The Maximum Number of Edges in a Three-Dimensional Grid-Drawing

An exact formula is given for the maximum number of edges in a graph that admits a three-dimensional grid-drawing contained in a given bounding box. A three-dimensional (straight-line) grid-drawing of a graph represents the vertices by distinct points in Z, and represents each edge by a line-segment between its endpoints that does not intersect any other vertex, and does not intersect any other edge except at the endpoints. A folklore result states that every (simple) graph has a three-dimensional grid-drawing (see [2]). We therefore are interested in grid-drawings with small ‘volume’. The bounding box of a three-dimensional grid-drawing is the axis-aligned box of minimum size that contains the drawing. By an X × Y × Z grid-drawing we mean a three-dimensional griddrawing, such that the edges of the bounding box contain X, Y , and Z grid-points, respectively. The volume of a three-dimensional grid-drawing is the number of grid-points in the bounding box; that is, the volume of an X×Y ×Z grid-drawing is XY Z. (This definition is formulated to ensure that a two-dimensional grid-drawing has positive volume.) Our main contribution is the following extremal result. Theorem 1. The maximum number of edges in an X × Y × Z grid-drawing is exactly (2X − 1)(2Y − 1)(2Z − 1)−XY Z . Proof. Consider an X × Y ×Z grid-drawing of a graph G with n vertices and m edges. Let P be the set of points (x, y, z) contained in the bounding box such that 2x, 2y, and 2z are all integers. Observe that |P | = (2X − 1)(2Y − 1)(2Z − 1). The midpoint of every edge of G is in P , and no two edges share a common midpoint. Hence m ≤ |P |. In addition, the midpoint of an edge does not intersect a vertex. Thus m ≤ |P | − n . (1) A drawing with the maximum number of edges has no edge that passes through a grid-point. Otherwise, sub-divide the edge, and place the new vertex at that grid-point. Thus n = XY Z, and m ≤ |P | −XY Z, as claimed. This bound is attained by the following construction. Associate a vertex with each grid-point in an X × Y × Z grid-box B. As illustrated in Figure 1, every vertex (x, y, z) is adjacent to each of (x ± 1, y, z), (x, y ± 1, z), (x, y, z ± 1), (x ± 1, y ± 1, z), (x ± 1, y, z ± 1), (x, y ± 1, z ± 1), and (x±1, y±1, z±1), unless such a grid-point is not in B. It is easily seen that no two edges intersect, except at a common endpoint. Furthermore, every point in P is either a vertex or the midpoint of an edge. Thus the number of edges is |P | −XY Z. ‡School of Computer Science, Carleton University, Ottawa, Ontario, Canada. E-mail: {jit,morin,davidw}@scs.carleton.ca §Département d’informatique et d’ingénierie, Université du Québec en Outaouais, Gatineau, Québec, Canada. E-mail: [email protected] ∗Research supported by NSERC.