Parameterized Pre-Coloring Extension and List Coloring Problems

Golovach, Paulusma, and Song [Inform. and Comput., 237 (2014), pp. 204--214] asked to determine the parameterized complexity of the following problems parameterized by $k$: 1. Given a graph $G$, a clique modulator $D$ (a clique modulator is a set of vertices, whose removal results in a clique) of size $k$ for $G$, and a list $L(v)$ of colors for every $v\in V(G)$, decide whether $G$ has a proper list coloring. 2. Given a graph $G$, a clique modulator $D$ of size $k$ for $G$, and a pre-coloring $\lambda_P: X \rightarrow Q$ for $X \subseteq V(G),$ decide whether $\lambda_P$ can be extended to a proper coloring of $G$ using only colors from $Q$. For problem 1 we design an ${\mathcal O}^*(2^k)$-time randomized algorithm and for problem 2 we obtain a kernel with at most $3k$ vertices. Banik et al. [in Proceedings of IWOCA 2019, Springer, Berlin, 2019, pp. 61--69] proved the following problem is fixed-parameter tractable and asked whether it admits a polynomial kernel: Given a graph $G$, an integer $k$, and a list $L(v)$ of exactly $n-k$ colors for every $v \in V(G),$ decide whether there is a proper list coloring for $G$. We obtain a kernel with ${\mathcal O}(k^2)$ vertices and colors and a compression to a variation of the problem with ${\mathcal O}(k)$ vertices and ${\mathcal O}(k^2)$ colors.