Oriented diameter of the complete tripartite graph (III)

TL;DR

The paper resolves the oriented-diameter problem for complete tripartite graphs ${\sf K}(3,p,q)$ with $5\le p\le q$, showing a tight threshold at $q=\binom{p+1}{\lfloor (p+1)/2\rfloor}$: the oriented diameter is $2$ when $q$ is below this bound and $3$ at or above it. It achieves this via a two-pronged strategy: (i) a structural, case-by-case analysis of diameter-two orientations by partitioning the middle part and applying combinatorial arguments (including Sperner-type bounds) to bound $q$ in each case (the “big part”); and (ii) explicit constructions of diameter-two strong orientations when $q$ lies below the threshold (the “small part”). Together these yield a complete determination of $f({\sf K}(3,p,q))$ for all $p\ge 5$, extending the prior results for ${\sf K}(3,3,q)$ and ${\sf K}(3,4,q)$ and solving Koh–Tay’s problem for ${\sf K}(3,p,q)$. The results enhance understanding of strong orientations in complete multipartite graphs and provide concrete diameter-two orientation recipes for network routing contexts.

Abstract

Given a bridgeless graph $G$, let $\mathbb{D}(G)$ be the set of all strong orientations of $G$, and define the oriented diameter $f(G)$ of $G$ to be the minimum of diameters $diam(D)$ among all the strong orientations $D\in \mathbb{D}(G)$, i.e., $f(G)=\min\{diam(D)\mid D\in \mathbb{D}(G)\}$. In this paper, we determine the oriented diameter of complete tripartite graph $K(3,p,q)$ for $p\geqslant 5$. Combining with the previous results, the oriented diameter of complete tripartite graph $K(3,p,q)$ are known.

Theorems & Definitions (34)

• Lemma 1.1: Sperner's Lemma
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3
• Lemma 4.4
• Lemma 4.5