On the number of edge disjoint cliques in graphs of given size


In this paper, we prove that any graph ofn vertices andt r−1(n)+m edges, wheret r−1(n) is the Turán number, contains (1−o(1)m edge disjointK r'sifm=o(n 2). Furthermore, we determine the maximumm such that every graph ofn vertices andt r−1(n)+m edges containsm edge disjointK r's ifn is sufficiently large.

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


  1. [1]

    B. Bollobás,Extremal Graph Theory, Academic Press, 1978.

  2. [2]

    B. Bollobás, On complete subgraphs of different orders,Math. Proc. Cambridge Philos. Soc. 79 (1976), 19–24.

    Google Scholar 

  3. [3]

    P. Erdős, Some unsolved problems in graph theory and combinatorial analysis,Combinatorial Mathematics and Its Applications (Proc. Conf., Oxford, 1969), Academic Press, 1971, 97–109.

  4. [4]

    P. Erdős, A. Goodman andL. Pósa, The representation of graphys by set intersections,Canad. J. Math. 18 (1966), 106–112.

    Google Scholar 

  5. [5]

    E. Győri, On the number of edge disjoint triangles in graphs of given size, Combinatorics,Proceedings of the 7-th Hungarian Combinatorial Colloquium, 1987, Eger, 267–276.

  6. [6]

    E. Győri andZ. Tura, Decomposition of graphs into complete subgraphs of given order,Studia Sci. Math. Hung. 22 (1987), 315–320.

    Google Scholar 

  7. [7]

    L. Lovász andM. Simonovits, On the number of complete subgraphs of a graph II,Studies in Pure Mathematics, To the memory of Paul Turán (eds. P. Erdös, L. Alpár, G. Halász, etc.), 1983, Akad. Kiadó, 459–495.

  8. [8]

    M. Simonovits, A method for solving extremal problems in graph theory, stability problems,Theory of Graphs, Academic Press, New York, 1968, 279–319.

    Google Scholar 

Download references

Author information



Additional information

Research partially supported by Hungarian National Foundation for Scientific Research Grant no. 1812.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Győri, E. On the number of edge disjoint cliques in graphs of given size. Combinatorica 11, 231–243 (1991). https://doi.org/10.1007/BF01205075

Download citation

AMS code classification (1980)

  • 05 C 35