DISTANCE THREE LABELINGS FOR DIRECT PRODUCTS OF THREE COMPLETE GRAPHS

Abstract

The distance 3 labeling number lambda(G) (j(0), j(1), j(2)) for a graph G = (V, E) is the smallest integer alpha such that there is a function f : V -> [0, alpha], satisfying vertical bar f(u) - f (v)vertical bar >= j(delta-1) for any pair of vertices u, v of distance delta <= 3. In this paper, we determine the distance 3 labeling number lambda(G)(j, k, 1) for the direct product G = K-n x K-m x K-2 (n >= m >= 3) of 3 complete graphs under various conditions on j and k. As a consequence, we have the radio number rn(G) = 2mn 1.