Complete subgraphs in multipartite graphs


Turán’s Theorem states that every graph G of edge density \({{\left\| G \right\|} \mathord{\left/ {\vphantom {{\left\| G \right\|} {\left( {_2^{\left| G \right|} } \right) > }}} \right. \kern-\nulldelimiterspace} {\left( {_2^{\left| G \right|} } \right) > }}\frac{{k - 2}} {{k - 1}}\) contains a complete graph K k and describes the unique extremal graphs. We give a similar Theorem for λ-partite graphs. For large λ, we find the minimal edge density d k λ , such that every λ-partite graph whose parts have pairwise edge density greater than d k λ contains a K k. It turns out that \(d_\ell ^k = \frac{{k - 2}} {{k - 1}}\) for large enough λ. We also describe the structure of the extremal graphs.

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


  1. [1]

    B. Bollobás: Extremal Graph Theory, Academic Press London (1978).

  2. [2]

    A. Bondy, J. Shen, S. Thomassé and C. Thomassen: Density conditions for triangles in multipartite graphs, Combinatorica 26 (2006), 121–131.

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    R. Diestel: Graph Theory, Springer-Verlag New York (1997).

Download references

Author information



Corresponding author

Correspondence to Florian Pfender.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pfender, F. Complete subgraphs in multipartite graphs. Combinatorica 32, 483–495 (2012).

Download citation

Mathematics Subject Classification (2000)

  • 05C35 (05C69)