A Graph Radio k-Coloring Algorithm

Abstract

For a positive integer k, a radio k-coloring of a simple connected graph G = (V(G), E(G)) is a mapping f  : V(G) → { 0,1,2,…} such that $|f(u)-f(v)|\geqslant k+1-d(u,v)$ for each pair of distinct vertices u and v of G, where d(u,v) is the distance between u and v in G. The span of a radio k-coloring f, rc _k (f), is the maximum integer assigned by it to some vertex of G. The radio k-chromatic number, rc _k (G) of G is $\displaystyle\min\{rc_{k}(f)\}$, where the minimum is taken over all radio k-colorings f of G. If k is the diameter of G, then rc _k (G) is known as the radio number of G. In this paper, we give an algorithm to find an upper bound of rc _k (G). We also give an algorithm that implement the result in [16,17] for lower bound of rc _k (G). We check that for cycle C _n , upper and lower bound obtained from these algorithms coincide with the exact value of radio number, when n is an even integer with $4\leqslant n\leqslant 400$. Also applying these algorithms we get the exact value of the radio number of several circulant graphs.