{"paper":{"title":"Gracefulness of two nested cycles: a first approach","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Joaqu\\'in Tey, Miguel Licona","submitted_at":"2024-11-20T02:51:06Z","abstract_excerpt":"It is known that if a plane graph is graceful (resp. near-graceful), then its semidual is conservative (resp. near-conservative). In this work we prove that the semidual of a plane graph of size $M$ consisting of two nested cycles is conservative if $M \\equiv 0,3 \\pmod 4$, and near-conservative otherwise. We also show that for a given integer $m_1 \\geq 3$, there exists $m^* > m_1$ such that for $m_2 \\geq m^*$, if $m_1+m_2 \\equiv 0,3 \\pmod 4$ (resp. $m_1+m_2 \\equiv 1,2 \\pmod 4$), then there exists a graceful (resp. near-graceful) plane graph consisting of two nested cycles with sizes $m_1$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.12998","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.12998/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"}