Weak Coloring Numbers of Intersection Graphs

TL;DR

This paper investigates the weak and strong coloring numbers of intersection graphs of geometric objects, providing bounds and revealing differences in their behavior across various classes and dimensions.

Contribution

It establishes bounds for weak coloring numbers of intersection graphs of well-behaved objects and highlights differences between weak and strong coloring numbers in specific graph classes.

Findings

Upper and lower bounds for weak coloring numbers are proved.

A class with polynomial strong but exponential weak coloring numbers is identified.

Dimension impacts weak coloring numbers differently for balls and hypercubes.

Abstract

Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with lower index 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, the first author together with 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 centrally symmetric 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).