Bounding the tripartite-circle crossing number of complete tripartite graphs

Abstract

A tripartite-circle drawing of the complete tripartite graph $K_{m,n,p}$ is a drawing in the plane, where each part of the 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 $\crN{3}(K_{n,n,n})$and the exact value for $K_{2,2,n}$. In contrast to 1- and 2-circle drawings which may attain the Harary-Hill bound, our results imply that optimal drawings of the complete graph do not contain balanced 3-circle drawings as subdrawings that do not cross any of the remaining edges.