On the Prime Labeling of Graphs

Abstract

A graph G with vertex set V and edge set E is said to have a prime labeling if its vertices can be labeled with distinct integers 1 2 V such that for every edge xy in E, the labels assigned to x and y are relatively prime. A graph is called prime if it has a prime labeling. In this paper, we introduce to consider the extremal prime labeling graphs with the maximum number of edges. In addition, by modelling the prime labeling problem, we obtain some prime graphs in the families of Cartesian and direct product of paths and cycles.