{"paper":{"title":"An algorithmic framework for colouring locally sparse graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Ewan Davies, Fran\\c{c}ois Pirot, Jean-S\\'ebastien Sereni, Ross J. Kang","submitted_at":"2020-04-15T15:30:38Z","abstract_excerpt":"We develop an algorithmic framework for graph colouring that reduces the problem to verifying a local probabilistic property of the independent sets.\n  With this we give, for any fixed $k\\ge 3$ and $\\varepsilon>0$, a randomised polynomial-time algorithm for colouring graphs of maximum degree $\\Delta$ in which each vertex is contained in at most $t$ copies of a cycle of length $k$, where $1/2\\le t\\le \\Delta^\\frac{2\\varepsilon}{1+2\\varepsilon}/(\\log\\Delta)^2$, with $\\lfloor(1+\\varepsilon)\\Delta/\\log(\\Delta/\\sqrt t)\\rfloor$ colours.\n  This generalises and improves upon several notable results inc"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2004.07151","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/2004.07151/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"}