标题： 对迪恩构造梯形素数标号的评论 显示英文标题

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

摘要： 一个阶为$m$的图上的素数标记是将$\{ 1, 2, \ldots, m \}$分配给图的顶点的一种分配，使得每对相邻顶点的标签互质。 阶为$2n$的梯子是$2 \times n$网格图图$P_2 \times P_n$。 在最近的一篇论文中，Dean 声称证明了每个梯子都有素数标记的素数梯子猜想。 我们指出 Dean 构造中的一个缺陷，表明需要更强的假设才能成立。 我们猜想这个更强的假设是正确的。 我们还提供了一个受 Dean 方法启发的替代构造，这表明如果偶数哥德巴赫猜想和 Lemoine 猜想的一个特定加强形式为真，则素数梯子猜想成立。 显示英文摘要

摘要： 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.