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