  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$.
