The $\mathbb{Z}$-Divisor Graph on $\mathbb{Q} \cap [1,7]$ is Planar and 3-Colorable

Abstract

We prove that a divisor graph on an interval of integers is a planar graph whenever the interval $[n,m]$ satisfies $m\leq 7n$, and that 7 is the greatest such real number for which we can guarantee that the graph is planar. We prove these graphs are 3-colorable.