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An embedding of a graph into R-3 is 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 R-3 is free if pi(1) (R-3 - 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.The first author (corresponding author) was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number 2010-0009794).The second author was supported by the National Research Foundation of Korea (NRF) grants 2011-0027989 and 2013R1A1A2059197

Linearly embedded graphs in 3–space with homotopically free exteriors