Two lower bounds for $p$-centered colorings

Given a graph $G$ and an integer $p$, a coloring $f:V(G)\to \mathbb{N}$ is \emph{$p$-centered} if for every connected subgraph $H$ of $G$, either $f$ uses more than $p$ colors on $H$ or there is a color that appears exactly once in $H$. The notion of $p$-centered colorings plays a central role in the theory of sparse graphs. In this note we show two lower bounds on the number of colors required in a $p$-centered coloring. First, we consider monotone classes of graphs whose shallow minors have average degree bounded polynomially in the radius, or equivalently (by a result of Dvo\v{r}'ak and Norin), admitting strongly sublinear separators. We construct such a class such that $p$-centered colorings require a number of colors super-polynomial in $p$. This is in contrast with a recent result of Pilipczuk and Siebertz, who established a polynomial upper bound in the special case of graphs excluding a fixed minor. Second, we consider graphs of maximum degree $\mathrm{\Delta }$. D\k{e}bski, Felsner, Micek, and Schr"{o}der recently proved that these graphs have $p$-centered colorings with $O({\mathrm{\Delta }}^{2-1\mathrm{/}p}p)$ colors. We show that there are graphs of maximum degree $\mathrm{\Delta }$ that require $\Omega(\Delta^{2-1/p} p \ln^{-1/p}\Delta)$colorsinany$p$-centered coloring, thus matching their upper bound up to a logarithmic factor.