{"paper":{"title":"Towards the Overfull Conjecture II","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guantao Chen, Jessica McDonald, Songling Shan","submitted_at":"2026-07-02T14:52:50Z","abstract_excerpt":"Let $G$ be a simple graph with maximum degree $\\Delta(G)$. A subgraph $H\\subseteq G$ is $\\Delta(G)$-overfull if $|E(H)|>\\Delta(G)\\left\\lfloor |V(H)|/2\\right\\rfloor$. In any edge coloring of $G$, each color class restricted to $H$ is a matching of size at most $\\left\\lfloor |V(H)|/2\\right\\rfloor$. Thus, if $G$ contains a $\\Delta(G)$-overfull subgraph, then $G$ cannot be edge-colored with only $\\Delta(G)$ colors. By Vizing's Theorem, $\\chi'(G)\\le \\Delta(G)+1$, and hence $G$ is class $2$. In 1986, Chetwynd and Hilton conjectured that whenever $\\Delta(G)>|V(G)|/3$, the converse also holds: every c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02270","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/2607.02270/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"}