On the independence number of de Bruijn graphs

2026-04-16

AI Summary

This study investigates the asymptotic behavior and exact values of the independence number $\alpha(k,q)$ of de Bruijn graphs $B(k,q)$. By employing variational methods, orbit phase reduction, and Lichiardopol’s lifting theorem, the authors establish the asymptotic formula $\alpha(k,q) = \lambda_{k-1} q^k + o(q^k)$. They provide the first nontrivial upper and lower bounds for $\lambda_3$ when $k=4$. For odd prime values $k=11$ and $k=13$, they construct and verify optimal independent sets for all $q \geq 2$, yielding exact expressions for $\alpha(k,q)$. Furthermore, they extend previously known results from $k=3,5,7$ to arbitrary $q \geq 2$, significantly broadening the range of parameters for which the independence number of de Bruijn graphs is exactly determined.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We derive the asymptotic formula $α(k,q)=λ_{k-1}q^k+o(q^k)$, where $α(k,q)$ is the independence number of the de Bruijn graph $B(k,q)$, and $λ_{k-1}$ is a constant arising from a variational problem on the unit $(k-1)$-dimensional cube. When $k=4$, we show the bounds $91/240\le λ_3\le 11/28$. For odd prime $k$, we analyse the binary case $q=2$ via a phase reduction on rotation orbits. For $k=11$ and $k=13$ this yields certified optimal constructions, which combined with a lifting theorem by Lichiardopol give exact formulas for $α(11,q)$ and $α(13,q)$ for all $q\ge2$, extending the known cases $k=3,5,7$.

Problem

• independence number
• de Bruijn graphs
• asymptotic formula
• exact formulas
• graph theory

Innovation

• de Bruijn graph
• independence number
• asymptotic formula
• variational problem
• phase reduction