Computer Science > Data Structures and Algorithms
[Submitted on 4 May 2023 (v1), last revised 22 Nov 2024 (this version, v3)]
Title:Coloring tournaments with few colors: Algorithms and complexity
View PDF HTML (experimental)Abstract: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 a new efficient decomposition lemma for tournaments, which we use to design polynomial-time algorithms to color various classes of tournaments with few colors, notably, to color a 2-colorable tournament with ten colors. We also use this lemma to prove equivalence between the problems of coloring 3-colorable tournaments and coloring 3-colorable graphs with constantly many 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.
Submission history
From: Alantha Newman [view email][v1] Thu, 4 May 2023 15:24:41 UTC (33 KB)
[v2] Fri, 8 Sep 2023 09:44:34 UTC (35 KB)
[v3] Fri, 22 Nov 2024 13:38:38 UTC (34 KB)
References & Citations
Bibliographic and Citation Tools
Bibliographic Explorer (What is the Explorer?)
Connected Papers (What is Connected Papers?)
Litmaps (What is Litmaps?)
scite Smart Citations (What are Smart Citations?)
Code, Data and Media Associated with this Article
alphaXiv (What is alphaXiv?)
CatalyzeX Code Finder for Papers (What is CatalyzeX?)
DagsHub (What is DagsHub?)
Gotit.pub (What is GotitPub?)
Hugging Face (What is Huggingface?)
Papers with Code (What is Papers with Code?)
ScienceCast (What is ScienceCast?)
Demos
Recommenders and Search Tools
Influence Flower (What are Influence Flowers?)
CORE Recommender (What is CORE?)
arXivLabs: experimental projects with community collaborators
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.