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( where α = 0.319. . . is an explicit constant and o(1) denotes a term tending to zero as

*H*) of the graph*H*and show that, if*H*has*t*vertices, then$$
c{\left( H \right)} = {\left( {\alpha \gamma {\left( H \right)} + o{\left( 1 \right)}} \right)}t{\sqrt {\log {\kern 1pt} {\kern 1pt} t} }
$$

*t*→∞. The extremal graphs are unions of pseudo-random graphs.If *H* has *t*^{1+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 ## Preview

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## Copyright information

© Springer-Verlag Berlin Heidelberg 2005