Antipodal number of some powers of cycles

For a simple connected graph GG and an integer kk with 1⩽k⩽1⩽k⩽ diam(GG), a radio kk-coloring of GG is an assignment ff of non-negative integers to the vertices of GG such that |f(u)−f(v)|⩾k+1−d(u,v)|f(u)−f(v)|⩾k+1−d(u,v) for each pair of distinct vertices uu and vv of GG, where diam(GG) is the diameter of GG and d(u,v)d(u,v) is the distance between uu and vv in GG. The span   of a radio kk-coloring ff is the largest integer assigned by ff to a vertex of GG, and the radio  kk-chromatic number   of GG, denoted by rck(G)rck(G), is the minimum of spans of all possible radio kk-colorings of GG. If k=diam(G)−1, then rck(G)rck(G) is known as the antipodal number of GG. In this paper, we give an upper and a lower bound of rck(Cnr) for all possible values of nn, kk and rr. Also we show that these bounds are sharp for antipodal number of Cnr for several values of nn and rr.