Antipodal number of some powers of cycles

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