A Comment on Dean's Construction of Prime Labelings on Ladders

TL;DR

This paper critiques Dean's proof of the Prime Ladder Conjecture, highlights the need for a stronger hypothesis, and proposes an alternative approach linking the conjecture to Goldbach and Lemoine's conjectures.

Contribution

It identifies a flaw in Dean's construction and suggests a new approach connecting the conjecture to well-known number theory conjectures.

Findings

Dean's proof has a flaw requiring a stronger hypothesis.

An alternative construction links the conjecture to Goldbach and Lemoine's conjectures.

The Prime Ladder Conjecture remains open but is connected to major number theory conjectures.

Abstract

A prime labeling on a graph of order $m$ is an assignment of $\{ 1, 2, \ldots, m \}$ to the vertices of the graph such that each pair of adjacent vertices has coprime labels. The ladder of order $2n$ is the $2 \times n$ grid graph graph $P_2 \times P_n$. In a recent paper, Dean claimed a proof of the Prime Ladder Conjecture that every ladder has a prime labeling. We point out a flaw in Dean's construction, showing that a stronger hypothesis is needed for it to hold. We conjecture that this stronger hypothesis is true. We also offer an alternative construction inspired by Dean's approach which shows that if the Even Goldbach Conjecture and a particular strengthening of Lemoine's Conjecture are true then the Prime Ladder Conjecture follows.