On the independence number of de Bruijn graphs

TL;DR

This paper derives an asymptotic formula for the independence number of de Bruijn graphs, analyzes specific cases, and provides exact formulas for certain parameters, advancing understanding of their combinatorial properties.

Contribution

The paper introduces a new asymptotic formula for the independence number of de Bruijn graphs and determines exact values for specific cases like k=11 and k=13.

Findings

Derived asymptotic formula for α(k,q) involving a variational constant

Established bounds for λ_3 when k=4

Provided exact formulas for α(11,q) and α(13,q) for all q ≥ 2

Abstract

We derive the asymptotic formula $\alpha(k,q)=\lambda_{k-1}q^k+o(q^k)$, where $\alpha(k,q)$ is the independence number of the de Bruijn graph $B(k,q)$, and $\lambda_{k-1}$ is a constant arising from a variational problem on the unit $(k-1)$-dimensional cube. When $k=4$, we show the bounds $91/240\le \lambda_3\le 11/28$. 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 $\alpha(11,q)$ and $\alpha(13,q)$ for all $q\ge2$, extending the known cases $k=3,5,7$.