Weak Coloring Numbers of Intersection Graphs

Yelena Yuditsky (Université libre de Bruxelles)

12-May-2022, 14:15-16:00 (23 months ago)

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, DvorĂ k, 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. 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).

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic


Copenhagen-Jerusalem Combinatorics Seminar

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