Limits for embedding distributions

TL;DR

This paper establishes asymptotic normality for embedding distributions of certain graph families, demonstrating a topological CLT and linking Euler-genus and crosscap-number limits, with applications to cacti and necklaces.

Contribution

It proves asymptotic normality of embedding distributions for specific graph families and connects Euler-genus and crosscap-number limits, extending topological graph theory results.

Findings

Embedding distributions of graphs with spiders are asymptotically normal.

Limits of Euler-genus and crosscap-number distributions coincide.

Cacti and necklaces have asymptotically normal embedding distributions.

Abstract

In this paper, we find and prove that, under some conditions, the embedding distributions of $H$-linear graph families with spiders are asymptotic normal distributions. It can been seen a version of central limit theorem in topological graph theory. We also prove that the limits of Euler-genus distributions is the same as limits of crosscap-number distributions. In addition, we show that the Euler-genus distributions (or crosscap-number distributions) of the cacti and necklaces are asymptotically normal distributions. In the end, some concrete examples are indicated.