On the complexity of some colorful problems parameterized by treewidth

• Published in: Information and computation Vol. 209; no. 2; pp. 143 - 153
• Main Authors: Fellows, Michael R., Fomin, Fedor V., Lokshtanov, Daniel, Rosamond, Frances, Saurabh, Saket, Szeider, Stefan, Thomassen, Carsten
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 01.02.2011; Elsevier
• Subjects: Applied sciences; Bounded treewidth; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Exact sciences and technology; Graph coloring; Graph theory; Information retrieval. Graph; Mathematics; Miscellaneous; Parameterized complexity; Sciences and techniques of general use; Theoretical computing; Bounded treewidth; Parameterized complexity; Graph coloring; Vertex; Computer theory; Color; Linear time; Chromatic number; Complexity; Assignment; Integer; Input; Graph colouring; Linear graph
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1016/j.ic.2010.11.026

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 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.