An intrinsic non-triviality of graphs

TL;DR

This paper explores the concept of intrinsically non-trivial graphs, proving they are intrinsically linked and identifying conditions under which they contain complex links or spatial handcuff graphs in all embeddings.

Contribution

It introduces the notion of intrinsically non-trivial graphs and establishes their intrinsic linking property, expanding understanding of graph embeddings in three-dimensional space.

Findings

Intrinsically non-trivial graphs are necessarily intrinsically linked.

Existence of graphs with embeddings containing either a non-splittable 3-component link or a split spatial handcuff graph.

Every spatial embedding of such graphs contains complex linked or handcuff subgraphs.

Abstract

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