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

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... dth 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 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. (Michael R. Fellows), fomin@ii.uib.no (Fedor V. Fomin), daniello@ii.uib.no (Daniel Lokshtanov), frances.rosamond@cdu.edu.au (Frances Rosamond), saket@imsc.res.in (Saket Saurabh), stefan@szeider.net (Stefan Szeider), C.Thomassen@mat.dtu.dk (Carsten Thomassen)