International Conference on Combinatorial Optimization and Applications

COCOA 2007: Combinatorial Optimization and Applications pp 366-377

# On the Complexity of Some Colorful Problems Parameterized by Treewidth

• Michael Fellows
• Fedor V. Fomin
• Daniel Lokshtanov
• Frances Rosamond
• Saket Saurabh
• Stefan Szeider
• Carsten Thomassen
Conference paper

DOI: 10.1007/978-3-540-73556-4_38

Volume 4616 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Fellows M. et al. (2007) On the Complexity of Some Colorful Problems Parameterized by Treewidth. In: Dress A., Xu Y., Zhu B. (eds) Combinatorial Optimization and Applications. COCOA 2007. Lecture Notes in Computer Science, vol 4616. Springer, Berlin, Heidelberg

## Abstract

We study the complexity of several coloring problems on graphs, parameterized by the treewidth t of the graph:

(1) The list chromatic numberχl(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]-hard, parameterized by the treewidth of G. The closely related Precoloring Extension problem is also shown to be W[1]-hard, parameterized by treewidth.

(3) 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 (t,r). We also show that a list-based variation, List Equitable Coloring is W[1]-hard for trees, parameterized by the number of colors on the lists.

### Topics

Parameterized Complexity Bounded Treewidth Graph Coloring

## Authors and Affiliations

• Michael Fellows
• 1
• Fedor V. Fomin
• 2
• Daniel Lokshtanov
• 2
• Frances Rosamond
• 1
• Saket Saurabh
• 2
• 3
• Stefan Szeider
• 4
• Carsten Thomassen
• 5
1. 1.University of Newcastle, NewcastleAustralia
2. 2.Department of Informatics, University of Bergen, BergenNorway
3. 3.Institute of Mathematical Sciences, ChennaiIndia
4. 4.Department of Computer Science, Durham University, DurhamU.K.
5. 5.Mathematics Institute, Danish Technical University, LyngbyDenmark