{"paper":{"title":"Coloring graphs with dense neighborhoods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Landon Rabern","submitted_at":"2012-09-17T13:25:36Z","abstract_excerpt":"It is shown that any graph with maximum degree $\\Delta$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\\frac{6k^2}{6k^2 + 1}\\Delta + k + 6$ is either $(\\Delta - k)$-colorable or contains a clique on more than $\\Delta - 2k$ vertices. In the $k=1$ case we improve the bound on the average degree to $\\frac23\\Delta + 4$ and the bound on the clique number to $\\Delta-1$. As corollaries, we show that every graph satisfies $\\chi \\leq \\max\\set{\\omega, \\Delta - 1, 4\\alpha}$ and every graph satisfies $\\chi \\leq \\max\\set{\\omega, \\Delta - 1, \\ceil{\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3646","kind":"arxiv","version":3},"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"}