Dushnik-Miller dimension of TD-Delaunay complexes

TL;DR

This paper investigates the Dushnik-Miller dimension of TD-Delaunay complexes in higher dimensions, disproving a conjecture that related their dimension to the Euclidean dimension for all d.

Contribution

It provides a counterexample for the conjecture that TD-Delaunay complexes in dimension 4 have Dushnik-Miller dimension 4, challenging previous assumptions.

Findings

Disproves the conjecture for d=4

Shows TD-Delaunay complexes in R^{d-1} have Dushnik-Miller dimension d

Establishes the limits of the relationship between Euclidean and Dushnik-Miller dimensions

Abstract

TD-Delaunay graphs, where TD stands for triangular distance, is a variation of the classical Delaunay triangulations obtained from a specific convex distance function. Bonichon et. al. noticed that every triangulation is the TD-Delaunay graph of a set of points in $\mathbb{R}^2$, and conversely every TD-Delaunay graph is planar. It seems natural to study the generalization of this property in higher dimensions. Such a generalization is obtained by defining an analogue of the triangular distance for $\mathbb{R}^d$. It is easy to see that TD-Delaunay complexes of $\mathbb{R}^{d-1}$ are of Dushnik-Miller dimension $d$. The converse holds for $d=2$ or $3$ and it was conjectured independently by Mary and Evans et. al. to hold for larger $d$. Here we disprove the conjecture already for $d = 4$.