Large Topological Cliques in Graphs Without a 4-Cycle

  • Authors:

  • Daniela Kühn;Deryk Osthus

  • Affiliations:

  • Mathematisches Seminar, Universität Hamburg, Bundesstraße 55, D – 20146 Hamburg, Germany (e-mail: kuehn@math.uni-hamburg.de);Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D – 10099 Berlin, Germany (e-mail: osthus@informatik.hu-berlin.de)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2004

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Abstract

Mader asked whether every $C_4$-free graph $G$ contains a subdivision of a complete graph whose order is at least linear in the average degree of $G$. We show that there is a subdivision of a complete graph whose order is almost linear. More generally, we prove that every $K_{s,t}$-free graph of average degree $r$ contains a subdivision of a complete graph of order $r^{\frac{1}{2}{+}\frac{1}{2(s-1)}-o(1)}$.