Efficient Grid-Drawings of Graphs in Three Dimensions

Using probabilistic methods, the authors obtain grid-drawings of graphs without crossings with low volume and small aspect ratio. They show that every D-degenerate graph on n vertices can be drawn in a [m]^3 grid 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).