Fixed-Parameter Tractability of $(n-k)$ List Coloring

Abstract

We consider the list-coloring problem from the perspective of parameterized complexity. The classical graph coloring problem is given an undirected graph and the goal is to color the vertices of the graph with minimum number of colors so that end points of each edge gets different colors. In list-coloring, each vertex is given a list of allowed colors with which it can be colored.

In parameterized complexity, the goal is to identify natural parameters in the input that are likely to be small and design an algorithm with time $f(k) n^c$ time where c is a constant independent of k, and k is the parameter. Such an algorithm is called a fixed-parameter tractable (fpt) algorithm. It is clear that the solution size as a parameter is not interesting for graph coloring, as the problem is $\text { NP}$-hard even for $k=3$. An interesting parameterization for graph coloring that has been studied is whether the graph can be colored with $n-k$ colors, where k is the parameter and n is the number of vertices. This is known to be fpt using the notion of crown reduction. Our main result is that this can be generalized for list-coloring as well. More specifically, we show that, given a graph with each vertex having a list of size $n-k$, it can be determined in $f(k)n^{O(1)}$ time, for some function f of k, whether there is a coloring that respects the lists.