Coprime and prime labelings of graphs

TL;DR

This paper investigates coprime and prime labelings of graphs, focusing on ladder graphs and complete bipartite graphs, exploring conditions for prime labelings and cyclic labelings.

Contribution

It provides new results on when ladder graphs are prime and introduces methods for cyclic labelings, extending coprime labeling theory to specific graph classes.

Findings

Ladder graphs can be prime under certain conditions.

Cyclic labelings are possible for some ladder graphs.

Coprime labelings are characterized for complete bipartite graphs.

Abstract

A coprime labeling of a simple graph of order $n$ is a labeling in which adjacent vertices are given relatively prime labels, and a graph is prime if the labels used can be taken to be the first $n$ positive integers. In this paper, we consider when ladder graphs are prime and when the corresponding labeling may be done in a cyclic manner around the vertices of the ladder. Furthermore, we discuss coprime labelings for complete bipartite graphs.