Linearly embedded graphs in 3-space with homotopically free exteriors

TL;DR

This paper explores conditions under which linear embeddings of graphs into three-dimensional space have free fundamental groups of their exteriors, revealing that small graphs with high minimal valency always have free embeddings, while larger or more connected graphs can have non-free embeddings.

Contribution

It establishes that all simple connected graphs with up to 6 vertices and minimal valency at least 3 have free linear embeddings, and shows that larger or highly connected graphs can have non-free embeddings.

Findings

Small graphs with high minimal valency always have free linear embeddings.

Larger or highly connected graphs can have non-free linear embeddings.

There are infinitely many graphs with non-free linear embeddings for any minimal valency or connectivity level.

Abstract

An embedding of a graph into $\mathbb{R}^3$ is said to be linear, if any edge of the graph is sent to be a line segment. And we say that an embedding $f$ of a graph $G$ into $\mathbb{R}^3$ is free, if $\pi_1(\mathbb{R}^3-f(G))$ is a free group. It was known that for any complete graph its linear embedding is always free. In this paper we investigate the freeness of linear embeddings considering the number of vertices. It is shown that for any simple connected graph with at most 6 vertices, if its minimal valency is at least 3, then its linear embedding is always free. On the contrary when the number of vertices is much larger than the minimal valency or connectivity, the freeness may not be an intrinsic property of such graphs. In fact we show that for any $n \geq 1$ there are infinitely many connected graphs with minimal valency $n$ which have non-free linear embeddings, and furthermore, that there are infinitely many $n$-connected graphs which have non-free linear embeddings.