  We establish a bound for the classic PUSH-PULL rumor spreading protocol on
arbitrary graphs, in terms of the vertex expansion of the graph. We show that
O(log^2(n)/\alpha) rounds suffice with high probability to spread a rumor from
a single node to all n nodes, in any graph with vertex expansion at least
\alpha. This bound matches the known lower bound, and settles the question on
the relationship between rumor spreading and vertex expansion asked by
Chierichetti, Lattanzi, and Panconesi (SODA 2010). Further, some of the
arguments used in the proof may be of independent interest, as they give new
insights, for example, on how to choose a small set of nodes in which to plant
the rumor initially, to guarantee fast rumor spreading.
