  In Defective Coloring we are given a graph $G$ and two integers $\chi_d$,
$\Delta^*$ and are asked if we can $\chi_d$-color $G$ so that the maximum
degree induced by any color class is at most $\Delta^*$. We show that this
natural generalization of Coloring is much harder on several basic graph
classes. In particular, we show that it is NP-hard on split graphs, even when
one of the two parameters $\chi_d$, $\Delta^*$ is set to the smallest possible
fixed value that does not trivialize the problem ($\chi_d = 2$ or $\Delta^* =
1$). Together with a simple treewidth-based DP algorithm this completely
determines the complexity of the problem also on chordal graphs. We then
consider the case of cographs and show that, somewhat surprisingly, Defective
Coloring turns out to be one of the few natural problems which are NP-hard on
this class. We complement this negative result by showing that Defective
Coloring is in P for cographs if either $\chi_d$ or $\Delta^*$ is fixed; that
it is in P for trivially perfect graphs; and that it admits a sub-exponential
time algorithm for cographs when both $\chi_d$ and $\Delta^*$ are unbounded.
