Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for a positive integer $k$, we seek a vertex ordering such that every vertex can (weakly, respectively, strongly) reach in $k$ steps only few vertices that precede it in the ordering. Both notions capture the sparsity of a graph or a graph class, and have interesting applications in the structural and algorithmic graph theory. Recently, Dvořák, McCarty, and Norin observed a natural volume-based upper bound for the strong coloring numbers of intersection graphs of well-behaved objects in $\mathbb{R}^d$, such as homothets of a compact convex object, or comparable axis-aligned boxes.
In this paper, we prove upper and lower bounds for the $k$-th weak coloring numbers of these classes of intersection graphs. As a consequence, we describe a natural graph class whose strong coloring numbers are polynomial in $k$, but the weak coloring numbers are exponential. We also observe a surprising difference in terms of the dependence of the weak coloring numbers on the dimension between touching graphs of balls (single-exponential) and hypercubes (double-exponential).
This is joint work with Zdeněk Dvořák, Jakub Pekárek, and Torsten Ueckerdt.