Channel Assignment of Triangular and Rhombic Honeycomb Networks Using Radio Labeling Techniques

Abstract

A radio labeling of a graph \(G=(V,E)\) is a function \({f} :{V}(G)\rightarrow {N}\) such that \(d(u,v)+| f(u)-f(v)| \geq 1+ \diam(G)\), where \(d(u,v)\) represents the shortest distance between the vertices \(u\) and \(v\) and \(\diam(G)\) is the diameter of \(G\). The span of a radio labeling \(f\) is defined as \({sp}(f)=\max\{| f(u)-f(v)|:u,v\in V(G)\}\). A radio number of \(G\) is the minimum span of all the radio labelings of $G$ and is denoted by \(rn(G)\). The radio number is used to optimize the assignment of frequency bands to channels in wireless communication networks. The honeycomb network is considered to be one of the most important network for placement of base stations in wireless communications networks. In this paper, the upper and lower bounds for the radio number of two well-known topologies of honeycomb network namely triangular and rhombic honeycomb networks are obtained. These bounds were graphically represented for easy understanding of the minimum and maximum spectrum needed for effective communication in a network.