On the Complexity of Some Colorful Problems Parameterized by Treewidth

Abstract. We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph: (1) The list chromatic number χl(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 Lv of colors, where each list has length at least r, there is a choice of one color from each vertex list Lv yielding a proper coloring of G. We show that the problem of determining whether χl(G) ≤ r, the LIST CHROMATIC NUMBER problem, is solvable in linear time for every fixed treewidth bound t. The method by which this is shown is new and of general applicability. (2) The LIST COLORING problem takes as input a graph G, together with an assignment to each vertex v of a set of colors Cv. The problem is to determine whether it is possible to choose a color for vertex v from the set of permitted colors Cv, for each vertex, so that the obtained coloring of G is proper. We show that this problem is W[1]-...

• Type: Conference
• Year: 2007
• Where: COCOA
• Authors: Michael R. Fellows, Fedor V. Fomin, Daniel Lokshtanov, Frances A. Rosamond, Saket Saurabh, Stefan Szeider, Carsten Thomassen
• Keywords: COCOA 2007, Coloring Problem, List Chromatic Number, Proper Coloring