{"paper":{"title":"Triangle-free Graphs with Large Minimum Common Degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fan Zhao, Jian Wang, Weihua Yang","submitted_at":"2024-08-10T13:12:27Z","abstract_excerpt":"Let $G$ be a graph. For $x\\in V(G)$, let $N(x)=\\{y\\in V(G)\\colon xy\\in E(G)\\}$. The minimum common degree of $G$, denoted by $\\delta_{2}(G)$, is defined as the minimum of $|N(x)\\cap N(y)|$ over all non-edges $xy$ of $G$. In 1982, H\\\"{a}ggkvist showed that every triangle-free graph with minimum degree greater than $\\lfloor\\frac{3n}{8}\\rfloor$ is homomorphic to a cycle of length 5. In this paper, we prove that every triangle-free graph with minimum common degree greater than $\\lfloor\\frac{n}{8}\\rfloor$ is homomorphic to a cycle of length 5, which implies H\\\"{a}ggkvist's result. The balanced blow"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.05547","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/2408.05547/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"}