Antipodal number of Cartesian products of complete graphs with cycles

Let G be a simple connected graph with diameter d, and k ∈ [1, d] be an integer. A radio k-coloring of graph G is a mapping g : V (G) → {0} ∪ N satisfying |g(u) − g(v)| ≥ 1 + k − d(u, v) for any pair of distinct vertices u and v of the graph G, where d(u, v) denotes distance between vertices u and v in G. The number max{g(u) : u ∈ V (G)} is known as the span of g and is denoted by rck(g). The radio k-chromatic number of graph G, denoted by rck(G), is defined as min{rck(g) : g is a radio k-coloring of G}. For k = d − 1, the radio k-coloring of graph G is called an antipodal coloring. So rcd−1(G) is called the antipodal number of G and is denoted by ac(G). Here, we study antipodal coloring of the Cartesian product of the complete graph Kr and cycle Cs, Kr□Cs, for r ≥ 4 and s ≥ 3. We determine the antipodal number of Kr□Cs, for even r ≥ 4 with s ≡ 1 (mod 4); and for any r ≥ 4 with s = 4t + 2, t odd. Also, for the remaining values of r and s, we give lower and upper bounds for ac(Kr□Cs).