Asymptotic Enumeration of Graph Classes with Many Components

TL;DR

This paper develops a probabilistic framework to asymptotically enumerate graph classes with many components, revealing phase transitions and explicit formulas for the number of such graphs with given vertices and components.

Contribution

It introduces a novel probabilistic approach using Boltzmann generators to derive asymptotic formulas for graph classes with many components, including phase transition analysis.

Findings

Asymptotic enumeration formulas with explicit functions of 

Identification of a critical * where behavior changes

Application of probabilistic methods to combinatorial enumeration

Abstract

We consider graph classes $\mathcal G$ in which every graph has components in a class $\mathcal{C}$ of connected graphs. We provide a framework for the asymptotic study of $\lvert\mathcal{G}_{n,N}\rvert$, the number of graphs in $\mathcal{G}$ with $n$ vertices and $N:=\lfloor\lambda n\rfloor$ components, where $\lambda\in(0,1)$. Assuming that the number of graphs with $n$ vertices in $\mathcal{C}$ satisfies \begin{align*} \lvert \mathcal{C}_n\rvert\sim b n^{-(1+\alpha)}\rho^{-n}n!, \quad n\to \infty \end{align*} for some $b,\rho>0$ and $\alpha>1$ -- a property commonly encountered in graph enumeration -- we show that \begin{align*} \lvert\mathcal{G}_{n,N}\rvert\sim c(\lambda) n^{f(\lambda)} (\log n)^{g(\lambda)} \rho^{-n}h(\lambda)^{N}\frac{n!}{N!}, \quad n\to \infty \end{align*} for explicitly given $c(\lambda),f(\lambda),g(\lambda)$ and $h(\lambda)$. These functions are piecewise continuous with a discontinuity at a critical value $\lambda^{*}$, which we also determine. The central idea in our approach is to sample objects of $\cal G$ randomly by so-called Boltzmann generators in order to translate enumerative problems to the analysis of iid random variables. By that we are able to exploit local limit theorems and large deviation results well-known from probability theory to prove our claims. The main results are formulated for generic combinatorial classes satisfying the SET-construction.