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Optimal b-Colourings and Fall Colourings in H-Free Graphs

Authors Jungho Ahn , Tala Eagling-Vose , Felicia Lucke , David Manlove , Fabricio Mendoza Granada , Daniël Paulusma

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ACM Subject Classification
  • Mathematics of computing → Graph theory
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Problems, reductions and completeness
Keywords
  • b-chromatic number
  • tight graph
  • fall achromatic number
  • fall chromatic number
  • H-free graph

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Abstract

In a colouring of a graph, a vertex is b-chromatic if it is adjacent to a vertex of every other colour. We consider four well-studied colouring problems: b-Chromatic Number, Tight b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number, which fit into a framework based on whether every colour class has (i) at least one b-chromatic vertex, (ii) exactly one b-chromatic vertex, or (iii) all of its vertices being b-chromatic. By combining known and new results, we fully classify the computational complexity of b-Chromatic Number, Fall Chromatic Number and Fall Achromatic Number in H-free graphs. For Tight b-Chromatic Number in H-free graphs, we develop a general technique to determine new graphs H, for which the problem is polynomial-time solvable, and we also determine new graphs H, for which the problem is still NP-complete. We show, for the first time, the existence of a graph H such that in H-free graphs, b-Chromatic Number is NP-hard, while Tight b-Chromatic Number is polynomial-time solvable.

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Jungho Ahn, Tala Eagling-Vose, Felicia Lucke, David Manlove, Fabricio Mendoza Granada, and Daniël Paulusma. Optimal b-Colourings and Fall Colourings in H-Free Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 2:1-2:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.2

Author Details

Jungho Ahn
  • Department of Computer Engineering, Inha University, Incheon, South Korea
Tala Eagling-Vose
  • Department of Computer Science, Durham University, UK
Felicia Lucke
  • ENS Lyon, France
David Manlove
  • School of Computing Science, University of Glasgow, UK
Fabricio Mendoza Granada
  • School of Computing Science, University of Glasgow, UK
Daniël Paulusma
  • Department of Computer Science, Durham University, UK

Funding

  • Ahn, Jungho: Supported by Leverhulme Trust Research Project Grant RPG-2024-182 and INHA University Research Grant.
  • Lucke, Felicia: Supported by Postdoc.Mobility Grant 230578.
  • Mendoza Granada, Fabricio: Supported by a scholarship from the College of Science and Engineering, University of Glasgow, and Becas Carlos Antonio Lopez (BECAL), Paraguay.
  • Paulusma, Daniël: Supported by Leverhulme Trust Research Project Grant RPG-2024-182.

Acknowledgements

The authors would like to thank the reviewers for their valuable comments.

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