Linearly embedded graphs in 3-space with homotopically free exteriors

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.