Neighborhood-Prime Labelings of Trees and Other Classes of Graphs

TL;DR

This paper investigates neighborhood-prime labelings, a variation of prime labelings, for various graph classes including trees like caterpillars, spiders, and firecrackers, focusing on label assignment conditions based on gcd in neighborhoods.

Contribution

It introduces the concept of neighborhood-prime labelings and explores their existence in specific graph classes such as polygonal snakes, books, and various types of trees.

Findings

Neighborhood-prime labelings exist for certain classes of trees.

Conditions for neighborhood-prime labelings depend on gcd properties in neighborhoods.

The paper extends prime labeling concepts to new graph classes.

Abstract

A neighborhood-prime labeling of a graph is a variation of a prime labeling in which the vertices are assigned labels from $1$ to $|V(G)|$ such that the $\gcd$ of the labels in the neighborhood of each non-degree $1$ vertex is equal to $1$. In this paper, we examine neighborhood-prime labelings of several classes of graphs such as polygonal snakes and books, with a focus on trees including caterpillars, spiders, and firecrackers.