Total Prime Labelings of Various Graphs

TL;DR

This paper introduces total prime labelings for various graphs, extending existing labelings by ensuring vertices and edges are labeled distinctly with specific coprimality conditions, and constructs such labelings for multiple graph classes.

Contribution

It develops new methods to construct total prime labelings for diverse graph classes, expanding the scope of prime labeling theory.

Findings

Snakes, books, prisms, prime trees, windmills are total prime graphs.

New constructions extend known prime labelings.

The paper demonstrates total prime labelings for multiple graph families.

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.