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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: 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, parameter-ized by the number of colors on the lists. 3. The list chromatic number χ 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.

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