Linearly embedded graphs in 3–space with homotopically free exteriors

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 $π_{1} (ℝ^{3} - f (G))$ 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 $n \geq 1$ 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.

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Keywords

linear embedding, complete graph, fundamental group, free

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M15, 05C10

References

Publication

Received: 17 June 2014

Revised: 31 August 2014

Accepted: 17 September 2014

Published: 22 April 2015

Authors

Youngsik Huh
Department of Mathematics College of Natural Sciences Hanyang University Seoul 133-791 South Korea

Jung Hoon Lee
Department of Mathematics and Institute of Pure and Applied Mathematics Chonbuk National University Jeonju 561-756 South Korea

Volume 15, issue 2 (2015)

Youngsik Huh and Jung Hoon Lee

Abstract