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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.01062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-08-05T13:11:33Z","cross_cats_sorted":[],"title_canon_sha256":"e139eef0b90f3c782b942eed70ae50c4feeef20a31c29a038fc63d9bcb4160f2","abstract_canon_sha256":"ad0c3e19ec4c11a982097bebfef035e6b18d38fa72f2c6ee69678922827f9a03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:35:44.149390Z","signature_b64":"dF5Mf7Iu0Q0w3UiOsX6UIHb9AeLdNItflwrNC9pDhQXE4QXeyCisgBmxMwl4Lhov7Ev2useDkZoSXY2Pgc4eAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b188cb4f37a7c3967c1f2661d4644fbb8125494a67e682f9e5a1eca27ed4054e","last_reissued_at":"2026-05-18T01:35:44.148915Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:35:44.148915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\{1,2,\\ldots,k\\}$. 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