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Local Improvement Gives Better Expanders

It has long been known that random regular graphs are with high probability good expanders. This was first established in the 1980s by Bollobás by directly calculating the probability that a set of vertices has small expansion and then applying the union bound. In this paper we improve on this analysis by relying on a simple high-level observation: if a graph contains a set of vertices with small expansion then it must also contain such a set of vertices that is locally optimal, that is, a set whose expansion cannot be made smaller by exchanging a vertex from the set with one from the set's complement. We show that the probability that a set of vertices satisfies this additional property is significantly smaller. Thus, after again applying the union bound, we obtain improved lower bounds on the expansion of random $Δ$-regular graphs for $Δ\ge 4$. In fact, the gains from this analysis increase as $Δ$ grows, a fact we explain by extending our technique to general $Δ$. Thus, in the end we obtain an improvement not only for some small special cases but on the general asymptotic bound on the expansion of $Δ$-regular graphs given by Bollobás.