{"paper":{"title":"Non-cyclic graphs of (non)orientable genus one","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Xuanlong Ma","submitted_at":"2015-12-03T03:16:32Z","abstract_excerpt":"Let $G$ be a finite non-cyclic group. The non-cyclic graph $\\Gamma_G$ of $G$ is the graph whose vertex set is $G\\setminus Cyc(G)$, two distinct vertices being adjacent if they do not generate a cyclic subgroup, where $Cyc(G)=\\{a\\in G: \\langle a,b\\rangle\\ \\text{is cyclic for each } b\\in G\\}$. In this paper, we classify all finite non-cyclic groups $G$ such that $\\Gamma_G$ has (non)orientable genus one."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00935","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"}