Minors in Expanding Graphs

Abstract.

We propose a unifying framework for studying extremal problems related to graph minors. This framework relates the existence of a large minor in a given graph to its expansion properties. We then apply the developed framework to several extremal problems and prove in particular that: (a) Every $K_{s,s^\prime}$-free graph G with average degree r ($2 \leq s \leq s^\prime$ are constants) contains a minor with average degree $cr^{1+ {\frac{1}{2(s-1)}}}$, for some constant $c = c(s, s^\prime) > 0$; (b) Every C_2k-free graph G with average degree r (k ≥ 2 is a constant) contains a minor with average degree $cr^{\frac{k+1}{2}}$, for some constant c = c(k) > 0. We also derive explicit lower bounds on the minor density in random, pseudo-random and expanding graphs.