The Extremal Function For Noncomplete Minors

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 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.