Golovach, P.A. and Paulusma, D. and Song, J. (2014) 'Closing complexity gaps for coloring problems on H-free graphs.', *Information and computation.*, 237 . pp. 204-214.

## Abstract

If a graph G contains no subgraph isomorphic to some graph H , then G is called H -free. A coloring of a graph G=(V,E) is a mapping c:V→{1,2,…} such that no two adjacent vertices have the same color, i.e., c(u)≠c(v) if uv∈E; if |c(V)|⩽k then c is a k -coloring. The Coloring problem is to test whether a graph has a coloring with at most k colors for some integer k . The Precoloring Extension problem is to decide whether a partial k -coloring of a graph can be extended to a k -coloring of the whole graph for some integer k . The List Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u). By imposing an upper bound ℓ on the size of each L(u) we obtain the ℓ -List Coloring problem. We first classify the Precoloring Extension problem and the ℓ -List Coloring problem for H -free graphs. We then show that 3-List Coloring is NP-complete for n -vertex graphs of minimum degree n−2, i.e., for complete graphs minus a matching, whereas List Coloring is fixed-parameter tractable for this graph class when parameterized by the number of vertices of degree n−2. Finally, for a fixed integer k>0, the Listk -Coloring problem is to decide whether a graph allows a coloring, such that every vertex u receives a color from some given set L(u) that must be a subset of {1,…,k}. We show that List 4-Coloring is NP-complete for P6-free graphs, where P6 is the path on six vertices. This completes the classification of Listk -Coloring for P6-free graphs.

Item Type: | Article |
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Keywords: | Graph coloring, Precoloring extension, List coloring, Forbidden induced subgraph. |

Full text: | (AM) Accepted Manuscript Download PDF (593Kb) |

Status: | Peer-reviewed |

Publisher Web site: | http://dx.doi.org/10.1016/j.ic.2014.02.004 |

Publisher statement: | NOTICE: this is the author’s version of a work that was accepted for publication in Information and computation. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Information and computation, 237, 2014, 10.1016/j.ic.2014.02.004 |

Date accepted: | No date available |

Date deposited: | 08 January 2015 |

Date of first online publication: | October 2014 |

Date first made open access: | No date available |

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