Linearly embedded graphs in 3–space with homotopically free exteriors

An embedding of a graph into ℝ³is said to be linear if any edge of the graph is sent to a line segment. And we say that an embedding f of a graph G into ℝ³is free if π₁(ℝ³– f (G)) is a free group. It is known that the linear embedding of any complete graph is always free. In this paper we investigate the freeness of linear embeddings by considering the number of vertices. It is shown that the linear embedding of any simple connected graph with at most 6 vertices whose minimal valency is at least 3 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 the graph. In fact we show that for any n ≥ 1 there are infinitely many connected graphs with minimal valency n which have nonfree linear embeddings and furthermore that there are infinitely many n–connected graphs which have nonfree linear embeddings.