{"paper":{"title":"A note on asymptotically optimal neighbour sum distinguishing colourings","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jakub Przyby{\\l}o","submitted_at":"2017-03-01T17:37:23Z","abstract_excerpt":"The least $k$ admitting a proper edge colouring $c:E\\to\\{1,2,\\ldots,k\\}$ of a graph $G=(V,E)$ without isolated edges such that $\\sum_{e\\ni u}c(e)\\neq \\sum_{e\\ni v}c(e)$ for every $uv\\in E$ is denoted by $\\chi'_{\\Sigma}(G)$. It has been conjectured that $\\chi'_{\\Sigma}(G)\\leq \\Delta + 2$ for every connected graph of order at least three different from the cycle $C_5$, where $\\Delta$ is the maximum degree of $G$. It is known that $\\chi'_{\\Sigma}(G) = \\Delta + O(\\Delta^\\frac{5}{6}\\ln^\\frac{1}{6}\\Delta)$ for a graph $G$ without isolated edges. We improve this upper bound to $\\chi'_{\\Sigma}(G) = \\D"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00406","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"}