Hamilton Paths in Certain Arithmetic Graphs

Abstract

For each integer $m\ge 1$, consider the graph $G_m$ whose vertex set is the set $\mathbb{N}=\{0,1,2,\dots \}$ of natural numbers and whose edges are the pairs $xy$ with $y=x+m$, $y=x-m$, $y=mx$, or $y=\frac{x}{m}$. Our aim in this note is to show that, for each $m$, the graph $G_m$ contains a Hamilton path. This answers a question of Lichiardopol.