An intrinsic nontriviality of graphs

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.