Antipodal number of some powers of cycles
For a simple connected graph G and an integer k with 1==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 radiok chromatic number of G, denoted by rc"k(G) is the minimum of spans of all possible radio k colorings of G. If k=diam(G) 1, then rc"k(G) is known as the antipodal number of G. In this paper, we give an upper and a lower bound of rc"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.
Discret. Math.