Hamilton Paths in Certain Arithmetic Graphs

Abstract

For each integer \(m \geq 1\), consider the graph \(G_m\) whose vertex set is the set \(\mathbb{N} = \{0,1,2,\ldots\}\) 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.