{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:BMTEF3G56Y6A52USW3JRTRZRYY","short_pith_number":"pith:BMTEF3G5","schema_version":"1.0","canonical_sha256":"0b2642ecddf63c0eea92b6d319c731c63a3bceb89c10aa7b3fa16ba3a39a5a60","source":{"kind":"arxiv","id":"1703.05672","version":1},"attestation_state":"computed","paper":{"title":"Distant total sum distinguishing index 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":"2017-03-16T15:28:03Z","abstract_excerpt":"Let $c:V\\cup E\\to\\{1,2,\\ldots,k\\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $\\Delta$. We say vertices $u,v\\in V$ are sum distinguished if $c(u)+\\sum_{e\\ni u}c(e)\\neq c(v)+\\sum_{e\\ni v}c(e)$. By $\\chi\"_{\\Sigma,r}(G)$ we denote the least integer $k$ admitting such a colouring $c$ for which every $u,v\\in V$, $u\\neq v$, at distance at most $r$ from each other are sum distinguished in $G$. For every positive integer $r$ an infinite family of examples is known with $\\chi\"_{\\Sigma,r}(G)=\\Omega(\\Delta^{r-1})$. In this paper we prove that $\\chi\"_{\\Sigma,r}(G)\\leq (2+o(1))\\De"},"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":"1703.05672","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-03-16T15:28:03Z","cross_cats_sorted":[],"title_canon_sha256":"e3b9ab9143c730bf4bb2b53ccffb8aa2c5522e01f03201bae19fc837ce17d947","abstract_canon_sha256":"5fc26ae265dd53798b220879d7f2250fee933ad7299c3f714790a81384a5f4b1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:56.028950Z","signature_b64":"wLJVN8uhgPhmWy2uFS0uejYnVI61/PaPSMr9ReeMckMwtZvHD4Zc+O6/65IhMItQlvN1o9EUQwI2PIPOAgvzCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b2642ecddf63c0eea92b6d319c731c63a3bceb89c10aa7b3fa16ba3a39a5a60","last_reissued_at":"2026-05-17T23:56:56.028270Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:56.028270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distant total sum distinguishing index 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":"2017-03-16T15:28:03Z","abstract_excerpt":"Let $c:V\\cup E\\to\\{1,2,\\ldots,k\\}$ be a proper total colouring of a graph $G=(V,E)$ with maximum degree $\\Delta$. We say vertices $u,v\\in V$ are sum distinguished if $c(u)+\\sum_{e\\ni u}c(e)\\neq c(v)+\\sum_{e\\ni v}c(e)$. By $\\chi\"_{\\Sigma,r}(G)$ we denote the least integer $k$ admitting such a colouring $c$ for which every $u,v\\in V$, $u\\neq v$, at distance at most $r$ from each other are sum distinguished in $G$. For every positive integer $r$ an infinite family of examples is known with $\\chi\"_{\\Sigma,r}(G)=\\Omega(\\Delta^{r-1})$. In this paper we prove that $\\chi\"_{\\Sigma,r}(G)\\leq (2+o(1))\\De"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.05672","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":"1703.05672","created_at":"2026-05-17T23:56:56.028389+00:00"},{"alias_kind":"arxiv_version","alias_value":"1703.05672v1","created_at":"2026-05-17T23:56:56.028389+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.05672","created_at":"2026-05-17T23:56:56.028389+00:00"},{"alias_kind":"pith_short_12","alias_value":"BMTEF3G56Y6A","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_16","alias_value":"BMTEF3G56Y6A52US","created_at":"2026-05-18T12:31:08.081275+00:00"},{"alias_kind":"pith_short_8","alias_value":"BMTEF3G5","created_at":"2026-05-18T12:31:08.081275+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/BMTEF3G56Y6A52USW3JRTRZRYY","json":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY.json","graph_json":"https://pith.science/api/pith-number/BMTEF3G56Y6A52USW3JRTRZRYY/graph.json","events_json":"https://pith.science/api/pith-number/BMTEF3G56Y6A52USW3JRTRZRYY/events.json","paper":"https://pith.science/paper/BMTEF3G5"},"agent_actions":{"view_html":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY","download_json":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY.json","view_paper":"https://pith.science/paper/BMTEF3G5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1703.05672&json=true","fetch_graph":"https://pith.science/api/pith-number/BMTEF3G56Y6A52USW3JRTRZRYY/graph.json","fetch_events":"https://pith.science/api/pith-number/BMTEF3G56Y6A52USW3JRTRZRYY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY/action/storage_attestation","attest_author":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY/action/author_attestation","sign_citation":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY/action/citation_signature","submit_replication":"https://pith.science/pith/BMTEF3G56Y6A52USW3JRTRZRYY/action/replication_record"}},"created_at":"2026-05-17T23:56:56.028389+00:00","updated_at":"2026-05-17T23:56:56.028389+00:00"}