0, and the above problem is calleddegenerate. One important degenerate extremal problem is the case whenC 2k , a cycle of 2k vertices, is forbidden. According to a theorem of P. Erdős, generalized by A. J. Bondy and M. Simonovits [32, ex (n, {C 2k })=O(n 1+1/k). In this paper we shall generalize this result and investigate some related questions.

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


  1. [1]

    C. Benson, Minimal regular graphs of girth eight and twelve,Canad. J. Math. 18 (1966), 1091–1094.

    MATH  MathSciNet  Google Scholar 

  2. [2]

    B. Bollobás,Extremal Graph Theory, Academic Press, (1978) p. XVII.

  3. [3]

    J. A. Bondy andM. Simonovits, Cycles of even length in graphs,J. Combin. Theory,16 B (1974), 97–105.

    MathSciNet  Google Scholar 

  4. [4]

    W. G. Brown, On graphs that do not contain a Thomsen graph,Canad. Math. Bull. 9 (1966), 281–185.

    MATH  MathSciNet  Google Scholar 

  5. [5]

    W. G. Brown, P. Erdős andM. Simonovits, Extremal problems for directed graphs,J. Combin Theory,15/1B (1973), 77–93.

    MATH  Google Scholar 

  6. [6]

    P. Erdős, Some Recent results on extremal problems in graph theory,Theory of Graphs, Intern. Symp. Rome, (1966), 118–123.

  7. [7]

    P. Erdős, On some new inequalities concerning extremal properties of graphs,Theory of Graphs, Proc. Coll. Tihany (1966), 77–81.

  8. [8]

    P. Erdős, On sequences of integers no one of which divides the product of two other related problems,Mitt. Forschunginst. Math. u. Mech. Tomsk 2 (1938), 74–82.

    Google Scholar 

  9. [9]

    P. Erdős, Problems and result in combinatorial analysis,Teorice Combinatorie Proc. Coll. Intern. Roma, settembre 1973.

  10. [10]

    P. Erdős andA. Rényi, On the evolution of random graphs,Magyar Tud. Akad. Mat. Kut. Int. Kőzl. 5 (1960), 17–61.

    Google Scholar 

  11. [11]

    P. Erdős, A. Rényi andV. T. Sós, On a problem of graph theory,Studia Sci. Math. Hungar. 1 (1966), 215–235.

    MathSciNet  Google Scholar 

  12. [12]

    P. Erdős andM. Simonovits, A limit theorem in graph theory,Studia Sci. Math. Hungar. 1 (1966), 51–57.

    MathSciNet  Google Scholar 

  13. [13]

    P. Erdős andM. Simonovits, Some extremal problems in graph theory,Coll. Math. Soc. J. Bolyai 4 (1969), 377–390.

    Google Scholar 

  14. [14]

    Hylten-Cavallius. On a combinatorial problem,Coll. Math. 6 (1958), 59–65.

    MATH  MathSciNet  Google Scholar 

  15. [15]

    T. Kővári, T. Sós andP. Turán, On a problem of Zarankiewicz,Coll. Math. 3 (1954), 50–57.

    Google Scholar 

  16. [16]

    M. Simonovits, A method for solving extremal problems in graph theory,Theory of Graphs Proc. Coll. Tihany, (1966), 279–319.

  17. [17]

    M. Simonovits, Paul Turán’s influence on graph theory,J. Graph Theory,2 (1977).

  18. [18]

    R. Singleton, On minimal graphs of maximum even girth,J. Combin. Theory,1 (1966), 306–332.

    MATH  MathSciNet  Google Scholar 

  19. [19]

    P. Turán, On an extremal problem in graph theory,Mat. Fiz. Lapok 48 (1941), 436–452

    MATH  MathSciNet  Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Faudree, R.J., Simonovits, M. On a class of degenerate extremal graph problems. Combinatorica 3, 83–93 (1983). https://doi.org/10.1007/BF02579343

Download citation

AMS subject classification (1980)

  • 05 C 35