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Combinatorica

, Volume 25, Issue 6, pp 725–753 | Cite as

The Extremal Function For Noncomplete Minors

  • Joseph Samuel Myers*
  • Andrew Thomason
Original Paper
We investigate the maximum number of edges that a graph G can have if it does not contain a given graph H as a minor (subcontraction). Let
$$ c{\left( H \right)} = \inf {\left\{ {c:e{\left( G \right)} \geqslant c{\left| G \right|}{\text{implies}}{\kern 1pt} {\kern 1pt} G \succ H} \right\}}. $$
We define a parameter γ(H) of the graph H and show that, if H has t vertices, then
$$ c{\left( H \right)} = {\left( {\alpha \gamma {\left( H \right)} + o{\left( 1 \right)}} \right)}t{\sqrt {\log {\kern 1pt} {\kern 1pt} t} } $$
where α = 0.319. . . is an explicit constant and o(1) denotes a term tending to zero as t→∞. The extremal graphs are unions of pseudo-random graphs.

If H has t1+τ edges then \( \gamma {\left( H \right)} \leqslant {\sqrt \tau } \), equality holding for almost all H and for all regular H. We show how γ(H) might be evaluated for other graphs H also, such as complete multi-partite graphs.

Mathematics Subject Classification (2000):

05C83 05C35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsCentre for Mathematical SciencesCambridge, CB3 0WBUnited Kingdom

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