Prime Vertex Labelings of Several Families of Graphs

TL;DR

This paper introduces new prime vertex labelings for various graph families, expanding the understanding of how vertices can be labeled with coprime integers while maintaining adjacency conditions.

Contribution

It provides the first known prime vertex labelings for cycle pendant stars, cycle chains, prisms, and generalized books.

Findings

Prime labelings exist for cycle pendant stars.

Prime labelings are established for cycle chains.

New labelings are found for prisms and generalized books.

Abstract

A simple and connected $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$, such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs including cycle pendant stars, cycle chains, prisms, and generalized books.