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

We consider the list-coloring problem from the perspective of parameterized complexity. In the classical graph coloring problem we are 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 get different colors. In list-coloring, each vertex is given a list of allowed colors with which it can be colored. 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 fixed parameter tractable. 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.