{"paper":{"title":"Perfect codes in generalized Fibonacci cubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michel Mollard (IF)","submitted_at":"2018-01-12T09:44:26Z","abstract_excerpt":"The {\\em Fibonacci cube} of dimension $n$, denoted as $\\Gamma\\_n$,  is the subgraph of the $n$-cube $Q\\_n$ induced by vertices with no consecutive 1's. In an article of 2016  Ashrafi and his co-authors proved the non-existence of perfect codes in  $\\Gamma\\_n$ for $n\\geq 4$. As an open problem the authors suggest to consider the existence of perfect codes in generalization of Fibonacci cubes. The most direct generalization is the family $\\Gamma\\_n(1^s)$ of subgraphs induced by strings without $1^s$ as a substring where $s\\geq 2$ is a given integer. We prove the existence of a perfect code in $\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.04106","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}