# On coloring numbers of graph powers

## TL;DR

This paper establishes bounds on the coloring numbers of graph powers, linking them to weak coloring numbers and graph parameters like maximum degree, tree width, and genus, with implications for graph coloring algorithms.

## Contribution

It provides new upper bounds on coloring numbers of graph powers based on weak coloring numbers and graph parameters, advancing theoretical understanding.

## Key findings

- Bound on col(G^p) in terms of wcol and degree
- Polynomial ratio between bounds and lower bounds for fixed tree width or genus
- Upper bound on col(G^2) based on maximum average degree and maximum degree

## Abstract

The weak $r$-coloring numbers $wcol_r(G)$ of a graph $G$ were introduced by the first two authors as a generalization of the usual coloring number $col(G)$, and have since found interesting theoretical and algorithmic applications. This has motivated researchers to establish strong bounds on these parameters for various classes of graphs. Let $G^p$ denote the $p$-th power of $G$. We show that, all integers $p >0$ and $\Delta \ge 3$ and graphs $G$ with $\Delta(G) \leq \Delta$ satisfy $col(G^p) \in O(p \cdot wcol_{\lceil p/2\rceil}(G)(\Delta-1)^{\lfloor p/2\rfloor})$; for fixed tree width or fixed genus the ratio between this upper bound and worst case lower bounds is polynomial in $p$. For the square of graphs $G$, we also show that, if the maximum average degree $2k-2 < mad(G) \leq 2k$, then $ col(G^2) \leq (2k-1)\Delta(G)+2k+1$.