Distributions of sparse spanning subgraphs in random graphs

TL;DR

This paper presents a general method for determining the distribution of various spanning subgraphs in random graphs, including those with specific degree sequences, triangle-free structures, Hamilton cycles, and collections of disjoint triangles.

Contribution

It introduces a unified approach to analyze the distribution of diverse spanning subgraphs in Erdős–Rényi random graphs, extending previous results to broader classes.

Findings

Derived distributions for spanning subgraphs with specified degree sequences

Extended analysis to triangle-free and Hamilton cycle subgraphs

Provided a framework applicable to various isomorphic subgraph collections

Abstract

We describe a general approach of determining the distribution of spanning subgraphs in the random graph $\G(n,p)$. In particular, we determine the distribution of spanning subgraphs of certain given degree sequences, which is a generalisation of the $d$-factors, of spanning triangle-free subgraphs, of (directed) Hamilton cycles and of spanning subgraphs that are isomorphic to a collection of vertex disjoint (directed) triangles.