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On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs

Authors Alex Koutsoutis, Kilian Krause, Chun-Hung Liu , Mirza Redzic , Torsten Ueckerdt

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LIPIcs.WG.2026.31.pdf
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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Extremal graph theory
Keywords
  • Tree-independence number
  • Tree-chromatic number
  • Treewidth

Metrics

Abstract

We investigate the relationship between graph parameters, which measure the complexity of the tree decompositions of a given graph. The treewidth tw(G) of a graph G measures the largest number of vertices required in a bag of every tree decomposition of G. Similarly, the tree-independence number tree-α(G) and the tree-chromatic number tree-χ(G) measure the largest independence number, respectively the largest chromatic number, required in a bag of every tree decomposition of G. Recently, Dallard, Milanič, and Štorgel asked (JCTB, 2024) whether for all graphs G it holds that tw(G)+1 ≤ tree-α(G) ⋅ tree-χ(G). We provide a negative answer for this question in a strong form: for every function f: {ℕ} → {ℕ}, there exists a graph G such that tw(G) > tree-α(G) ⋅ f(tree-χ(G)). On the other hand, we complement this result with an upper bound, by showing that tw(G)+1 ≤ tree-α(G)² ⋅ tree-χ(G) for every graph G.

Cite As Get BibTex

Alex Koutsoutis, Kilian Krause, Chun-Hung Liu, Mirza Redzic, and Torsten Ueckerdt. On the Relation Between Treewidth, Tree-Independence Number, and Tree-Chromatic Number of Graphs. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 31:1-31:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.WG.2026.31

Author Details

Alex Koutsoutis
  • Texas A&M University, College Station, TX, USA
Kilian Krause
  • Karlsruhe Institute of Technology, Germany
Chun-Hung Liu
  • Texas A&M University, College Station, TX, USA
Mirza Redzic
  • Karlsruhe Institute of Technology, Germany
Torsten Ueckerdt
  • Karlsruhe Institute of Technology, Germany

Funding

  • Liu, Chun-Hung: Partially supported by NSF under CAREER award DMS-2144042.
  • Redzic, Mirza: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 462679611.
  • Ueckerdt, Torsten: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 532213793.

References

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