Vertex k-Prime Labeling of Theta Graphs

Abstract

Objectives: To analyse some theta related graphs that admit vertex k-prime labeling for each positive integer k. Methods: In this study, vertices of the graphs are assigned with k, k+1,…,k+|V|-1 such that each pair of labels of adjacent vertices are relatively prime. Justifications for the proof are given. Findings: We examine the theta related graphs such as generalised theta graphs, uniform theta graphs, centralised uniform theta graphs for m = 1 are vertex k-prime. In addition, we introduce another structure of theta graph known as centralised generalised theta graph and show that vertex k-prime labeling exists for the graph. Novelty: Vertex k-prime labeling is a new variant of prime labeling and theta families of graphs exhibiting the labeling is a new finding. Another structure of theta graph known as centralised generalised theta graph is introduced and proved that vertex k-prime labeling exists for the graph.