Multilevel and Antipodal Labelings for Families of Graphs

Abstract

A radio k-labeling c of a graph G is a mapping c : V (G) → Z+ ∪ {0}, such that d(x, y) + |c(x) − c(y)| ≥ k + 1 holds for every two distinct vertices x and y of G, where d(x, y) is the distance between any two vertices x and y of G. The span of a radio k-labeling c is denoted by sp(c) and defined as max{|c(x) − c(y)| : x, y ∈ V (G)}. The radio labeling is a radio klabeling when k = diam(G). In other words, a radio labeling is a one-to-one function c : V (G) → Z+ ∪ {0}, such that |c(x) − c(y)| ≥ diam(G) + 1 − d(x, y) for any pair of vertices x, y in G. The radio number of G denoted by rn(G), is the lowest span taken over all radio labelings of the graph. When k = diam(G) − 1, a radio klabeling is called a radio antipodal labeling. An antipodal labeling for a graph G is a function c : V (G) → {0, 1, 2, ...}, so that d(x, y) + |c(x) − c(y)| ≥ diam(G) for all x, y ∈ G. The radio antipodal number for G denoted by an(G), is the minimum span of an antipodal labeling admitted by G. In this thesis, we investigate the exact value of the radio number and radio antipodal number for different family of graphs. Further more, we also determine the lower bound of the radio number for some cases.