Family sizes for complete multipartite graphs

TL;DR

This paper investigates the growth patterns of specific graph families, particularly complete multipartite graphs, revealing that most stabilize in size despite increasing vertices, with only a few growing exponentially.

Contribution

It provides the first analysis of family sizes for complete multipartite graphs, identifying stabilization points and exponential growth cases.

Findings

Most families stabilize after a certain size

Three families grow exponentially

Family sizes are unaffected by adding vertices beyond a point

Abstract

The obstruction set for graphs with knotless embeddings is not known, but a recent paper of Goldberg, Mattman, and Naimi indicates that it is quite large. Almost all known obstructions fall into four Triangle-Y families and they ask if there is an efficient way of finding or estimating the size of such graph families. Inspired by this question, we investigate the family size for complete multipartite graphs. Aside from three families that appear to grow exponentially, these families stabilize: after a certain point, increasing the number of vertices in a fixed part does not change family size.