{"paper":{"title":"Toward Vu's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Abhishek Dhawan, Abhishek Methuku, Michael C. Wigal, Peter Bradshaw","submitted_at":"2025-08-22T22:37:22Z","abstract_excerpt":"In 2002, Vu conjectured that graphs of maximum degree $\\Delta$ and maximum codegree at most $\\zeta \\Delta$ have chromatic number at most $(\\zeta+o(1))\\Delta$. Despite its importance, the conjecture has remained widely open. The only direct progress so far has been obtained in the ``dense regime,'' when $\\zeta$ is close to $1$, by Hurley, de Verclos, and Kang.\n  In this paper we provide the first progress in the sparse regime $\\zeta \\ll 1$, the case of primary interest to Vu. We show that there exists $\\zeta_0 > 0$ such that for all $\\zeta \\in [\\log^{-32}\\Delta,\\zeta_0]$, the following holds: i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.16818","kind":"arxiv","version":2},"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/2508.16818/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"}