Year: 2026
Type: preprint
Abstract: For positive integers $n$ and $d$, the $d$-dimensional grid of side length $n$ is the graph $G(n,d)$ with vertex set $[n]^d$, in which two vertices are adjacent if they differ by $1$ in exactly one coordinate and agree in all the other coordinates. We determine, for every fixed $d$, the asymptotic growth in $r$ of the $r$th strong coloring numbers of these grids. More precisely for $n$ large enough, ${\rm scol}_r(G(n,d)) = Ω_d\bigl(r^{d-1}\bigr)$ as $r\to\infty$. This matches an upper bound given by a volume packing argument, which holds more generally for touching graphs of unit balls in $\mathbb R^d$. For the lower bound, we construct in $G(n,d)$ a bramble of order $Ω_d(n^{d-1})$ with each element of radius at most $dn$. This construction was found by OpenAI's GPT-5.6 Sol.
Source: arXiv (Cornell University)
Authors: Micek, Piotr, Ueckerdt, Torsten
Institutions: Jagiellonian University, Karlsruhe Institute of Technology
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Award: ProduktStruktur von Graphen
OpenAlex ID: W7222610787