Grid-drawings of graphs in three-dimensions

Using probabilistic methods, we obtain grid-drawings of graphs without crossings with low volume and small aspect ratio. We show that every $D$-degenerate graph on $n$ vertices can be drawn in $[m]^3$ where $m = O(D^{5/3} n^{1/3}\log^{4/3}n)$. In particular, every graph of bounded maximum degree can be drawn in a grid with volume $O(n \log^{4}n)$.