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.
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