## Abstract

A k-colouring c of a graph G is a mapping V (G) → {1, 2, . . . k} such that c(u) 6= c(v) whenever u and v are adjacent. The corresponding decision problem is Colouring. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. Hence, every injective colouring is a star colouring and every star colouring is an acyclic colouring. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring (the last problem is also known as L(1, 1)-Labelling). A classical complexity result on Colouring is a well-known dichotomy for H-free graphs, which was established twenty years ago (in this context, a graph is H-free if and only if it does not contain H as an induced subgraph). Moreover, this result has led to a large collection of results, which helped us to better understand the complexity of Colouring. In contrast, there is no systematic study into the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring despite numerous algorithmic and structural results that have appeared over the years. We initiate such a systematic complexity study, and similar to the study of Colouring we use the class of H-free graphs as a testbed. We prove the following results: 1. We give almost complete classifications for the computational complexity of Acyclic Colouring, Star Colouring and Injective Colouring for H-free graphs. 2. If the number of colours k is fixed, that is, not part of the input, we give full complexity classifications for each of the three problems for H-free graphs. From our study we conclude that for fixed k the three problems behave in the same way, but this is no longer true if k is part of the input. To obtain several of our results we prove stronger complexity results that in particular involve the girth of a graph and the class of line graphs.

Original language | English |
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Title of host publication | 28th Annual European Symposium on Algorithms, ESA 2020 |

Editors | Fabrizio Grandoni, Grzegorz Herman, Peter Sanders |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771627 |

DOIs | |

Publication status | Published - 1 Aug 2020 |

Event | 28th Annual European Symposium on Algorithms, ESA 2020 - Virtual, Pisa, Italy Duration: 7 Sept 2020 → 9 Sept 2020 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 173 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 28th Annual European Symposium on Algorithms, ESA 2020 |
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Country/Territory | Italy |

City | Virtual, Pisa |

Period | 7/09/20 → 9/09/20 |

### Bibliographical note

Funding Information:Funding Jan Bok: Supported by GAUK 1198419 and SVV–2020–260578. Nikola Jedli˘cková: Supported by GAUK 1198419 and SVV–2020–260578. Daniël Paulusma: Supported by the Leverhulme Trust (RPG-2016-258).

Publisher Copyright:

© Jan Bok, Nikola Jedlicková, Barnaby Martin, Daniël Paulusma, and Siani Smith

## Keywords

- Acyclic colouring
- Dichotomy
- H-free
- Injective colouring
- Star colouring