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Combinatorica

, Volume 26, Issue 2, pp 121–131 | Cite as

Density Conditions For Triangles In Multipartite Graphs

  • Adrian BondyEmail author
  • Jian Shen
  • Stéphan Thomassé
  • Carsten Thomassen
Original Paper

We consider the problem of finding a large or dense triangle-free subgraph in a given graph G. In response to a question of P. Erdős, we prove that, if the minimum degree of G is at least 9|V (G)|/10, the largest triangle-free subgraphs are precisely the largest bipartite subgraphs in G. We investigate in particular the case where G is a complete multipartite graph. We prove that a finite tripartite graph with all edge densities greater than the golden ratio has a triangle and that this bound is best possible. Also we show that an infinite-partite graph with finite parts has a triangle, provided that the edge density between any two parts is greater than 1/2.

Mathematics Subject Classification (2000):

05C35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Adrian Bondy
    • 1
    Email author
  • Jian Shen
    • 2
  • Stéphan Thomassé
    • 3
  • Carsten Thomassen
    • 4
  1. 1.Laboratoire LaPCS, UFR de MathématiquesUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance
  2. 2.Department of MathematicsTexas State UniversitySan Marcos,TX 78666USA
  3. 3.Laboratoire LaPCS, UFR de MathématiquesUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance
  4. 4.Institute of Mathematics, Building 303DTULyngbyDenmark

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