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 Zn,s, s ≥ 1, n ≥ s, have vertex set {(i, j) | i = 1, 2 and j = 1, . . . , n} and edge set This research was initiated under the auspices of an MAA (SUMMA) Research Experience for Undergraduates program funded by NSA, NSF, and Moodys, and hosted at CSU Channel Islands during Summer, 2006. We are grateful to all for the opportunities provided. 46 P. Martinez, J. Ortiz, M. Tomova and C. Wyels {((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 Zn,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.