Grid-drawings of graphs in three-dimensions

TL;DR

This work studies three-dimensional grid-drawings of graphs with the aim of minimizing grid volume while avoiding edge crossings. It develops a probabilistic framework that leverages counting bounds for collinear and coplanar tuples, establishing that any $D$-degenerate graph with $n$ vertices and $k$ edges admits a grid-drawing in $[m]^3$ with $m^3 = O(D k \log n)$; in particular, graphs of bounded maximum degree have grid volume $O(n \log n)$. A key methodological contribution is the $t$-blowup construction combined with a degeneracy ordering and a greedy embedding to realize the drawing in $[m]^3$, supported by precise combinatorial bounds on affine-subspace configurations. These results advance the understanding of 3D grid-drawings, providing near-linear volume guarantees for broad graph classes and resolving, up to logarithmic factors, a long-standing open problem of Pach, Thiele, and Tóth on drawings with low volume.

Abstract

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 $m^3 = O(D^2 n\log n)$. In particular, every graph of bounded maximum degree can be drawn in a grid with volume $O(n \log n)$.

Theorems & Definitions (10)

• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof
• Proposition 4
• proof
• Theorem 5
• proof
• proof : Proof of Theorem \ref{['blowupThm']}