On the Existence of Knots and Links in the Complete Directed Graph on Six Vertices

Abstract

In 1983, John H. Conway and Cameron Gordon published "Knots and Links in Spatial Graphs" in the Journal of Graph Theory. In this paper, Conway and Gordon proved that the complete graph on six vertices is intrinsically linked. That is, any embedding of the complete graph on six vertices contains at least one pair of disjoint triangles that forms a nontrivial link. The complete directed graph on six vertices is defined to be the graph consisting of six vertices such that each pair of distinct vertices is connected by exactly one pair of edges with the property that this pair forms a 2-cycle. Given that the complete graph on six vertices is intrinsically linked, we conjecture that the complete directed graph on six vertices is as well. In the pages that follow, we present a collection of results that we obtained through our efforts to prove that the complete directed graph on six vertices is intrinsically linked. These results include several cases in which we have shown that embeddings of the complete directed graph on six vertices contain one or more pairs of disjoint directed triangles. We also consider the subgraphs of the complete directed graph on six vertices and the number of pairs of disjoint directed triangles that may be contained in a single subgraph.