Radio numbers for generalized prism graphs

A radio labeling is an assignment c:V(G) → N such that every distinct pair of vertices u,v satisfies the inequality d(u,v) + |c(u)-c(v)| ≥ 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 ≥ 1, n ≥ s, have vertex set {(i,j) | i = 1,2 and j = 1,...,n} and edge set {((i,j),(i,j ±1))} ∪ {((1,i),(2,i+σ)) | σ = -⌊(s-1)/2⌋...,0,...,⌊s/2⌋}. 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.