{"paper":{"title":"On the structure of dense graphs with given odd girth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Shipeng Wang, Xingyan Lu","submitted_at":"2026-07-05T14:14:28Z","abstract_excerpt":"A classical theorem of Andr\\'asfai, Erd\\H{o}s, and S\\'os states that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\\delta(G)>\\frac{2n}{2k+1}$ is bipartite (i.e., homomorphic to $K_2$). Messuti and Schacht proved that the same odd girth condition with $\\delta(G)>\\frac{3n}{4k}$ forces a homomorphism to $C_{2k+1}$.\n  In this paper, we strengthen the above results by showing that every $n$-vertex graph $G$ with odd girth at least $2k+1$ and minimum degree $\\delta(G)>\\frac{4n}{6k-1}$ is homomorphic to the M\\\"obius ladder on $4k$ vertices. This answers a question of M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.04323","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/2607.04323/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"}