{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:GMFLHI67FZ5O3KVZMNNFGWHDAD","short_pith_number":"pith:GMFLHI67","canonical_record":{"source":{"id":"1904.02482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-04T11:09:32Z","cross_cats_sorted":[],"title_canon_sha256":"5d36b0885fd63b3626f4a4e67502b1fe10fe04b9f284ce84f599ef95332b3999","abstract_canon_sha256":"7f1b9bb0fca58bdd83f238b91922cf49c1b5db81442b384355db2db4d5c8db3e"},"schema_version":"1.0"},"canonical_sha256":"330ab3a3df2e7aedaab9635a5358e300e08f4728ced7c54de15c45cafb4ddd50","source":{"kind":"arxiv","id":"1904.02482","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.02482","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"arxiv_version","alias_value":"1904.02482v1","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.02482","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"pith_short_12","alias_value":"GMFLHI67FZ5O","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"GMFLHI67FZ5O3KVZ","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"GMFLHI67","created_at":"2026-05-18T12:33:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:GMFLHI67FZ5O3KVZMNNFGWHDAD","target":"record","payload":{"canonical_record":{"source":{"id":"1904.02482","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-04T11:09:32Z","cross_cats_sorted":[],"title_canon_sha256":"5d36b0885fd63b3626f4a4e67502b1fe10fe04b9f284ce84f599ef95332b3999","abstract_canon_sha256":"7f1b9bb0fca58bdd83f238b91922cf49c1b5db81442b384355db2db4d5c8db3e"},"schema_version":"1.0"},"canonical_sha256":"330ab3a3df2e7aedaab9635a5358e300e08f4728ced7c54de15c45cafb4ddd50","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:23.504697Z","signature_b64":"X28J8mOsPEil395u8pp7PQWD6zKhC3JaWEP3mmsxRJCTjeHUre9QhI4AD3JCUrUSzV1tVaG0mLCv8PSVQu58Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"330ab3a3df2e7aedaab9635a5358e300e08f4728ced7c54de15c45cafb4ddd50","last_reissued_at":"2026-05-17T23:49:23.504246Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:23.504246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1904.02482","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":"2026-05-17T23:49:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TdGTNckz/d4M8u+FG8o6sa+7HbbnfQPVgiWlxW7QaqEHphBq6K9elKm4OeP5g1+sFd7hK91r0bSfTUoil2C1Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:36:22.880849Z"},"content_sha256":"26e880bf520154a62a8e71b3a1cca8b19cdc51012f2ac15b4511233236d0dbbd","schema_version":"1.0","event_id":"sha256:26e880bf520154a62a8e71b3a1cca8b19cdc51012f2ac15b4511233236d0dbbd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:GMFLHI67FZ5O3KVZMNNFGWHDAD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Extension Degree Conditions for Fractional Factor","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Juan L.G. Guirao, Weifan Wang, Wei Gao","submitted_at":"2019-04-04T11:09:32Z","abstract_excerpt":"In Gao's previous work, the authors determined several graph degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if $b=f(x)=g(x)=a$ for all vertices $x$ in $G$. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference $\\Delta$ between $g(x)$ and $f(x)$ for every vertex $x$ in $G$. These obtained new degree conditions reformulate Gao's previous conclusions, and show how $\\Delta$ acts in th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.02482","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:49:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pmo/V4JgsYZEcrrOHeO5wTUSqbgRxoRL6KF+PeTNDqcQhnB5Tvm8Rxun+1OSyD1XplLkcNnY3LVPK7kRlVYpBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T03:36:22.881197Z"},"content_sha256":"1db03835c9eb80b2013d85b0bd87fd6885e2bc4d7f1645c2cbe632e59fad0eef","schema_version":"1.0","event_id":"sha256:1db03835c9eb80b2013d85b0bd87fd6885e2bc4d7f1645c2cbe632e59fad0eef"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/GMFLHI67FZ5O3KVZMNNFGWHDAD/bundle.json","state_url":"https://pith.science/pith/GMFLHI67FZ5O3KVZMNNFGWHDAD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/GMFLHI67FZ5O3KVZMNNFGWHDAD/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":"2026-06-01T03:36:22Z","links":{"resolver":"https://pith.science/pith/GMFLHI67FZ5O3KVZMNNFGWHDAD","bundle":"https://pith.science/pith/GMFLHI67FZ5O3KVZMNNFGWHDAD/bundle.json","state":"https://pith.science/pith/GMFLHI67FZ5O3KVZMNNFGWHDAD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/GMFLHI67FZ5O3KVZMNNFGWHDAD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:GMFLHI67FZ5O3KVZMNNFGWHDAD","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":"7f1b9bb0fca58bdd83f238b91922cf49c1b5db81442b384355db2db4d5c8db3e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-04T11:09:32Z","title_canon_sha256":"5d36b0885fd63b3626f4a4e67502b1fe10fe04b9f284ce84f599ef95332b3999"},"schema_version":"1.0","source":{"id":"1904.02482","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1904.02482","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"arxiv_version","alias_value":"1904.02482v1","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1904.02482","created_at":"2026-05-17T23:49:23Z"},{"alias_kind":"pith_short_12","alias_value":"GMFLHI67FZ5O","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_16","alias_value":"GMFLHI67FZ5O3KVZ","created_at":"2026-05-18T12:33:18Z"},{"alias_kind":"pith_short_8","alias_value":"GMFLHI67","created_at":"2026-05-18T12:33:18Z"}],"graph_snapshots":[{"event_id":"sha256:1db03835c9eb80b2013d85b0bd87fd6885e2bc4d7f1645c2cbe632e59fad0eef","target":"graph","created_at":"2026-05-17T23:49:23Z","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"},"paper":{"abstract_excerpt":"In Gao's previous work, the authors determined several graph degree conditions of a graph which admits fractional factor in particular settings. It was revealed that these degree conditions are tight if $b=f(x)=g(x)=a$ for all vertices $x$ in $G$. In this paper, we continue to discuss these degree conditions for admitting fractional factor in the setting that several vertices and edges are removed and there is a difference $\\Delta$ between $g(x)$ and $f(x)$ for every vertex $x$ in $G$. These obtained new degree conditions reformulate Gao's previous conclusions, and show how $\\Delta$ acts in th","authors_text":"Juan L.G. Guirao, Weifan Wang, Wei Gao","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-04T11:09:32Z","title":"The Extension Degree Conditions for Fractional Factor"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1904.02482","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":"sha256:26e880bf520154a62a8e71b3a1cca8b19cdc51012f2ac15b4511233236d0dbbd","target":"record","created_at":"2026-05-17T23:49:23Z","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":"7f1b9bb0fca58bdd83f238b91922cf49c1b5db81442b384355db2db4d5c8db3e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-04-04T11:09:32Z","title_canon_sha256":"5d36b0885fd63b3626f4a4e67502b1fe10fe04b9f284ce84f599ef95332b3999"},"schema_version":"1.0","source":{"id":"1904.02482","kind":"arxiv","version":1}},"canonical_sha256":"330ab3a3df2e7aedaab9635a5358e300e08f4728ced7c54de15c45cafb4ddd50","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"330ab3a3df2e7aedaab9635a5358e300e08f4728ced7c54de15c45cafb4ddd50","first_computed_at":"2026-05-17T23:49:23.504246Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:23.504246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"X28J8mOsPEil395u8pp7PQWD6zKhC3JaWEP3mmsxRJCTjeHUre9QhI4AD3JCUrUSzV1tVaG0mLCv8PSVQu58Dg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:23.504697Z","signed_message":"canonical_sha256_bytes"},"source_id":"1904.02482","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:26e880bf520154a62a8e71b3a1cca8b19cdc51012f2ac15b4511233236d0dbbd","sha256:1db03835c9eb80b2013d85b0bd87fd6885e2bc4d7f1645c2cbe632e59fad0eef"],"state_sha256":"9f9ffcf194e2556c3646cb6d922043fc78adc9dea3d6d84e688084c1fb018afd"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"TdW9fZvHEKTictnNXW05mIM+1y7qX00Afq+OCfTQWC88CWQlOrYQ+6TTeWK1sj83N5exLNxptqgsKonRCVIIBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T03:36:22.883306Z","bundle_sha256":"f9685d54699008fe4b14ac66abc4f12a308ff392cdb33d84fd3ef13a5c559aff"}}