Maximal knotless graphs

TL;DR

This paper investigates the properties of maximal knotless graphs in three-dimensional space, establishing bounds on their edges, constructing examples, and classifying small cases.

Contribution

It provides new bounds on the number of edges in maximal knotless graphs, constructs infinite families, and classifies all such graphs with nine vertices and 20 edges.

Findings

Maximal knotless graphs have at least 7/4 times the number of vertices in edges.

Constructed an infinite family of maximal knotless graphs with fewer than 5/2 times the vertices in edges.

Classified all maximal knotless graphs with nine vertices and 20 edges.

Abstract

A graph is maximal knotless if it is edge maximal for the property of knotless embedding in $R^3$. We show that such a graph has at least $\frac74 |V|$ edges, and construct an infinite family of maximal knotless graphs with $|E| < \frac52|V|$. With the exception of $|E| = 22$, we show that for any $|E| \geq 20$ there exists a maximal knotless graph of size $|E|$. We classify the maximal knotless graphs through nine vertices and 20 edges. We determine which of these maxnik graphs are the clique sum of smaller graphs and construct an infinite family of maxnik graphs that are not clique sums.