Asymptotic normality of some graph sequences

TL;DR

This paper proves that the number of partitions of a graph's vertices into independent sets follows a normal distribution asymptotically, extending previous results from simple graphs to more complex classes.

Contribution

It confirms and extends Do and Galvin's conjecture on asymptotic normality for graph Stirling numbers, including graphs co-chromatic with quasi-threshold graphs.

Findings

Asymptotic normality holds for a broader class of graphs.

The proof combines multiple advanced mathematical techniques.

The result generalizes known cases from edgeless graphs to complex graph classes.

Abstract

For a simple finite graph G denote by {G \brace k} the number of ways of partitioning the vertex set of G into k non-empty independent sets (that is, into classes that span no edges of G). If E_n is the graph on n vertices with no edges then {E_n \brace k} coincides with {n \brace k}, the ordinary Stirling number of the second kind, and so we refer to {G \brace k} as a graph Stirling number. Harper showed that the sequence of Stirling numbers of the second kind, and thus the graph Stirling sequence of E_n, is asymptotically normal --- essentially, as n grows, the histogram of ({E_n \brace k})_{k \geq 0}, suitably normalized, approaches the density function of the standard normal distribution. In light of Harper's result, it is natural to ask for which sequences (G_n)_{n \geq 0} of graphs is there asymptotic normality of ({G_n \brace k})_{k \geq 0}. Do and Galvin conjectured that if for each n, G_n is acylic and has n vertices, then asymptotic normality occurs, and they gave a proof under the added condition that G_n has no more than o(\sqrt{n/\log n}) components. Here we settle Do and Galvin's conjecture in the affirmative, and significantly extend it, replacing "acyclic" in their conjecture with "co-chromatic with a quasi-threshold graph, and with negligible chromatic number". Our proof combines old work of Navon and recent work of Engbers, Galvin and Hilyard on the normal order problem in a Weyl algebra, and work of Kahn on the matching polynomial of a graph.