Complete subgraphs in multipartite graphs

Abstract

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.

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Correspondence to Florian Pfender.

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Pfender, F. Complete subgraphs in multipartite graphs. Combinatorica 32, 483–495 (2012). https://doi.org/10.1007/s00493-012-2425-5

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Mathematics Subject Classification (2000)

  • 05C35 (05C69)