Graph decomposition and parity

TL;DR

This paper investigates the distribution of subgraph counts in random graphs, providing conditions for uniform distribution modulo q, and characterizes the asymptotic behavior for certain graph classes.

Contribution

It introduces a graph decomposition approach to analyze subgraph distributions and determines asymptotic distributions for specific graph families in G(n,p).

Findings

Asymptotic distribution of two-component graphs in G(n,p) is characterized.

Infinite families of multi-component graphs with uniform distribution are identified.

A negative result shows no simple proof exists for uniform distribution in all graphs.

Abstract

Motivated by a recent extension of the zero-one law by Kolaitis and Kopparty, we study the distribution of the number of copies of a fixed disconnected graph in the random graph $G(n,p)$. We use an idea of graph decompositions to give a sufficient condition for this distribution to tend to uniform modulo $q$. We determine the asymptotic distribution of all fixed two-component graphs in $G(n,p)$ for all $q$, and we give infinite families of many-component graphs with a uniform asymptotic distribution for all $q$. We also prove a negative result, that no simple proof of uniform asymptotic distribution for arbitrary graphs exists.