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Let $G$ be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let $\\mu_G(v)=\\max\\{\\mu_G(u,v):u\\in V(G)\\setminus\\{v\\}\\}$, where $\\mu_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\\rightarrow\\mathbb{N}\\setminus\\{0,1\\}$, if $d_G(v)\\ge a(v)+b(v)+2\\mu_G(v)-3$ for each $v\\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such th"},"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":"2009.02175","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-04T13:19:36Z","cross_cats_sorted":[],"title_canon_sha256":"455c4de50f01636ca3f23093143e1210775f1e93fca5d6b6c716b89607e1abdc","abstract_canon_sha256":"ee0b37a28647ea00810beae0aaf873a45a41a5529757c472aa0042b3b1699282"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:33:04.131974Z","signature_b64":"HIqjbEjNLEco9KwF1Qk9NmLfwneG5gPlQfK9cl8u35D4I+cpL3z/meCXmrlfkeCUrxOyWeSJH0q5smMrzjgIDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e4324bba0c78b9dbb1d05dfe5dbb8434a43de669efcd4b86166f2e610a049ba","last_reissued_at":"2026-07-05T01:33:04.131633Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:33:04.131633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A generalization of Stiebitz-type results on graph decomposition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chunlei Zu, Qinghou Zeng","submitted_at":"2020-09-04T13:19:36Z","abstract_excerpt":"In this paper, we consider the decomposition of multigraphs under minimum degree constraints and give a unified generalization of several results by various researchers. Let $G$ be a multigraph in which no quadrilaterals share edges with triangles and other quadrilaterals and let $\\mu_G(v)=\\max\\{\\mu_G(u,v):u\\in V(G)\\setminus\\{v\\}\\}$, where $\\mu_G(u,v)$ is the number of edges joining $u$ and $v$ in $G$. We show that for any two functions $a,b:V(G)\\rightarrow\\mathbb{N}\\setminus\\{0,1\\}$, if $d_G(v)\\ge a(v)+b(v)+2\\mu_G(v)-3$ for each $v\\in V(G)$, then there is a partition $(X,Y)$ of $V(G)$ such th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.02175","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2009.02175/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2009.02175","created_at":"2026-07-05T01:33:04.131690+00:00"},{"alias_kind":"arxiv_version","alias_value":"2009.02175v1","created_at":"2026-07-05T01:33:04.131690+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.02175","created_at":"2026-07-05T01:33:04.131690+00:00"},{"alias_kind":"pith_short_12","alias_value":"BZBSJO5AY6FZ","created_at":"2026-07-05T01:33:04.131690+00:00"},{"alias_kind":"pith_short_16","alias_value":"BZBSJO5AY6FZ3OY5","created_at":"2026-07-05T01:33:04.131690+00:00"},{"alias_kind":"pith_short_8","alias_value":"BZBSJO5A","created_at":"2026-07-05T01:33:04.131690+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/BZBSJO5AY6FZ3OY5AXP6LW5YIN","json":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN.json","graph_json":"https://pith.science/api/pith-number/BZBSJO5AY6FZ3OY5AXP6LW5YIN/graph.json","events_json":"https://pith.science/api/pith-number/BZBSJO5AY6FZ3OY5AXP6LW5YIN/events.json","paper":"https://pith.science/paper/BZBSJO5A"},"agent_actions":{"view_html":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN","download_json":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN.json","view_paper":"https://pith.science/paper/BZBSJO5A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2009.02175&json=true","fetch_graph":"https://pith.science/api/pith-number/BZBSJO5AY6FZ3OY5AXP6LW5YIN/graph.json","fetch_events":"https://pith.science/api/pith-number/BZBSJO5AY6FZ3OY5AXP6LW5YIN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/action/storage_attestation","attest_author":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/action/author_attestation","sign_citation":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/action/citation_signature","submit_replication":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/action/replication_record"}},"created_at":"2026-07-05T01:33:04.131690+00:00","updated_at":"2026-07-05T01:33:04.131690+00:00"}