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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 $c:V(G) \rightarrow \mathbb Z_+$ such that $$d(u,v)+c(u)c(v)\geq 1+\text{diam}(G)$$ 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 $C_n\square C_n$.) Comment: To appear in Ars Combinatoria, 15 pages 
Article: THE RADIO NUMBER OF CnCn
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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. of CnC n.)
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