Universality of cutoff for graphs with an added random matching

Abstract

We establish universality of cutoff for simple random walk on a class of random graphs defined as follows. Given a finite graph $\mathit{G}=(\mathit{V},\mathit{E})$ with $|\mathit{V}|$ even we define a random graph ${\mathit{G}}^{\ast }=(\mathit{V},\mathit{E}\cup {\mathit{E}}^{\prime })$ obtained by picking ${\mathit{E}}^{\prime }$ to be the (unordered) pairs of a random perfect matching of V. We show that for a sequence of such graphs ${\mathit{G}}_{\mathit{n}}$ of diverging sizes and of uniformly bounded degree, if the minimal size of a connected component of ${\mathit{G}}_{\mathit{n}}$ is at least 3 for all n, then the random walk on ${\mathit{G}}_{\mathit{n}}^{\ast }$ exhibits cutoff w.h.p. This provides a simple generic operation of adding some randomness to a given graph, which results in cutoff.