Radio number of Hamming graphs of diameter 3

Abstract

For $G$ a simple, connected graph, a vertex labeling $f:V(G)\to {\mathbb{Z}}_{+}$ is called a $\text{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|\ge \mathrm{diam}(G)+1-d(u,v)$ for all distinct vertices $u,v\in V(G)$. The $\text{radio number}$ 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 $\text{radio graceful graph}$. We determine the radio number of all diameter $3$ Hamming graphs and show that an infinite subset of them is radio graceful.