Radiatic Dimension of a Graph

摘要

Let G(V,E) be a simple, finite, connected graph. An injective mapping f : V (G) ! Z+ such that for every two distinct vertices u, vε V (G), |f(u) ? f(v)|>diam(G) + 1 ? d(u, v) is called a radio labeling of G. The radio number of f, denoted by rn(f) is the maximum number assigned to any vertex of G. The radio number of G, is the minimum value of rn(f) taken over all radio labelings f of G. A graph G on n vertices is radio graceful if and only if rn(G) = n. In this paper, we define the radiatic dimension of G to be the smallest positive integer k, such that the sequence of injective functions fi :V (G) ! {1, 2, 3, . . . , n}, 1 < i < k, satisfy the condition that for every two distinct vertices u, vε V (G), |fi(u)?fi(v)|> diam(G)+1?d(u, v) for some i and denote it by rd(G). Hence a graph is radio graceful if and only if rd(G) = 1. In this paper we study the radiatic dimension of some standard graphs and characterize graphs of diameter 2 that are radio graceful.