Radio numbers for generalized prism graphs

TL;DR

This paper determines the radio numbers for generalized prism graphs with parameters s=1, 2, and 3, and introduces techniques potentially applicable to other graph families.

Contribution

It provides exact radio numbers for specific generalized prism graphs and develops new methods for calculating radio numbers in broader graph classes.

Findings

Radio numbers for Z_{n,1}, Z_{n,2}, Z_{n,3} are explicitly determined.

New techniques for calculating radio numbers are introduced.

Methods may be applicable to other graph families.

Abstract

A radio labeling is an assignment $c:V(G) \rightarrow \textbf{N}$ such that every distinct pair of vertices $u,v$ satisfies the inequality $d(u,v)+|c(u)-c(v)|\geq \diam(G)+1$. The span of a radio labeling is the maximum value. The radio number of $G$, $rn(G)$, is the minimum span over all radio labelings of $G$. Generalized prism graphs, denoted $Z_{n,s}$, $s \geq 1$, $n\geq s$, have vertex set $\{(i,j)\,|\, i=1,2 \text{and} j=1,...,n\}$ and edge set $\{((i,j),(i,j \pm 1))\} \cup \{((1,i),(2,i+\sigma))\,|\,\sigma=-\left\lfloor\frac{s-1}{2}\right\rfloor\,\ldots,0,\ldots,\left\lfloor\frac{s}{2}\right\rfloor\}$. In this paper we determine the radio number of $Z_{n,s}$ for $s=1,2$ and $3$. In the process we develop techniques that are likely to be of use in determining radio numbers of other families of graphs.