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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"},"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":"1703.00406","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-01T17:37:23Z","cross_cats_sorted":[],"title_canon_sha256":"3c0121faf10ea63881607d8abab1bfb1b0dc3c901105a2b2009b6b807aa9136b","abstract_canon_sha256":"3ae92b4a7639f39c0244dad7e1558662de080339c9e13022a1d99b6022352285"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:56.036647Z","signature_b64":"zKZPnDhWla0J8tKs7GQPkJNXOx71tbr97WC13yg45BISTlzXcrDQnudM73NTnicPJihtD2sg3FhCeCnWo0rZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"311d601d181104881a297f9926d893cab839b5dc36609968257bc2d4e03b6925","last_reissued_at":"2026-05-17T23:56:56.035978Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:56.035978Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1703.00406","created_at":"2026-05-17T23:56:56.036082+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.00406v1","created_at":"2026-05-17T23:56:56.036082+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.00406","created_at":"2026-05-17T23:56:56.036082+00:00"},{"alias_kind":"pith_short_12","alias_value":"GEOWAHIYCECI","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_16","alias_value":"GEOWAHIYCECIQGRJ","created_at":"2026-05-18T12:31:15.632608+00:00"},{"alias_kind":"pith_short_8","alias_value":"GEOWAHIY","created_at":"2026-05-18T12:31:15.632608+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK","json":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK.json","graph_json":"https://pith.science/api/pith-number/GEOWAHIYCECIQGRJP6MSNWETZK/graph.json","events_json":"https://pith.science/api/pith-number/GEOWAHIYCECIQGRJP6MSNWETZK/events.json","paper":"https://pith.science/paper/GEOWAHIY"},"agent_actions":{"view_html":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK","download_json":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK.json","view_paper":"https://pith.science/paper/GEOWAHIY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.00406&json=true","fetch_graph":"https://pith.science/api/pith-number/GEOWAHIYCECIQGRJP6MSNWETZK/graph.json","fetch_events":"https://pith.science/api/pith-number/GEOWAHIYCECIQGRJP6MSNWETZK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK/action/storage_attestation","attest_author":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK/action/author_attestation","sign_citation":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK/action/citation_signature","submit_replication":"https://pith.science/pith/GEOWAHIYCECIQGRJP6MSNWETZK/action/replication_record"}},"created_at":"2026-05-17T23:56:56.036082+00:00","updated_at":"2026-05-17T23:56:56.036082+00:00"}