Linearly Free Graphs

TL;DR

This paper investigates the property of graphs called linear freeness, which depends on their embeddings in three-dimensional space, and establishes conditions under which graphs maintain this property, with applications to complete and bipartite graphs.

Contribution

It generalizes Nicholson's concept of linear freeness, providing new criteria for when a graph's linear embedding is free and exploring how this property is preserved under graph enlargements.

Findings

Complete graphs are linearly free.

Graphs with minimal valency ≥ 3 and fewer than 8 vertices are linearly free.

Complete bipartite graphs K_{n,m} are linearly free for n,m ≤ 6.

Abstract

In this paper we are interested in an intrinsic property of graphs which is derived from their embeddings into the Euclidean 3-space $\mathbb{R}^3$. An embedding of a graph into $\mathbb{R}^3$ is said to be linear, if it sends every edge 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. Lastly a simple connected graph is said to be linearly free if every its linear embedding is free. In 1980s it was proved that every complete graph is linearly free, by Nicholson. In this paper, we develop Nicholson's arguments into a general notion, and establish a sufficient condition for a linear embedding to be free. As an application of the condition we give a partial answer for a question: how much can the complete graph $K_n$ be enlarged so that the linear freeness is preserved and the clique number does not increase? And an example supporting our answer is provided. As the second application it is shown that a simple connected graph of minimal valency at least $3$ is linearly free, if it has less than 8 vertices. The conditional inequality is strict, because we found a graph with $8$ vertices which is not linearly free. It is also proved that for $n, m \leq 6$ the complete bipartite graph $K_{n,m}$ is linearly free.