GD 2004: Graph Drawing pp 395-402

# No-Three-in-Line-in-3D

• Attila Pór
• David R. Wood
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3383)

## Abstract

The no-three-in-line problem, introduced by Dudeney in 1917, asks for the maximum number of points in the n× n grid with no three points collinear. In 1951, Erdös proved that the answer is Θ(n). We consider the analogous three-dimensional problem, and prove that the maximum number of points in the n× n× n grid with no three collinear is Θ(n2). This result is generalised by the notion of a 3D drawing of a graph. Here each vertex is represented by a distinct gridpoint in ℤ3, such that the line-segment representing each edge does not intersect any vertex, except for its own endpoints. Note that edges may cross. A 3D drawing of a complete graph Kn is nothing more than a set of n gridpoints with no three collinear. A slight generalisation of our first result is that the minimum volume for a 3D drawing of Kn is Θ(n3/2). This compares favourably to Θ(n3) when edges are not allowed to cross. Generalising the construction for Kn, we prove that every k-colourable graph on n vertices has a 3D drawing with $$\mathcal{O}(n\sqrt{k})$$ volume. For the k-partite Turán graph, we prove a lower bound of Ω((kn)3/4).

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