On the complexity of some colorful problems parameterized by treewidth

In this paper, we study the complexity of several coloring problems on graphs, parameterized by the treewidth of the graph.

1.

The List Coloring problem takes as input a graph G, together with an assignment to each vertex v of a set of colors C_v. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors C_v, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth.

2.

An equitable coloring of a graph G is a proper coloring of the vertices where the numbers of vertices having any two distinct colors differs by at most one. We show that the problem is hard for W[1], parameterized by the treewidth plus the number of colors. We also show that a list-based variation, List Equitable Coloring is W[1]-hard for forests, parameterized by the number of colors on the lists.

3.

The list chromatic numberχ_l(G) of a graph G is defined to be the smallest positive integer r, such that for every assignment to the vertices v of G, of a list L_v of colors, where each list has length at least r, there is a choice of one color from each vertex list L_v yielding a proper coloring of G. We show that the problem of determining whether χ_l(G)?r, the List Chromatic Number problem, is solvable in linear time on graphs of constant treewidth.