Asymptotic normality of some graph sequences
David Galvin

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Combinatorial Mathematics · Limits and Structures in Graph Theory · Graph theory and applications
