On the independence number of de Bruijn graphs
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910134805987328 |
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| author | Majer, Pietro Novaga, Matteo |
| author_facet | Majer, Pietro Novaga, Matteo |
| contents | 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 $k=3,5,7$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_14671 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the independence number of de Bruijn graphs Majer, Pietro Novaga, Matteo Combinatorics Information Theory 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 $k=3,5,7$. |
| title | On the independence number of de Bruijn graphs |
| topic | Combinatorics Information Theory |
| url | https://arxiv.org/abs/2604.14671 |