{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:BZBSJO5AY6FZ3OY5AXP6LW5YIN","short_pith_number":"pith:BZBSJO5A","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"},"canonical_sha256":"0e4324bba0c78b9dbb1d05dfe5dbb8434a43de669efcd4b86166f2e610a049ba","source":{"kind":"arxiv","id":"2009.02175","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.02175","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"arxiv_version","alias_value":"2009.02175v1","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.02175","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"pith_short_12","alias_value":"BZBSJO5AY6FZ","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"BZBSJO5AY6FZ3OY5","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"BZBSJO5A","created_at":"2026-07-05T01:33:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:BZBSJO5AY6FZ3OY5AXP6LW5YIN","target":"record","payload":{"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"},"canonical_sha256":"0e4324bba0c78b9dbb1d05dfe5dbb8434a43de669efcd4b86166f2e610a049ba","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"},"source_kind":"arxiv","source_id":"2009.02175","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:33:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VY7kLiPtlKvjTE7y6nCb7cQwtcZAIgFF0nMHB6diHSSYWtUR9VhZeQfF/I717RkJfWlh783inq20XWR/63FSAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T20:38:39.462495Z"},"content_sha256":"a87f3461a9254651f382a3a65902038d5329a7944bd05b55771f837a3e4092f8","schema_version":"1.0","event_id":"sha256:a87f3461a9254651f382a3a65902038d5329a7944bd05b55771f837a3e4092f8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:BZBSJO5AY6FZ3OY5AXP6LW5YIN","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T01:33:04Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SX9i9+W5iwfxhckk3br8dQ6b1kbYuyNT3/rLopWsZ7t9r3s7PC4q9xSYMUz1KoO4077lXZIs7RaRHKQ+pcNeCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-15T20:38:39.463501Z"},"content_sha256":"a0406e5fa7f2d1c97bc89ff8c36a7d039f98e448741b10fbddbcadae62c9d8c2","schema_version":"1.0","event_id":"sha256:a0406e5fa7f2d1c97bc89ff8c36a7d039f98e448741b10fbddbcadae62c9d8c2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/bundle.json","state_url":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-15T20:38:39Z","links":{"resolver":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN","bundle":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/bundle.json","state":"https://pith.science/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BZBSJO5AY6FZ3OY5AXP6LW5YIN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:BZBSJO5AY6FZ3OY5AXP6LW5YIN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"ee0b37a28647ea00810beae0aaf873a45a41a5529757c472aa0042b3b1699282","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-04T13:19:36Z","title_canon_sha256":"455c4de50f01636ca3f23093143e1210775f1e93fca5d6b6c716b89607e1abdc"},"schema_version":"1.0","source":{"id":"2009.02175","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2009.02175","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"arxiv_version","alias_value":"2009.02175v1","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2009.02175","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"pith_short_12","alias_value":"BZBSJO5AY6FZ","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"pith_short_16","alias_value":"BZBSJO5AY6FZ3OY5","created_at":"2026-07-05T01:33:04Z"},{"alias_kind":"pith_short_8","alias_value":"BZBSJO5A","created_at":"2026-07-05T01:33:04Z"}],"graph_snapshots":[{"event_id":"sha256:a0406e5fa7f2d1c97bc89ff8c36a7d039f98e448741b10fbddbcadae62c9d8c2","target":"graph","created_at":"2026-07-05T01:33:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2009.02175/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Chunlei Zu, Qinghou Zeng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-04T13:19:36Z","title":"A generalization of Stiebitz-type results on graph decomposition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2009.02175","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:a87f3461a9254651f382a3a65902038d5329a7944bd05b55771f837a3e4092f8","target":"record","created_at":"2026-07-05T01:33:04Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ee0b37a28647ea00810beae0aaf873a45a41a5529757c472aa0042b3b1699282","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2020-09-04T13:19:36Z","title_canon_sha256":"455c4de50f01636ca3f23093143e1210775f1e93fca5d6b6c716b89607e1abdc"},"schema_version":"1.0","source":{"id":"2009.02175","kind":"arxiv","version":1}},"canonical_sha256":"0e4324bba0c78b9dbb1d05dfe5dbb8434a43de669efcd4b86166f2e610a049ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e4324bba0c78b9dbb1d05dfe5dbb8434a43de669efcd4b86166f2e610a049ba","first_computed_at":"2026-07-05T01:33:04.131633Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:33:04.131633Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"HIqjbEjNLEco9KwF1Qk9NmLfwneG5gPlQfK9cl8u35D4I+cpL3z/meCXmrlfkeCUrxOyWeSJH0q5smMrzjgIDg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:33:04.131974Z","signed_message":"canonical_sha256_bytes"},"source_id":"2009.02175","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a87f3461a9254651f382a3a65902038d5329a7944bd05b55771f837a3e4092f8","sha256:a0406e5fa7f2d1c97bc89ff8c36a7d039f98e448741b10fbddbcadae62c9d8c2"],"state_sha256":"c90d8708c28e94c31465888f78776c8bd8618072dff800fe9cb050ca541b8801"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G7wMagBzFGa2gTEPPIa/YVAzEKSAfzedubSU44dN8XDjAtpM1A/VXKflVhOQzCMNevqAXPUf6NgzSir/oz/GBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-15T20:38:39.470566Z","bundle_sha256":"2d09fbc4cf80b2291938e437f6d29ec393b86c92058f1f3bb3a3dd962e1f89cf"}}