Golovach, P.A. and Paulusma, Daniel and Song, J. (2012) 'Closing complexity gaps for coloring problems on H-free graphs.', in *Algorithms and computation : 23rd International Symposium, ISAAC 2012, Taipei, Taiwan, 19-21 December 2012 ; proceedings.* Berlin ; Heidelberg: Springer, pp. 14-23. Lecture notes in computer science., 7676 (7676).

## 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 List k-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 List k-Coloring for P6-free graphs.

Item Type: | Book chapter |
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Keywords: | Graph coloring, Precoloring extension, List coloring, Forbidden. |

Full text: | Full text not available from this repository. |

Publisher Web site: | http://dx.doi.org/10.1007/978-3-642-35261-4_5 |

Date accepted: | No date available |

Date deposited: | No date available |

Date of first online publication: | 2012 |

Date first made open access: | No date available |

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