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

Published online by Cambridge University Press:  28 April 2004

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

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 13 , Issue 3 , May 2004 , pp. 361 - 371
Copyright
2004 Cambridge University Press

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