Capturing links in spatial complete graphs

TL;DR

This paper characterizes minimally linked sets of disjoint cycle pairs in graphs, showing they are equivalent to Petersen family graphs and constructing specific minimally linked Hamiltonian cycle pairs in complete graphs.

Contribution

It provides a characterization of minimally linked cycle sets in graphs and constructs explicit examples in complete graphs for various cycle lengths.

Findings

All pairs of disjoint cycles are minimally linked only in Petersen family graphs.

Constructed minimally linked Hamiltonian cycle pairs in complete graphs for p,q ≥ 3.

Identified the minimal size of linked sets in complete graphs as at most eighteen elements.

Abstract

We say that a set of pairs of disjoint cycles $\Lambda(G)$ of a graph $G$ is linked if for any spatial embedding $f$ of $G$ there exists an element $\lambda$ of $\Lambda(G)$ such that the $2$-component link $f(\lambda)$ is nonsplittable, and also say minimally linked if none of its proper subsets are linked. In this paper, (1) we show that the set of all pairs of disjoint cycles of $G$ is minimally linked if and only if $G$ is essentially same as a graph in the Petersen family, and (2) for any two integers $p,q\ge 3$, we exhibit a minimally linked set of Hamiltonian $(p,q)$-pairs of cycles of the complete graph $K_{p+q}$ with at most eighteen elements.