  In the \probrFix problem, we are given a graph $G$, a (non-proper)
vertex-coloring $c : V(G) \to [r]$, and a positive integer $k$. The goal is to
decide whether a proper $r$-coloring $c'$ is obtainable from $c$ by recoloring
at most $k$ vertices of $G$. Recently, Junosza-Szaniawski, Liedloff, and
Rz{\k{a}}{\.z}ewski [SOFSEM 2015] asked whether the problem has a polynomial
kernel parameterized by the number of recolorings $k$. In a full version of the
manuscript, the authors together with Garnero and Montealegre, answered the
question in the negative: for every $r \geq 3$, the problem \probrFix does not
admit a polynomial kernel unless $\NP \subseteq \coNP / \poly$. Independently
of their work, we give an alternative proof of the theorem. Furthermore, we
study the complexity of \probrFixSwap, where the only difference from \probrFix
is that instead of $k$ recolorings we have a budget of $k$ color swaps. We show
that for every $r \geq 3$, the problem \probrFixSwap is $\W[1]$-hard whereas
\probrFix is known to be FPT. Moreover, when $r$ is part of the input, we
observe both \probFix and \probFixSwap are $\W[1]$-hard parameterized by
treewidth. We also study promise variants of the problems, where we are
guaranteed that a proper $r$-coloring $c'$ is indeed obtainable from $c$ by
some finite number of swaps. For instance, we prove that for $r=3$, the
problems \probrFixPromise and \probrFixSwapPromise are $\NP$-hard for planar
graphs. As a consequence of our reduction, the problems cannot be solved in
$2^{o(\sqrt{n})}$ time unless the Exponential Time Hypothesis (ETH) fails.
