  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 $\Delta$. D\k{e}bski, Felsner,
Micek, and Schr\"{o}der recently proved that these graphs have $p$-centered
colorings with $O(\Delta^{2-1/p} p)$ colors. We show that there are graphs of
maximum degree $\Delta$ that require $\Omega(\Delta^{2-1/p} p
\ln^{-1/p}\Delta)$ colors in any $p$-centered coloring, thus matching their
upper bound up to a logarithmic factor.
