{"paper":{"title":"Colouring Graphs with Sparse Neighbourhoods: Bounds and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Luke Postle, Marthe Bonamy, Thomas Perrett","submitted_at":"2018-10-15T21:27:19Z","abstract_excerpt":"Let $G$ be a graph with chromatic number $\\chi$, maximum degree $\\Delta$ and clique number $\\omega$. Reed's conjecture states that $\\chi \\leq \\lceil (1-\\varepsilon)(\\Delta + 1) + \\varepsilon\\omega \\rceil$ for all $\\varepsilon \\leq 1/2$. It was shown by King and Reed that, provided $\\Delta$ is large enough, the conjecture holds for $\\varepsilon \\leq 1/130,000$. In this article, we show that the same statement holds for $\\varepsilon \\leq 1/26$, thus making a significant step towards Reed's conjecture. We derive this result from a general technique to bound the chromatic number of a graph where n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06704","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":""},"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"}