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

TL;DR

This paper characterizes the typical structure of large bounded-degree graphs and proves they satisfy a logical limit law, with distinctions between labeled and unlabeled cases depending on the maximum degree.

Contribution

It provides a detailed asymptotic structural description of graphs with bounded maximum degree and establishes their adherence to a logical limit law.

Findings

Graphs with maximum degree R have a predictable typical structure as n grows.

Such graphs satisfy a labeled limit law for first-order logic.

Unlabeled limit law holds when R ≥ 5.

Abstract

For any fixed integer $R \geq 2$ 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 $R \geq 5$ then also an unlabelled limit law holds.