On the generalized coloring numbers

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.