An embedding of a graph into
is said to be
linear if any edge of the graph is sent to a line segment. And we say that an
embedding
of a graph
into
is
free if
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
there are infinitely many connected graphs with minimal valency
which
have nonfree linear embeddings and furthermore that there are infinitely many
–connected
graphs which have nonfree linear embeddings.
Keywords
linear embedding, complete graph, fundamental group, free