{"paper":{"title":"Asymptotically optimal neighbour sum distinguishing total colourings of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jakub Przyby{\\l}o","submitted_at":"2015-08-05T13:11:33Z","abstract_excerpt":"Consider a simple graph $G=(V,E)$ of maximum degree $\\Delta$ and its proper total colouring $c$ with the elements of the set $\\{1,2,\\ldots,k\\}$. The colouring $c$ is said to be \\emph{neighbour sum distinguishing} if for every pair of adjacent vertices $u$, $v$, we have $c(u)+\\sum_{e\\ni u}c(e)\\neq c(v)+\\sum_{e\\ni v}c(e)$. The least integer $k$ for which it exists is denoted by $\\chi\"_{\\sum}(G)$, hence $\\chi\"_{\\sum}(G) \\geq \\Delta+1$. On the other hand, it has been daringly conjectured that just one more label than presumed in the famous Total Colouring Conjecture suffices to construct such tota"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.01062","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"}