Bounding the tripartite-circle crossing number of complete tripartite graphs

TL;DR

This paper investigates the minimum crossings in tripartite-circle drawings of complete tripartite graphs, providing bounds that show 3-circle drawings are not always optimal compared to 1- and 2-circle cases.

Contribution

It establishes new upper and lower bounds for crossings in tripartite-circle drawings, highlighting differences from simpler circle drawing cases.

Findings

Balanced 3-circle drawings are not optimal for complete graphs.

Bounds improve understanding of crossing minimization in tripartite graphs.

Results contrast with known bounds for 1- and 2-circle drawings.

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 $K_{m,n,p}$. 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.