Subdivisions of $K_{r+2}$ in Graphs of Average Degree at Least $r+\varepsilon$ and Large but Constant Girth

  • Authors:

  • Daniela Kühn;Deryk Osthus

  • Affiliations:

  • Freie Universität Berlin, Institut für Mathematik, Arnimallee 2–6, D – 14195 Berlin, Germany (e-mail: dkuehn@math.fu-berlin.de);Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D – 10099 Berlin, Germany (e-mail: osthus@informatik.fu-berlin.de)

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

Quantified Score

Hi-index 0.00

Visualization

Abstract

We show that for every $\varepsilon\,{}\,0$ there exists an $r_0\,{=}\,r_0(\varepsilon)$ such that, for all integers $r\,{\ge}\, r_0$, every graph of average degree at least $r+\varepsilon$ and girth at least 1000 contains a subdivision of $K_{r+2}$. Combined with a result of Mader this implies that, for every $\varepsilon\,{}\,0$, there exists an $f(\varepsilon)$ such that, for all $r\,{\ge}\, 2$, every graph of average degree at least $r+\varepsilon$ and girth at least $f(\varepsilon)$ contains a subdivision of $K_{r+2}$. We also prove a more general result concerning subdivisions of arbitrary graphs.