Topological minors in graphs of large girth

  • Authors:

  • Daniela Kühn;Deryk Osthus

  • Affiliations:

  • Mathematisches Seminar der Universität Hamburg, Bundesstraße 55, 20146 Hamburg, Germany;Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2002

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Abstract

We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of Kr+1 and that for r ≥ 435 a girth of at least 15 suffices. This implies that the conjecture of Hajós that every graph of chromatic number at least r contains a subdivision of Kr (which is false in general) is true for graphs of girth at least 186 (or 15 if r ≥ 436). More generally, we show that for every graph H of maximum degree Δ(H)≥2, every graph G of minimum degree at least max{δ(H), 3} and girth at least 166 log|H|/logΔ(H) contains a subdivision of H. This bound on the girth of G is best possible up to the value of the constant and improves a result of Mader, who gave a bound linear in |H|.