On the independence number of de Bruijn graphs
This advances the theoretical understanding of de Bruijn graph independence numbers, a combinatorial problem with applications in coding theory and network design.
The authors derive an asymptotic formula for the independence number of de Bruijn graphs, providing exact formulas for specific cases (k=11,13) and bounds for k=4.
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 $k=3,5,7$.