Coloring and Covering Nowhere Dense Graphs

Abstract

In [M. Grohe, S. Kreutzer, and S. Siebertz, J. ACM, 64 (2017), 17] it was shown that nowhere dense classes of graphs admit sparse neighborhood covers of small degree. We show that a monotone graph class admits sparse neighborhood covers if and only if it is nowhere dense. The existence of such covers for nowhere dense classes is established through bounds on so-called weak coloring numbers. The core results of this paper are various lower and upper bounds on the weak coloring numbers and other, closely related, generalized coloring numbers. We prove tight bounds for these numbers on graphs of bounded treewidth. We clarify and tighten the relation between the density of shallow minors and the various generalized coloring numbers. These upper bounds are complemented by new, stronger exponential lower bounds on the weak and strong coloring numbers, and by superpolynomial lower bounds on the weak coloring numbers on classes of polynomial expansion. Finally, we show that computing weak $r$-coloring numbers is NP-complete for all $r\geq 3$.