Divisor graphs have arbitrary order and size

Abstract

A divisor graph $G$ is an ordered pair $(V, E)$ where $V \subset \mathbbm{Z}$ and for all $u \neq v \in V$, $u v \in E$ if and only if $u \mid v$ or $v \mid u$. A graph which is isomorphic to a divisor graph is also called a divisor graph. In this note, we will prove that for any $n \geqslant 1$ and $0 \leqslant m \leqslant \binom{n}{2}$ then there exists a divisor graph of order $n$ and size $m$. We also present a simple proof of the characterization of divisor graphs which is due to Chartran, Muntean, Saenpholpant and Zhang.