Question 88
A graph $G$ is called a *divisibility graph* if its vertices can be assigned distinct positive integers such that between two vertices assigned $u, v$ there is an edge if and only if $\frac{u}{v}$ or $\frac{v}{u}$ is a positive integer. Show that for any positive integer $n$ and $0 \le e \le \frac{n(n-1)}{2}$, there is a divisibility graph with $n$ vertices and $e$ edges.
(Danila Cherkashin)
(Danila Cherkashin)