Complements of non-separating planar graphs
Andrei Pavelescu, Elena Pavelescu · arXiv (Cornell University) · 2021
We prove that the complement of a non-separating planar graph of order at least nine is intrinsically linked. We also prove that the complement of a non-separating planar graph of order at least 10 is intrinsically knotted. We show these lower bounds on the orders are the best possible. We show that for a maximal non-separating planar graph with $n\ge 7$ vertices, its complement $cG$ is $(n-7)-$apex. We conclude that the Colin de Verdiere invariant for such graphs satisfies $\mu(cG)\le n-4$. We use maximal non-separating planar graphs to build examples of maximal linklessly embeddable graphs, and examples of maximal knotlessly embeddable graphs.