Multilevel Distance Labelings for Paths and Cycles

For a graph G, let diam(G) denote the diameter of G. For any two vertices u and v in G, let d(u, v) denote the distance between u and v. A multi-level distance labeling (or distance labeling) for G is a function f that assigns to each vertex of G a non-negative integer such that for any vertices u and v, |f(u)− f(v)| ≥ diam(G) − dG(u, v) + 1. The span of f is the largest number in f(V ). The radio number of G, denoted by rn(G), is the minimum span of a distance labeling for G. In this paper, we completely determine the radio numbers for paths and cycles. Supported in part by the National Science Foundation under grant DMS 9805945. Supported in part by the National Science Council, Taiwan, under grant NSC 912115-M-110-003.