Colouring random Hasse diagrams and box-Delaunay graphs

TL;DR

The paper studies two closely related graph families derived from random points in $[0,1]^d$: box-Delaunay graphs and Hasse diagrams under the coordinatewise order. It establishes that, for fixed dimension $d\ge2$ and whp, the chromatic number scales as $\chi(G)=(\log n)^{d-1+o(1)}$ and the independence number scales as $\alpha(G)=n/(\log n)^{d-1+o(1)}$, with sharp constants in dimension $d=2$ (where $\chi(G) \asymp \log n/\log\log n$ and $\alpha(G) \asymp n\log\log n/\log n$). The results extend prior work of Chen, Pach, Szegedy and Tardos and resolve Tomon’s conjecture in higher dimensions, by providing tight asymptotics (up to polylog factors) for random box-Delaunay/Hasse graphs. The methodology combines Poissonisation, maximum-degree and local-sparsity analyses, and a dimension-wise digit-revelation (or suitable-pair) framework to transfer 2D techniques to arbitrary fixed dimension, yielding sharp or near-sharp bounds on both colorability and independence. These insights advance understanding of non-local geometric random graphs and have implications for related coloring problems, including conflict-free colouring in axis-aligned boxes.

Abstract

Fix $d\ge2$ and consider a uniformly random set $P$ of $n$ points in $[0,1]^{d}$. Let $G$ be the Hasse diagram of $P$ (with respect to the coordinatewise partial order), or alternatively let $G$ be the Delaunay graph of $P$ with respect to axis-parallel boxes (where we put an edge between $u,v\in P$ whenever there is an axis-parallel box containing $u,v$ and no other points of $P$). In each of these two closely related settings, we show that the chromatic number of $G$ is typically $(\log n)^{d-1+o(1)}$ and the independence number of $G$ is typically $n/(\log n)^{d-1+o(1)}$. When $d=2$, we obtain bounds that are sharp up to constant factors: the chromatic number is typically of order $\log n/\log\log n$ and the independence number is typically of order $n\log\log n/\log n$. These results extend and sharpen previous bounds by Chen, Pach, Szegedy and Tardos. In addition, they provide new bounds on the largest possible chromatic number (and lowest possible independence number) of a $d$-dimensional box-Delaunay graph or Hasse diagram, in particular resolving a conjecture of Tomon.

Figures (2)

• Figure 1: An illustration of the 2-dimensional case. The points $x(1)$, $x(2)$, $x(3)$ are denoted by crosses, and other points that could possibly contribute to $X$ (i.e., points $x\in P_{\mathrm{Po}}$ with $\mathrm R[o,x]\cap P_{\mathrm{Po}}=\emptyset$) are denoted by black dots. In this example, only the bottom-most two dots actually contribute to $X$. The points $x(1),x(2),x(3)$ ensure that no point in the grey area contributes to $X$.
• Figure 2: The solid points are neighbours of $p$ and the grey points are the common neighbours of $p$ and $q$.

Theorems & Definitions (42)

• Definition
• Definition
• Theorem
• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Remark 2.2
• Remark 2.3
• Theorem 2.4
• Lemma 2.5