Oriented diameter of the complete tripartite graph (II)

TL;DR

The paper determines the oriented diameter for two families of complete tripartite graphs: $K(3,3,q)$ and $K(3,4,q)$. It combines constructive orientations yielding diameter $2$ for specific ranges of $q$ with rigorous case analyses to rule out diameter-$2$ orientations beyond those ranges, establishing precise thresholds. Specifically, $f(K(3,3,q))=2$ when $3\le q\le 6$ and $f(K(3,3,q))=3$ for $q>6$, while $f(K(3,4,q))=2$ for $4\le q\le 11$ and $f(K(3,4,q))=3$ for $q>11$. These results complete the classification of oriented diameters for these complete tripartite cases and advance the broader classification for $K(3,p,q)$.

Abstract

For a graph $G$, let $\mathbb{D}(G)$ denote the set of all strong orientations of $G$, and the oriented diameter of $G$ is $f(G)=\min \{diam(D) \mid D \in \mathbb{D}(G)\}$, which is the minimum value of the diameters $diam(D)$ where $D \in \mathbb{D}(G)$. In this paper, we determine the oriented diameter of complete tripartite graphs $K(3,3, q)$ and $K(3,4, q)$, these are special cases that arise in determining the oriented diameter of $K(3, p, q)$.

Theorems & Definitions (5)

• Lemma 2.1
• Lemma 3.1
• Theorem 3.1
• Lemma 4.1
• Theorem 4.1