Small minors in dense graphs

Abstract

A fundamental result in structural graph theory states that every graph with large average degree contains a large complete graph as a minor. We prove this result with the extra property that the minor is small with respect to the order of the whole graph. More precisely, we describe functions f and h such that every graph with n vertices and average degree at least f(t) contains a Kt-model with at most h(t)⋅logn vertices. The logarithmic dependence on n is best possible (for fixed t). In general, we prove that f(t)≤2t−1+ε. For t≤4, we determine the least value of f(t); in particular, f(3)=2+ε and f(4)=4+ε. For t≤4, we establish similar results for graphs embedded on surfaces, where the size of the Kt-model is bounded (for fixed t).