### Abstract

Original language | Undefined |
---|---|

Pages (from-to) | 9-19 |

Number of pages | 11 |

Journal | Theoretical computer science |

Volume | 414 |

Issue number | 1 |

DOIs | |

Publication status | Published - 13 Jan 2012 |

### Keywords

- MSC-05C
- Linear forest
- EWI-21077
- METIS-293161
- Forbidden induced subgraph
- Graph coloring
- IR-82685

### Cite this

*Theoretical computer science*,

*414*(1), 9-19. https://doi.org/10.1016/j.tcs.2011.10.005

}

*Theoretical computer science*, vol. 414, no. 1, pp. 9-19. https://doi.org/10.1016/j.tcs.2011.10.005

**Updating the complexity status of coloring graphs without a fixed induced linear forest.** / Broersma, Haitze J.; Golovach, P.A.; Paulusma, Daniël; Song, Jian.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Updating the complexity status of coloring graphs without a fixed induced linear forest

AU - Broersma, Haitze J.

AU - Golovach, P.A.

AU - Paulusma, Daniël

AU - Song, Jian

N1 - eemcs-eprint-21077

PY - 2012/1/13

Y1 - 2012/1/13

N2 - A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph Pk denotes a path on k vertices. The ℓ-Coloring problem is the problem to decide whether a graph can be colored with at most ℓ colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8-free graphs. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-Coloring is NP-complete for P9-free graphs, and a result of Woeginger and Sgall, who showed that 5-Coloring is NP-complete for P8-free graphs. Additionally, we prove that the precoloring extension version of 4-Coloring is NP-complete for P7-free graphs, but that the precoloring extension version of 3-Coloring can be solved in polynomial time for (P2+P4)-free graphs, a subclass of P7-free graphs. Here P2+P4 denotes the disjoint union of a P2 and a P4. We denote the disjoint union of s copies of a P3 by sP3 and involve Ramsey numbers to prove that the precoloring extension version of 3-Coloring can be solved in polynomial time for sP3-free graphs for any fixed s. Combining our last two results with known results yields a complete complexity classification of (precoloring extension of) 3-Coloring for H-free graphs when H is a fixed graph on at most 6 vertices: the problem is polynomial-time solvable if H is a linear forest; otherwise it is NP-complete.

AB - A graph is H-free if it does not contain an induced subgraph isomorphic to the graph H. The graph Pk denotes a path on k vertices. The ℓ-Coloring problem is the problem to decide whether a graph can be colored with at most ℓ colors such that adjacent vertices receive different colors. We show that 4-Coloring is NP-complete for P8-free graphs. This improves a result of Le, Randerath, and Schiermeyer, who showed that 4-Coloring is NP-complete for P9-free graphs, and a result of Woeginger and Sgall, who showed that 5-Coloring is NP-complete for P8-free graphs. Additionally, we prove that the precoloring extension version of 4-Coloring is NP-complete for P7-free graphs, but that the precoloring extension version of 3-Coloring can be solved in polynomial time for (P2+P4)-free graphs, a subclass of P7-free graphs. Here P2+P4 denotes the disjoint union of a P2 and a P4. We denote the disjoint union of s copies of a P3 by sP3 and involve Ramsey numbers to prove that the precoloring extension version of 3-Coloring can be solved in polynomial time for sP3-free graphs for any fixed s. Combining our last two results with known results yields a complete complexity classification of (precoloring extension of) 3-Coloring for H-free graphs when H is a fixed graph on at most 6 vertices: the problem is polynomial-time solvable if H is a linear forest; otherwise it is NP-complete.

KW - MSC-05C

KW - Linear forest

KW - EWI-21077

KW - METIS-293161

KW - Forbidden induced subgraph

KW - Graph coloring

KW - IR-82685

U2 - 10.1016/j.tcs.2011.10.005

DO - 10.1016/j.tcs.2011.10.005

M3 - Article

VL - 414

SP - 9

EP - 19

JO - Theoretical computer science

JF - Theoretical computer science

SN - 0304-3975

IS - 1

ER -