THE RADIO NUMBER OF CnCn

Abstract. Radio labeling is a variation of Hale’s channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph G subject to certain constraints involving the distances between the vertices. Specifically, a radio labeling of a connected graph G is a function c: V (G) → N such that d(u, v) + |c(u) − c(v) | ≥ 1 + diam(G) for every two distinct vertices u and v of G. The span of a radio labeling is the maximum integer assigned to a vertex. The radio number of a graph G is the minimum span, taken over all radio labelings of G. This paper establishes the radio number of the Cartesian product of a cycle graph with itself (i.e. o