2021/03/31 by Dvořák, Zdeněk, Pekárek, Jakub, Ueckerdt, Torsten +1
#Combinatorics (math.CO) #Computational Geometry (cs.CG) #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics
paper · doi:10.48550/arxiv.2103.17094
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 ℝd, 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).