{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:MZYPBYWNQLOPAFHG73G2O4U35I","short_pith_number":"pith:MZYPBYWN","schema_version":"1.0","canonical_sha256":"6670f0e2cd82dcf014e6fecda7729bea2e6560f3dd1799678ece3edc0d07a731","source":{"kind":"arxiv","id":"1806.09357","version":1},"attestation_state":"computed","paper":{"title":"The Tutte's condition in terms of graph factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David G.L. Wang, Hongliang Lu","submitted_at":"2018-06-25T10:08:35Z","abstract_excerpt":"Let $G$ be a connected general graph of even order, with a function $f\\colon V(G)\\to\\Z^+$. We obtain that $G$ satisfies the Tutte's condition \\[ o(G-S)\\le \\sum_{v\\in S}f(v)\\qquad\\text{for any nonempty set $S\\subset V(G)$}, \\] with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\\colon V(G)\\to 2^\\N$ such that $H(v)\\in \\{J_f(v),\\,J_f^+(v)\\}$ for each $v\\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a "},"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":"1806.09357","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-25T10:08:35Z","cross_cats_sorted":[],"title_canon_sha256":"10fca947b0422af32701739f21eb659f5ae5b73737e6b572e9307cc92016aa1d","abstract_canon_sha256":"e59adeb7e9dbab7034df133edd47e1caffee1872f1acc4f570a48cc70d2ee090"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:29.585122Z","signature_b64":"kAZy/2VeugfdPxj3WTTgJVwbGaLCsRFPt1QGaQ7zFyx64SbbXTqmQC9zhkP/uQ04kGmQVAS143jSDqjoLlYXBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6670f0e2cd82dcf014e6fecda7729bea2e6560f3dd1799678ece3edc0d07a731","last_reissued_at":"2026-05-18T00:12:29.584664Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:29.584664Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Tutte's condition in terms of graph factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David G.L. Wang, Hongliang Lu","submitted_at":"2018-06-25T10:08:35Z","abstract_excerpt":"Let $G$ be a connected general graph of even order, with a function $f\\colon V(G)\\to\\Z^+$. We obtain that $G$ satisfies the Tutte's condition \\[ o(G-S)\\le \\sum_{v\\in S}f(v)\\qquad\\text{for any nonempty set $S\\subset V(G)$}, \\] with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\\colon V(G)\\to 2^\\N$ such that $H(v)\\in \\{J_f(v),\\,J_f^+(v)\\}$ for each $v\\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09357","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":"1806.09357","created_at":"2026-05-18T00:12:29.584735+00:00"},{"alias_kind":"arxiv_version","alias_value":"1806.09357v1","created_at":"2026-05-18T00:12:29.584735+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.09357","created_at":"2026-05-18T00:12:29.584735+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZYPBYWNQLOP","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZYPBYWNQLOPAFHG","created_at":"2026-05-18T12:32:40.477152+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZYPBYWN","created_at":"2026-05-18T12:32:40.477152+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/MZYPBYWNQLOPAFHG73G2O4U35I","json":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I.json","graph_json":"https://pith.science/api/pith-number/MZYPBYWNQLOPAFHG73G2O4U35I/graph.json","events_json":"https://pith.science/api/pith-number/MZYPBYWNQLOPAFHG73G2O4U35I/events.json","paper":"https://pith.science/paper/MZYPBYWN"},"agent_actions":{"view_html":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I","download_json":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I.json","view_paper":"https://pith.science/paper/MZYPBYWN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1806.09357&json=true","fetch_graph":"https://pith.science/api/pith-number/MZYPBYWNQLOPAFHG73G2O4U35I/graph.json","fetch_events":"https://pith.science/api/pith-number/MZYPBYWNQLOPAFHG73G2O4U35I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I/action/storage_attestation","attest_author":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I/action/author_attestation","sign_citation":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I/action/citation_signature","submit_replication":"https://pith.science/pith/MZYPBYWNQLOPAFHG73G2O4U35I/action/replication_record"}},"created_at":"2026-05-18T00:12:29.584735+00:00","updated_at":"2026-05-18T00:12:29.584735+00:00"}