#### Volume 9, issue 1 (2009)

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An intrinsic nontriviality of graphs

### Ryo Nikkuni

Algebraic & Geometric Topology 9 (2009) 351–364
##### Abstract

We say that a graph is intrinsically nontrivial if every spatial embedding of the graph contains a nontrivial spatial subgraph. We prove that an intrinsically nontrivial graph is intrinsically linked, namely every spatial embedding of the graph contains a nonsplittable $2$–component link. We also show that there exists a graph such that every spatial embedding of the graph contains either a nonsplittable $3$–component link or an irreducible spatial handcuff graph whose constituent $2$–component link is split.

 Dedicated to Professor Akio Kawauchi on his 60th birthday
##### Keywords
spatial graph, intrinsically linked, spatial handcuff graph
Primary: 57M15
Secondary: 57M25
##### Publication
Received: 30 July 2008
Revised: 29 January 2009
Accepted: 31 January 2009
Published: 23 February 2009
##### Authors
 Ryo Nikkuni Institute of Human and Social Sciences Faculty of Teacher Education Kanazawa University Kakuma-machi, Kanazawa, Ishikawa, 920-1192 Japan http://www.ed.kanazawa-u.ac.jp/~nick/index-e.html