{"paper":{"title":"On the independence number of de Bruijn graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The independence number of the de Bruijn graph B(k,q) equals λ_{k-1} q^k plus a smaller term, where λ_{k-1} is the value of a variational problem on the unit (k-1)-cube.","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.CO","authors_text":"Matteo Novaga, Pietro Majer","submitted_at":"2026-04-16T06:28:49Z","abstract_excerpt":"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 $k=11,13,17$ this yields compact orbit-marker certificates for optimal constructions. Combined with a lifting theorem by Lichiardopol, these certificates give exact formulas for $\\"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We derive the asymptotic formula α(k,q)=λ_{k-1}q^k+o(q^k), where λ_{k-1} is a constant arising from a variational problem on the unit (k-1)-dimensional cube. ... 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≥2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The variational problem on the (k-1)-cube correctly captures the asymptotic density of maximum independent sets, and the phase-reduction constructions for odd-prime k are optimal in the binary case so that the cited lifting theorem extends them exactly to all q.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"α(k,q) = λ_{k-1} q^k + o(q^k) where λ_{k-1} arises from a variational problem on the (k-1)-cube; exact formulas hold for k=11,13 via phase reduction and lifting, with bounds 91/240 ≤ λ_3 ≤ 11/28 for k=4.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The independence number of the de Bruijn graph B(k,q) equals λ_{k-1} q^k plus a smaller term, where λ_{k-1} is the value of a variational problem on the unit (k-1)-cube.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"92c13cf0069f61e3c4215f223f17f5cd7733f28efe2f4203f82860a42046b3e0"},"source":{"id":"2604.14671","kind":"arxiv","version":2},"verdict":{"id":"5774d9a3-b8e5-4081-afce-5f747c6a8621","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T11:22:28.857098Z","strongest_claim":"We derive the asymptotic formula α(k,q)=λ_{k-1}q^k+o(q^k), where λ_{k-1} is a constant arising from a variational problem on the unit (k-1)-dimensional cube. ... 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≥2.","one_line_summary":"α(k,q) = λ_{k-1} q^k + o(q^k) where λ_{k-1} arises from a variational problem on the (k-1)-cube; exact formulas hold for k=11,13 via phase reduction and lifting, with bounds 91/240 ≤ λ_3 ≤ 11/28 for k=4.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The variational problem on the (k-1)-cube correctly captures the asymptotic density of maximum independent sets, and the phase-reduction constructions for odd-prime k are optimal in the binary case so that the cited lifting theorem extends them exactly to all q.","pith_extraction_headline":"The independence number of the de Bruijn graph B(k,q) equals λ_{k-1} q^k plus a smaller term, where λ_{k-1} is the value of a variational problem on the unit (k-1)-cube."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.14671/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}