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

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 $a,b\leq 2$.

Figures (11)

• Figure 1: The vertices on the circles $\textsc{m}$ and $\textsc{n}$ are labeled clockwise; the vertices of the circle $\textsc{p}$ are labeled counterclockwise.
• Figure 2: The definitions of $x_{i}(\textsc{a,b})$ and $y_{i}(\textsc{a,b})$ illustrated for three cases.
• Figure 3: Crossing-minimal tripartite-circle drawings of $K_{1,2,n}$ for $n=2,3,4$; crossing-optimal drawings of the complete graph $K_{n+3}$ are obtained by adding gray edges.
• Figure 4: A crossing-minimal tripartite-circle drawing of $K_{1,2,n}$ for all $n\geq 4$ and $2\leq t \leq \lceil(n-1)/2\rceil$.
• Figure 5: A simple tripartite-circle drawing of $K_{1,2,n}$ to illustrate \ref{['lem:lowerbound']}.

Theorems & Definitions (31)

• Proposition 0
• Theorem 1
• Theorem 2
• Theorem 3: Theorem 7, tripartite_JGT
• Proposition 3
• proof
• Theorem 3
• Lemma 4
• proof
• Lemma 5