Improved bounds for weak coloring numbers

TL;DR

This paper establishes new bounds for weak coloring numbers, showing their growth rate in relation to graph parameters like treewidth and planarity, with implications for graph theory and related parameters.

Contribution

It provides the first tight bounds for weak coloring numbers in graphs with fixed treewidth and improves bounds for planar and outerplanar graphs.

Findings

Maximum r-th weak coloring number for treewidth k is Θ(r^{k-1} log r)

Lower bound for planar graphs improved to Ω(r^2 log r)

Exact bound for outerplanar graphs is Θ(r log r)

Abstract

Weak coloring numbers generalize the notion of degeneracy of a graph. They were introduced by Kierstead \& Yang in the context of games on graphs. Recently, several connections have been uncovered between weak coloring numbers and various parameters studied in graph minor theory and its generalizations. In this note, we show that for every fixed $k\geq1$, the maximum $r$-th weak coloring number of a graph with simple treewidth $k$ is $\Theta(r^{k-1}\log r)$. As a corollary, we improve the lower bound on the maximum $r$-th weak coloring number of planar graphs from $\Omega(r^2)$ to $\Omega(r^2\log r)$, and we obtain a tight bound of $\Theta(r\log r)$ for outerplanar graphs.