Strong coloring numbers of grids

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, ${\rm scol}_r(G(n,d)) = \Omega_d\bigl(r^{d-1}\bigr)$ 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 $\Omega_d(n^{d-1})$ with each element of radius at most $dn$. This construction was found by OpenAI's GPT-5.6 Sol.