Radio number of Hamming graphs of diameter 3

TL;DR

This paper determines the radio number of all diameter 3 Hamming graphs and identifies an infinite subset that are radio graceful, advancing understanding of radio labelings in graph theory.

Contribution

It provides the first complete determination of the radio number for diameter 3 Hamming graphs and characterizes those that are radio graceful.

Findings

Radio number of diameter 3 Hamming graphs is explicitly determined.

An infinite subset of these graphs are proven to be radio graceful.

Results contribute to graph labeling theory and applications in network frequency assignment.

Abstract

For $G$ a simple, connected graph, a vertex labeling $f:V(G)\rightarrow \mathbb{Z}_+$ is called a $\textit{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|\geq \operatorname{diam}(G) + 1 - d(u,v)$ for all distinct vertices $u,v\in V(G)$. The $\textit{radio number}$ of $G$ is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,...,|V(G)|\}$ exists, $G$ is called a $\textit{radio graceful graph}$. We determine the radio number of all diameter $3$ Hamming graphs and show that an infinite subset of them is radio graceful.