Characterizing Large Clique Number in Tournaments

2026-02-10

AI Summary

This study addresses the structural characterization of tournaments with large clique numbers, aiming to establish effective certification methods for such parameters. By integrating tournament theory, backedge graph analysis, and extremal combinatorial techniques, the work proves that every tournament with a sufficiently large clique number necessarily contains a bounded-size substructure belonging to one of two fundamental families of tournaments. This result resolves an open problem posed by Aboulker et al. and establishes a finite verification mechanism for large clique numbers in tournaments, thereby deepening the understanding of unavoidable substructures in tournaments with high chromatic number.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Aboulker, Aubian, Charbit, and Lopes (2023) defined the clique number of a tournament to be the minimum clique number of one of its backedge graphs. Here we show that if $T$ is a tournament of sufficiently large clique number, then $T$ contains a subtournament of large clique number from one of two simple families of tournaments. In particular, large clique number is always certified by a bounded-size set. This answers a question of Aboulker, Aubian, Charbit, and Lopes (2023), and gives new insight into a line of research initiated by Kim and Kim (2018) into unavoidable subtournaments in tournaments with large dichromatic number.

Problem

• tournament
• clique number
• subtournament
• backedge graph
• unavoidable structure

Innovation

• tournament
• clique number
• unavoidable subtournament
• backedge graph
• dichromatic number