## Abstract

For a graph G and a tree-decomposition (T,B) of G, the chromatic number of (T,B) is the maximum of χ(G[B]), taken over all bags B∈B. The tree-chromatic number of G is the minimum chromatic number of all tree-decompositions (T,B) of G. The path-chromatic number of G is defined analogously. In this article, we introduce an operation that always increases the path-chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path-chromatic number and tree-chromatic number are different. This settles a question of Seymour (J Combin Theory Ser B 116 (2016), 229–237). Our results also imply that the path-chromatic numbers of the Mycielski graphs are unbounded.

Original language | English |
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Pages (from-to) | 213-222 |

Number of pages | 10 |

Journal | Journal of Graph Theory |

Volume | 86 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1 Oct 2017 |

Externally published | Yes |

## Keywords

- chromatic number
- Mycielski graphs
- path-decompositions
- tree-decompositions