Random graphs with bounded maximum degree: asymptotic structure and a logical limit law

Abstract

For any fixed integer $R\ge 2$ we characterise the typical structure of undirected graphs with vertices $1,...,n$ and maximum degree $R$, as $n$ tends to infinity. The information is used to prove that such graphs satisfy a labelled limit law for first-order logic. If $R\ge 5$ then also an unlabelled limit law holds.

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