Total Prime Labelings of Various Graphs

Abstract

A total prime labeling of a graph of order $n$ is an extension of a prime labeling in which we distinctly label the vertices and edges. The goal of the labeling is for adjacent vertex labels to be relatively prime, and for each vertex of degree at least two, the greatest common divisor of the labels on its incident edges is equal to 1. In this paper, we construct total prime labelings by extending known prime and minimum coprime labelings and by developing new constructions for various classes of graphs. In particular, we show that snakes, books, prisms, prime trees, certain families of windmills, and other families of graphs are total prime.