Local limit theorem for joint subgraph counts

TL;DR

This paper establishes a local limit theorem for joint subgraph counts in Erdős-Rényi graphs, describing their distribution as a nonlinear transformation of a multivariate normal, and applies it to graph enumeration problems.

Contribution

It extends previous work by proving a local limit theorem for joint subgraph counts, linking them to a multivariate normal distribution and solving open problems in graph enumeration.

Findings

Joint subgraph counts follow a nonlinear transformed multivariate normal distribution.

Results confirm the existence and enumeration of proportional graphs.

Answers to several open questions in graph theory are provided.

Abstract

Extending a previous result of the first two authors, we prove a local limit theorem for the joint distribution of subgraph counts in the Erd\H{o}s-R\'{e}nyi random graph $G(n,p)$. This limit can be described as a nonlinear transformation of a multivariate normal distribution, where the components of the multivariate normal correspond to the graph factors of Janson. As an application, we show a number of results concerning the existence and enumeration of proportional graphs and related concepts, answering various questions of Janson and collaborators in the affirmative.