A lower bound for the radio number of graphs

Abstract

A radio labeling of a graph G is a mapping φ: V(G) → {0, 1, 2, …} such that (Formula presented) for every pair of distinct vertices u, v of G, where diam(G) and d(u, v) are the diameter of G and distance between u and v in G, respectively. The radio number (G) of G is the smallest number k such that G has radio labeling with (Formula presented). In this paper, we slightly improve the lower bound for the radio number of graphs given by Das et al. in [5] and, give necessary and sufficient condition to achieve the lower bound. Using this result, we determine the radio number for cartesian product of paths Pn and the Peterson graph P. We give a short proof for the radio number of cartesian product of paths Pn and complete graphs Km given by Kim et al. in [6].