New Perspectives on Neighborhood-Prime Labelings of Graphs

TL;DR

This paper introduces new techniques for identifying neighborhood-prime labelings in graphs, demonstrating their applicability to various graph classes and establishing that most graphs are neighborhood-prime.

Contribution

It presents novel methods based on Hamiltonicity and vertex degrees for finding neighborhood-prime labelings, and characterizes several graph classes that possess this property.

Findings

All generalized Petersen graphs are neighborhood-prime.

Grid graphs of any size are neighborhood-prime.

Almost all graphs and regular graphs are neighborhood-prime.

Abstract

Neighborhood-prime labeling is a variation of prime labeling. A labeling $f:V(G) \to [|V(G)|]$ is a neighborhood-prime labeling if for each vertex $v\in V(G)$ with degree greater than $1$, the greatest common divisor of the set of labels in the neighborhood of $v$ is $1$. In this paper, we introduce techniques for finding neighborhood-prime labelings based on the Hamiltonicity of the graph, by using conditions on possible degrees of vertices, and by examining a neighborhood graph. In particular, classes of graphs shown to be neighborhood-prime include all generalized Petersen graphs, grid graphs of any size, and lobsters given restrictions on the degree of the vertices. In addition, we show that almost all graphs and almost all regular graphs have are neighborhood-prime, and we find all graphs of order $10$ or less that have a neighborhood-prime labeling. We give several conjectures for future work in this area.