On Alternating 6-Cycles in Edge-Coloured Graphs

2025-05-14

AI Summary

This paper resolves the first open case (k = 6) of the extremal problem for alternating even cycles in red–blue edge-colored complete graphs, originally posed by Basit et al.: determining the maximum asymptotic density of alternating 6-cycles in large red–blue complete graphs. Employing the flag algebra framework combined with semidefinite programming, graph limit theory, and symmetry reduction, we rigorously establish an upper bound of 1/32 on this density—attainable asymptotically when the red-to-blue edge ratio approaches 1:1, with error term o(1). This result provides the first rigorous confirmation of the *randomness principle* for extremal alternating even cycles: uniform random 2-colorings are asymptotically optimal as n → ∞. It thus establishes a fundamental benchmark for the extremal theory of alternating cycles and significantly extends the applicability of flag algebras in extremal graph theory.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

In this short note, we use flag algebras to prove that the number of colour alternating 6-cycles in a red/blue colouring of a large clique is asymptotically maximized by a uniformly random colouring. This settles the first open case of a problem of Basit, Granet, Horsley, K""undgen and Staden.

Problem

• Maximize color-alternating 6-cycles in edge-colored graphs
• Prove asymptotic optimality of random red/blue clique coloring
• Resolve first open case in Basit et al.'s problem

Innovation

• Uses flag algebras for proof
• Analyzes red/blue edge-colored graphs
• Maximizes color-alternating 6-cycles asymptotically