Combinatorics Seminar- Yelena Yuditsky

16:15-18:00

Speaker: Yelena Yuditsky

Title: Weak Coloring Numbers of Intersection Graphs

Abstract: 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 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, Dvoràk, McCarty, and Norin observed a natural volume-based upper bound for the strong coloring numbers
of intersection graphs of well-behaved objects in R^d, such as homothets of a compact convex object, or comparable axis-aligned boxes.

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).