Minimum coprime graph labelings

TL;DR

This paper investigates the minimum coprime number for various graph classes, providing exact values, resolving conjectures, and analyzing asymptotic behavior for random graphs.

Contribution

It determines the minimum coprime number for specific graph classes, resolves existing conjectures, and offers asymptotic results for Erdős-Rényi random graphs.

Findings

Exact minimum coprime numbers for certain graph classes

Resolution of three conjectures in the literature

Asymptotic behavior for Erdős-Rényi random graphs

Abstract

A coprime labeling of a graph $G$ is a labeling of the vertices of $G$ with distinct integers from $1$ to $k$ such that adjacent vertices have coprime labels. The minimum coprime number of $G$ is the least $k$ for which such a labeling exists. In this paper, we determine the minimum coprime number for several well-studied classes of graphs, including the coronas of complete graphs with empty graphs and the joins of two paths. In particular, we resolve a conjecture of Seoud, El Sonbaty, and Mahran and two conjectures of Asplund and Fox. We also provide an asymptotic for the minimum coprime number of the Erd\H{o}s-R\'enyi random graph.