  $\chi$-bounded classes are studied here in the context of star colorings and
more generally $\chi_p$-colorings. This leads to natural extensions of the
notion of bounded expansion class and to structural characterization of these.
In this paper we solve two conjectures related to star coloring boundedness.
One of the conjectures is disproved and in fact we determine which weakening
holds true. We give structural characterizations of (strong and weak)
$\chi_p$-bounded classes. On the way, we generalize a result of Wood relating
the chromatic number of a graph to the star chromatic number of its
$1$-subdivision. As an application of our characterizations, among other
things, we show that for every odd integer $g>3$ even hole-free graphs $G$
contain at most $\varphi(g,\omega(G))\,|G|$ holes of length $g$.
