Capturing links in spatial complete graphs

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.