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