On Radio Labeling of Diameter N-2 and Caterpillar Graphs

Abstract

. Radio labeling of graphs is a specific type of graph labeling. The basic type of graph labeling is vertex coloring; this is where the vertices of a graph G are assigned different colors so that adjacent vertices are not given the same color. A k-coloring of a graph G is a coloring that uses k colors. The chromatic number of a graph G is the minimum value for k such that a k-coloring exists for G [2]. Radio labeling is a type of graph labeling that evolved as a way to use graph theory to try to solve the channel assignment problem: how to assign radio channels to radio transmitters so that two transmitters that are relatively close to one another do not have frequencies that cause interference between them. This problem of channel assignment was first put into a graph theoretic context by Hale [6]. In terms of graph theory, the vertices of a graph represent the locations of the radio transmitters, or radio stations, with the labels of the vertices corresponding to channels or frequencies assigned to the stations. Different restrictions on labelings of graphs have been studied to address the channel assignment problem. Radio labeling of a simple connected graph G is a labeling f : V (G) → Z such that for every pair of distinct vertices u and v of G, distance(u, v) + |f(u) − f(v)| ≥ diameter(G) + 1. The radio number of G is the smallest number m such that there exists a radio labeling f with f(v) ≤ m for all v in V (G). The radio numbers of certain families of graphs have already been found. Bounds and radio numbers of some tree graphs have been determined. Daphne Der-