  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.
