  A graph on $n$ vertices is equitably $k$-colorable if it is $k$-colorable and
every color is used either $\left\lfloor n/k \right\rfloor$ or $\left\lceil n/k
\right\rceil$ times.
  Such a problem appears to be considerably harder than vertex coloring, being
$\mathsf{NP\text{-}Complete}$ even for cographs and interval graphs.
  In this work, we prove that it is $\mathsf{W[1]\text{-}Hard}$ for block
graphs and for disjoint union of split graphs when parameterized by the number
of colors; and $\mathsf{W[1]\text{-}Hard}$ for $K_{1,4}$-free interval graphs
when parameterized by treewidth, number of colors and maximum degree,
generalizing a result by Fellows et al. (2014) through a much simpler
reduction.
  Using a previous result due to Dominique de Werra (1985), we establish a
dichotomy for the complexity of equitable coloring of chordal graphs based on
the size of the largest induced star.
  Finally, we show that \textsc{equitable coloring} is $\mathsf{FPT}$ when
parameterized by the treewidth of the complement graph.
