Abstract

. A three-dimensional grid drawing of a graph G is a placement of the vertices at distinct integer points so that the straight-line segments representing the edges of G are pairwise non-crossing. It is shown that for any fixed r 2, every r-colorable graph of n vertices has a threedimensional grid drawing that fits into a box of volume O(n 2 ). The order of magnitude of this bound cannot be improved. 1 Introduction In a grid drawing of a graph, the vertices are represented by distinct points with integer coordinates and the edges are represented by straight-line segments connecting the corresponding pairs of points. Grid drawings in the plane have a vast literature [BE]. In particular, it is known that every planar graph of n vertices has a two-dimensional grid drawing that fits into a rectangle of area O(n 2 ), and this bound is asymptotically tight [FP],[S]. The possibility of three-dimensional representations of graphs was suggested by software engineers [MR]. The analysis of...

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