Radio Number for Ninth Power Paths

RM 211

Let G be a connected graph. The distance between two vertices u and v in G is defined by the length of the shortest path in G between u and v, which we denote d(u,v). The diameter of G, denoted diam(G), is the maximum distance between any two vertices in G. A radio labeling of G is a function f that assigns each vertex a distinct non-negative integer such that |f(u)-f(v)| greater than or equal to [diam(G)-d(u,v)+1] holds for any two distinct vertices u and v in G. The span of f is the difference between the largest and smallest channels used. The radio number of G, denoted rn(G), is defined as the minimum span of all radio labelings of G. f is said to be an optimal radio labeling of G if the span of f equals the radio number of G. The ninth power of G is a graph constructed from G by adding edges between vertices of distance nine or less apart in G. We will deal with finding the radio number for ninth power n-vertex path graphs.