The tripartite-circle crossing number of graphs with two small partition classes

TL;DR

This paper determines the exact tripartite-circle crossing number for complete tripartite graphs with small partition classes, specifically for graphs where two of the partitions have at most two vertices.

Contribution

The paper provides the exact crossing number for $K_{a,b,n}$ with $a,b \,\leq 2$, advancing understanding of graph drawing complexities with constrained vertex placements.

Findings

Exact crossing number for $K_{a,b,n}$ with $a,b\leq 2$ determined.

Establishes minimal crossings in tripartite-circle drawings for specific graph classes.

Contributes to graph drawing theory and crossing number calculations.

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. The tripartite-circle crossing number of a tripartite graph is the minimum number of edge crossings among all its tripartite-circle drawings. We determine the exact value of the tripartite-circle crossing number of $K_{a,b,n}$, where $a,b\leq 2$.