Achieving All Radio Numbers

Abstract

For a connected graph G, a radio labeling is a function c: V(G) → ℤ+ such that for every pair of vertices (u, v) in V(G), the radio condition is satisfied:The span of a radio labeling c is the largest integer in the image of c. The radio number of a graph G is the smallest integer M such that span(c) = M for some radio labeling c. It is known that a graph of n vertices has a radio number of at least n and at most , where r is determined by the parity of n. This paper defines and examines three-parameter graphs known as Sok graphs. We show that for all but two integers between the known minimum and maximum, there exists a Sok graph whose radio number is that integer. The results of this work entirely settle the question of what the possible radio numbers are for graphs of order n.