Complete Minors in Cubic Graphs with Few Short Cycles and Random Cubic Graphs

Abstract

We first prove that for any fixed \(k\), a cubic graph with few short cycles contains a \(K_{k}\)-minor. This is a direct generalization of a result on girth by Thomassen. We then use this theorem to show that for any fixed \(k\), a random cubic graph contains a \(K_{k}\)-minor asymptotically almost surely.