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 $u\ne v\in V$, $uv\in E$ 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 $n⩾1$ and $0⩽m⩽\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ 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.

AWOCA 2006

Cited by in corpus (1)

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