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).