Intrinsic linking in directed graphs

Abstract

We extend the notion of intrinsic linking to directed graphs. We give methods of constructing intrinsically linked directed graphs, as well as complicated directed graphs that are not intrinsically linked. We prove that the double directed version of a graph $G$ is intrinsically linked if and only if $G$ is intrinsically linked. One Corollary is that $\overline{J_{6}}$, the complete symmetric directed graph on 6 vertices (with 30 directed edges), is intrinsically linked. We further extend this to show that it is possible to find a subgraph of $\overline{J_{6}}$ by deleting 6 edges that is still intrinsically linked, but that no subgraph of $\overline{J_{6}}$ obtained by deleting 7 edges is intrinsically linked. We also show that $\overline{J_{6}}$ with an arbitrary edge deleted is intrinsically linked, but if the wrong two edges are chosen, $\overline{J_{6}}$ with two edges deleted can be embedded linklessly.