On coloring numbers of graph powers

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 pth power of G. We show that, all integers p>0 and Δ≥3 and graphs G with Δ(G)≤Δ satisfy col(G^p)∈O(p⋅wcol_⌈p∕2⌉(G)(Δ−1)^⌊p∕2⌋); 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)≤2k, then col(G²)≤(2k−1)Δ(G)+2k+1.