{"paper":{"title":"An improved procedure for colouring graphs of bounded local density","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Eoin Hurley, R\\'emi de Joannis de Verclos, Ross J. Kang","submitted_at":"2020-07-15T17:49:16Z","abstract_excerpt":"We develop an improved bound for the chromatic number of graphs of maximum degree $\\Delta$ under the assumption that the number of edges spanning any neighbourhood is at most $(1-\\sigma)\\binom{\\Delta}{2}$ for some fixed $0<\\sigma<1$. The leading term in the reduction of colours achieved through this bound is best possible as $\\sigma\\to0$. As two consequences, we advance the state of the art in two longstanding and well-studied graph colouring conjectures, the Erd\\H{o}s-Ne\\v{s}et\\v{r}il conjecture and Reed's conjecture. We prove that the strong chromatic index is at most $1.772\\Delta^2$ for any"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.07874","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2007.07874/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"}