Radio Number of Hamming Graphs of Diameter 3

Abstract

For $G$ a simple, connected graph, a vertex labeling $f:V(G)\to \Z_+$ is called a \emph{radio labeling of $G$} if it satisfies $|f(u)-f(v)|\geq\diam(G)+1-d(u,v)$ for all distinct vertices $u,v\in V(G)$. The \emph{radio number of $G$} is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto $\{1,2,\dots,|V(G)|\}$ exists, $G$ is called a \emph{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.

Keywords

• radio labeling
• $l(3
• 2
• 1)$-labeling
• radio graceful graph
• hamming graph