Computing the Radio Number via Multilevel Distance Labelings for Connected Graphs

Suppose G is a connected graph. For any two vertices s and t, let dG (s,t) denote the distance between s and t in G. The diameter of G is the maximum distance between any pair of vertices, and it is denoted by diamG. A multilevel distance labeling is a function VG⟶Z+, such that for any two vertices s≠t, we have ds,t+fs−ft≥diamG+1. The condition ds,t+fs−ft≥diamG+1 will be referred to as the multilevel distance labeling or the radio condition. The span of f is the largest positive integer in the range of. The minimum span of a multilevel distance labeling for G is called the radio number, represented by G. In this article, we compute the multilevel distance labeling of some graphs.