On the generalized coloring numbers

TL;DR

This paper reviews the concept of generalized coloring numbers, which extend the traditional coloring number to measure graph sparsity more precisely, and discusses recent combinatorial and algorithmic advances, including improved bounds.

Contribution

The paper provides a comprehensive overview of generalized coloring numbers, introduces new proofs for uniform orders, and improves bounds for graphs with excluded topological minors.

Findings

Unified framework for generalized coloring numbers

New simple proof for uniform orders

Improved bounds for graphs with excluded topological minors

Abstract

The \emph{coloring number} $\mathrm{col}(G)$ of a graph $G$, which is equal to the \emph{degeneracy} of $G$ plus one, provides a very useful measure for the uniform sparsity of $G$. The coloring number is generalized by three series of measures, the \emph{generalized coloring numbers}. These are the \emph{$r$-admissibility} $\mathrm{adm}_r(G)$, the \emph{strong $r$-coloring number} $\mathrm{col}_r(G)$ and the \emph{weak $r$-coloring number} $\mathrm{wcol}_r(G)$, where $r$ is an integer parameter. The generalized coloring numbers measure the edge density of bounded-depth minors and thereby provide an even more uniform measure of sparsity of graphs. They have found many applications in graph theory and in particular play a key role in the theory of bounded expansion and nowhere dense graph classes introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. We overview combinatorial and algorithmic applications of the generalized coloring numbers, emphasizing new developments in this area. We also present a simple proof for the existence of uniform orders and improve known bounds, e.g., for the weak coloring numbers on graphs with excluded topological minors.