Bounding the tripartite‐circle crossing number of complete tripartite graphs

Abstract

A tripartite-circle drawing of a tripartite graph is a drawing in the plane, where each part of a vertex partition is placed on one of three disjoint circles, and the edges do not cross the circles. We present upper and lower bounds on the minimum number of crossings in tripartite-circle drawings of K m n p , , and the exact value for K n 2,2, . In contrast to 1-and 2-circle drawings, which may attain the Harary-Hill bound, our results imply that balanced restricted 3-circle drawings of the complete graph are not optimal.

On a conjecture by Anthony Hill

“…Then we obtain yet another kinds of Hill drawings (six-partite or eight-partite). Let us note that Camacho et al [5] proved that there is no tripartite version of cylindrical drawings yielding the Hill bound.…”