### Abstract

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, togetherwith an assignment to each vertex v of a set of colors Cv. The problem is to determinewhether 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.

2. An equitable coloring of a graph G is a proper coloring of the verticeswhere the numbers of vertices having any two distinct colors differs by at most one.We show that the problem is hard forW[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 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 fromeach 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 on graphs of constant treewidth.

1. The List Coloring problem takes as input a graph G, togetherwith an assignment to each vertex v of a set of colors Cv. The problem is to determinewhether 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.

2. An equitable coloring of a graph G is a proper coloring of the verticeswhere the numbers of vertices having any two distinct colors differs by at most one.We show that the problem is hard forW[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 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 fromeach 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 on graphs of constant treewidth.

Original language | English |
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Pages (from-to) | 143-153 |

Number of pages | 11 |

Journal | Information and Computation |

Volume | 209 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2011 |

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## Cite this

Fellows, M., Fomin, F., Lokshtanov, D., Rosamond, F., Saurabh, S., Szeider, S., & Thomassen, C. (2011). On the complexity of some colorful problems parameterized by treewidth.

*Information and Computation*,*209*(2), 143-153. https://doi.org/10.1016/j.ic.2010.11.026