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Coloring Tournaments with Few Colors: Algorithms and Complexity

Authors Felix Klingelhoefer, Alantha Newman

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  • 14 pages

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Author Details

Felix Klingelhoefer
  • Laboratoire G-SCOP (Univ. Grenoble Alpes), Grenoble, France
Alantha Newman
  • Laboratoire G-SCOP (CNRS, Univ. Grenoble Alpes), Grenoble, France


We thank Louis Esperet for useful discussions and for his encouragement.

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Felix Klingelhoefer and Alantha Newman. Coloring Tournaments with Few Colors: Algorithms and Complexity. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 71:1-71:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


A k-coloring of a tournament is a partition of its vertices into k acyclic sets. Deciding if a tournament is 2-colorable is NP-hard. A natural problem, akin to that of coloring a 3-colorable graph with few colors, is to color a 2-colorable tournament with few colors. This problem does not seem to have been addressed before, although it is a special case of coloring a 2-colorable 3-uniform hypergraph with few colors, which is a well-studied problem with super-constant lower bounds. We present an efficient decomposition lemma for tournaments and show that it can be used to design polynomial-time algorithms to color various classes of tournaments with few colors, including an algorithm to color a 2-colorable tournament with ten colors. For the classes of tournaments considered, we complement our upper bounds with strengthened lower bounds, painting a comprehensive picture of the algorithmic and complexity aspects of coloring tournaments.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Approximation algorithms
  • Tournaments
  • Graph Coloring
  • Algorithms
  • Complexity


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