Capturing links in spatial complete graphs

2021/05/24 by Ryo Nikkuni, Nikkuni, Ryo
Engineering · Computer Science · Mathematics · #graph theory and CDMA systems #Advanced Graph Theory Research #Limits and Structures in Graph Theory

paper · pdf · doi:10.48550/arxiv.2105.11297

Abstract

We say that a set of pairs of disjoint cycles Λ(G) of a graph G is linked if for any spatial embedding f of G there exists an element λ of Λ(G) such that the 2-component link f(λ) 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≥ 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.

Related

• Generalization of the Conway-Gordon theorem and intrinsic linking on complete graphs
• Intrinsically Linked Graphs with Knotted Components
• Linkages in Polytope Graphs
• Knotting and linking in the Petersen family
• The Complement Problem for Linklessly Embeddable Graphs