The Radio Number of Cn Cn
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) → Z+ such that d(u,v)+|c(u)-c(v)|≥ 1+diam(G) for every two distinct vertices u and v of G (where d(u,v) is the distance between u and v). 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., of Cn Cn.)
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