{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:IW2UB2WLLIPHZPQSPCQGBHKQWY","short_pith_number":"pith:IW2UB2WL","canonical_record":{"source":{"id":"1507.07066","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-25T04:43:56Z","cross_cats_sorted":[],"title_canon_sha256":"8503f1e62c94c967067b1affd01195eb6cf4b07ed251f2502be81d60fa5c8074","abstract_canon_sha256":"eda4d21ff33132add0e216f0da2d7f9507540d2fa3e1ea5695f2c3ddcb12be47"},"schema_version":"1.0"},"canonical_sha256":"45b540eacb5a1e7cbe1278a0609d50b611253c8453a358f64406d30f59fa2148","source":{"kind":"arxiv","id":"1507.07066","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07066","created_at":"2026-05-18T01:36:17Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07066v1","created_at":"2026-05-18T01:36:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07066","created_at":"2026-05-18T01:36:17Z"},{"alias_kind":"pith_short_12","alias_value":"IW2UB2WLLIPH","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IW2UB2WLLIPHZPQS","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IW2UB2WL","created_at":"2026-05-18T12:29:25Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:IW2UB2WLLIPHZPQSPCQGBHKQWY","target":"record","payload":{"canonical_record":{"source":{"id":"1507.07066","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-25T04:43:56Z","cross_cats_sorted":[],"title_canon_sha256":"8503f1e62c94c967067b1affd01195eb6cf4b07ed251f2502be81d60fa5c8074","abstract_canon_sha256":"eda4d21ff33132add0e216f0da2d7f9507540d2fa3e1ea5695f2c3ddcb12be47"},"schema_version":"1.0"},"canonical_sha256":"45b540eacb5a1e7cbe1278a0609d50b611253c8453a358f64406d30f59fa2148","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:17.344564Z","signature_b64":"rFrwNWSG3bNt2AXVG9RCeByWbSk2KZWH8hWFWPwWLdrpICA0aJwBebKZncU2MBElRU4dsz62DrzmplEyMAYhDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45b540eacb5a1e7cbe1278a0609d50b611253c8453a358f64406d30f59fa2148","last_reissued_at":"2026-05-18T01:36:17.343951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:17.343951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.07066","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":"2026-05-18T01:36:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gFb3G6ar07xkuQB7NemNlZCwXfwC6DZyB041GoDWPEC7xsaPMm9dTAMHqFogLXVMN5DuHHC9j4FxGFbmphgFCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T19:39:44.215222Z"},"content_sha256":"a4b00448c1747fae36671e94ec55091b971a0f740970a49e65203b663510dbce","schema_version":"1.0","event_id":"sha256:a4b00448c1747fae36671e94ec55091b971a0f740970a49e65203b663510dbce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:IW2UB2WLLIPHZPQSPCQGBHKQWY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Path-factors involving paths of order seven and nine","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michitaka Furuya, Yoshimi Egawa","submitted_at":"2015-07-25T04:43:56Z","abstract_excerpt":"In this paper, we show the following two theorems (here $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$): (i)~If a graph $G$ satisfies $c_{1}(G-X)+\\frac{1}{3}c_{3}(G-X)+\\frac{1}{3}c_{5}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{7}\\}$-factor. (ii)~If a graph $G$ satisfies $c_{1}(G-X)+c_{3}(G-X)+\\frac{2}{3}c_{5}(G-X)+\\frac{1}{3}c_{7}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{9}\\}$-factor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07066","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":"2026-05-18T01:36:17Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xYftIRcFzmJd1Aak4ayGXovJPNk9JpBM5mggVQm3Vnft2BlF+3saSxvPu6osv087wrYHlzjdJ85etkMkGLXHCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T19:39:44.215587Z"},"content_sha256":"b201c5d5787f9911e3cbaa0dcf15b890f55982defc87d81094b6b378f81f9700","schema_version":"1.0","event_id":"sha256:b201c5d5787f9911e3cbaa0dcf15b890f55982defc87d81094b6b378f81f9700"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IW2UB2WLLIPHZPQSPCQGBHKQWY/bundle.json","state_url":"https://pith.science/pith/IW2UB2WLLIPHZPQSPCQGBHKQWY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IW2UB2WLLIPHZPQSPCQGBHKQWY/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":"2026-06-30T19:39:44Z","links":{"resolver":"https://pith.science/pith/IW2UB2WLLIPHZPQSPCQGBHKQWY","bundle":"https://pith.science/pith/IW2UB2WLLIPHZPQSPCQGBHKQWY/bundle.json","state":"https://pith.science/pith/IW2UB2WLLIPHZPQSPCQGBHKQWY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IW2UB2WLLIPHZPQSPCQGBHKQWY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IW2UB2WLLIPHZPQSPCQGBHKQWY","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":"eda4d21ff33132add0e216f0da2d7f9507540d2fa3e1ea5695f2c3ddcb12be47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-25T04:43:56Z","title_canon_sha256":"8503f1e62c94c967067b1affd01195eb6cf4b07ed251f2502be81d60fa5c8074"},"schema_version":"1.0","source":{"id":"1507.07066","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.07066","created_at":"2026-05-18T01:36:17Z"},{"alias_kind":"arxiv_version","alias_value":"1507.07066v1","created_at":"2026-05-18T01:36:17Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.07066","created_at":"2026-05-18T01:36:17Z"},{"alias_kind":"pith_short_12","alias_value":"IW2UB2WLLIPH","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IW2UB2WLLIPHZPQS","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IW2UB2WL","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:b201c5d5787f9911e3cbaa0dcf15b890f55982defc87d81094b6b378f81f9700","target":"graph","created_at":"2026-05-18T01:36:17Z","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 this paper, we show the following two theorems (here $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$): (i)~If a graph $G$ satisfies $c_{1}(G-X)+\\frac{1}{3}c_{3}(G-X)+\\frac{1}{3}c_{5}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{7}\\}$-factor. (ii)~If a graph $G$ satisfies $c_{1}(G-X)+c_{3}(G-X)+\\frac{2}{3}c_{5}(G-X)+\\frac{1}{3}c_{7}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{9}\\}$-factor.","authors_text":"Michitaka Furuya, Yoshimi Egawa","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-25T04:43:56Z","title":"Path-factors involving paths of order seven and nine"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07066","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:a4b00448c1747fae36671e94ec55091b971a0f740970a49e65203b663510dbce","target":"record","created_at":"2026-05-18T01:36:17Z","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":"eda4d21ff33132add0e216f0da2d7f9507540d2fa3e1ea5695f2c3ddcb12be47","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-25T04:43:56Z","title_canon_sha256":"8503f1e62c94c967067b1affd01195eb6cf4b07ed251f2502be81d60fa5c8074"},"schema_version":"1.0","source":{"id":"1507.07066","kind":"arxiv","version":1}},"canonical_sha256":"45b540eacb5a1e7cbe1278a0609d50b611253c8453a358f64406d30f59fa2148","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45b540eacb5a1e7cbe1278a0609d50b611253c8453a358f64406d30f59fa2148","first_computed_at":"2026-05-18T01:36:17.343951Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:17.343951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rFrwNWSG3bNt2AXVG9RCeByWbSk2KZWH8hWFWPwWLdrpICA0aJwBebKZncU2MBElRU4dsz62DrzmplEyMAYhDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:17.344564Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.07066","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a4b00448c1747fae36671e94ec55091b971a0f740970a49e65203b663510dbce","sha256:b201c5d5787f9911e3cbaa0dcf15b890f55982defc87d81094b6b378f81f9700"],"state_sha256":"3115ff789ccb9859aa7e89d629eb532898ef35101c7e9564783dfd86d31064bf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IHpHszbB6VM0jlWlw5O1kEM4G6fy88R7Q2k/kQOJunmZjkIn+uej+onxOoxiw4ARaO0aZKBevh6iqTXovgZUBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T19:39:44.217596Z","bundle_sha256":"51e0544ad7c2b199b91ad1b0bf13908c64ac61b52153c3a56a30500d328abf81"}}