{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:3CMEGZVXLL2TIMIEG75L5NZGJE","short_pith_number":"pith:3CMEGZVX","canonical_record":{"source":{"id":"1706.06945","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-21T15:00:30Z","cross_cats_sorted":[],"title_canon_sha256":"3a078f06ef679eb02b429f0c3b218678200f68beafbf7720baa485e292487d2f","abstract_canon_sha256":"41e145b24c9465ac9ef70074cbe2ceb7d452f80539c8cd6972ecf9d37efc25a0"},"schema_version":"1.0"},"canonical_sha256":"d8984366b75af534310437fabeb7264932badefe135997eb2889fc752b0dc9ae","source":{"kind":"arxiv","id":"1706.06945","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.06945","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"arxiv_version","alias_value":"1706.06945v1","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.06945","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"pith_short_12","alias_value":"3CMEGZVXLL2T","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3CMEGZVXLL2TIMIE","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3CMEGZVX","created_at":"2026-05-18T12:30:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:3CMEGZVXLL2TIMIEG75L5NZGJE","target":"record","payload":{"canonical_record":{"source":{"id":"1706.06945","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-21T15:00:30Z","cross_cats_sorted":[],"title_canon_sha256":"3a078f06ef679eb02b429f0c3b218678200f68beafbf7720baa485e292487d2f","abstract_canon_sha256":"41e145b24c9465ac9ef70074cbe2ceb7d452f80539c8cd6972ecf9d37efc25a0"},"schema_version":"1.0"},"canonical_sha256":"d8984366b75af534310437fabeb7264932badefe135997eb2889fc752b0dc9ae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:56.425827Z","signature_b64":"xsakwEfo2UmbdQVJrLlm56LDYd0sW5qzxEcCsd+aLYitctS6tqEQhCpgUVPKFT5ICJLDLx6TRCLXLNMgiPH2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d8984366b75af534310437fabeb7264932badefe135997eb2889fc752b0dc9ae","last_reissued_at":"2026-05-18T00:41:56.425265Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:56.425265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1706.06945","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":"2026-05-18T00:41:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tllqqSmsR/UeODvoO1rP4UdxsJ7Ozg0wzXFxZDroZ3SYPVYHc8/EG8JopFaMabLtuqkKtmfDElwtC6ta+4OtDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T15:58:08.533228Z"},"content_sha256":"c608a3efd2690abff241b114fae202341355cf8ad034658ef3f51fb81e61a28f","schema_version":"1.0","event_id":"sha256:c608a3efd2690abff241b114fae202341355cf8ad034658ef3f51fb81e61a28f"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:3CMEGZVXLL2TIMIEG75L5NZGJE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On vertex-disjoint paths in regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jie Han","submitted_at":"2017-06-21T15:00:30Z","abstract_excerpt":"Let $c\\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\\lfloor 1/c \\rfloor$ paths whose union covers all but at most $o(n)$ vertices of $G$. The constant $\\lfloor 1/c \\rfloor$ is best possible when $1/c\\notin \\mathbb{N}$ and off by $1$ otherwise. Moreover, if in addition $G$ is bipartite, then the number of paths can be reduced to $\\lfloor 1/(2c) \\rfloor$, which is best possible."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.06945","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":"2026-05-18T00:41:56Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/0cXsio5QDrqATq5v7EJZ4SVyRfB9OEeYuXNqeeBMOPaan6dlt7T2nqItauDmydcrdel2qZG8eT0qcv0u4+NDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T15:58:08.533589Z"},"content_sha256":"3856daf9c2dc875c715f2ad930d2a6daa1b9a2e54df159555f3ed25ce259b25b","schema_version":"1.0","event_id":"sha256:3856daf9c2dc875c715f2ad930d2a6daa1b9a2e54df159555f3ed25ce259b25b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3CMEGZVXLL2TIMIEG75L5NZGJE/bundle.json","state_url":"https://pith.science/pith/3CMEGZVXLL2TIMIEG75L5NZGJE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3CMEGZVXLL2TIMIEG75L5NZGJE/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":"2026-06-04T15:58:08Z","links":{"resolver":"https://pith.science/pith/3CMEGZVXLL2TIMIEG75L5NZGJE","bundle":"https://pith.science/pith/3CMEGZVXLL2TIMIEG75L5NZGJE/bundle.json","state":"https://pith.science/pith/3CMEGZVXLL2TIMIEG75L5NZGJE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3CMEGZVXLL2TIMIEG75L5NZGJE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:3CMEGZVXLL2TIMIEG75L5NZGJE","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":"41e145b24c9465ac9ef70074cbe2ceb7d452f80539c8cd6972ecf9d37efc25a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-21T15:00:30Z","title_canon_sha256":"3a078f06ef679eb02b429f0c3b218678200f68beafbf7720baa485e292487d2f"},"schema_version":"1.0","source":{"id":"1706.06945","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1706.06945","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"arxiv_version","alias_value":"1706.06945v1","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1706.06945","created_at":"2026-05-18T00:41:56Z"},{"alias_kind":"pith_short_12","alias_value":"3CMEGZVXLL2T","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_16","alias_value":"3CMEGZVXLL2TIMIE","created_at":"2026-05-18T12:30:58Z"},{"alias_kind":"pith_short_8","alias_value":"3CMEGZVX","created_at":"2026-05-18T12:30:58Z"}],"graph_snapshots":[{"event_id":"sha256:3856daf9c2dc875c715f2ad930d2a6daa1b9a2e54df159555f3ed25ce259b25b","target":"graph","created_at":"2026-05-18T00:41:56Z","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":"Let $c\\in (0, 1]$ be a real number and let $n$ be a sufficiently large integer. We prove that every $n$-vertex $c n$-regular graph $G$ contains a collection of $\\lfloor 1/c \\rfloor$ paths whose union covers all but at most $o(n)$ vertices of $G$. The constant $\\lfloor 1/c \\rfloor$ is best possible when $1/c\\notin \\mathbb{N}$ and off by $1$ otherwise. Moreover, if in addition $G$ is bipartite, then the number of paths can be reduced to $\\lfloor 1/(2c) \\rfloor$, which is best possible.","authors_text":"Jie Han","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-21T15:00:30Z","title":"On vertex-disjoint paths in regular graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.06945","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:c608a3efd2690abff241b114fae202341355cf8ad034658ef3f51fb81e61a28f","target":"record","created_at":"2026-05-18T00:41:56Z","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":"41e145b24c9465ac9ef70074cbe2ceb7d452f80539c8cd6972ecf9d37efc25a0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-06-21T15:00:30Z","title_canon_sha256":"3a078f06ef679eb02b429f0c3b218678200f68beafbf7720baa485e292487d2f"},"schema_version":"1.0","source":{"id":"1706.06945","kind":"arxiv","version":1}},"canonical_sha256":"d8984366b75af534310437fabeb7264932badefe135997eb2889fc752b0dc9ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d8984366b75af534310437fabeb7264932badefe135997eb2889fc752b0dc9ae","first_computed_at":"2026-05-18T00:41:56.425265Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:56.425265Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xsakwEfo2UmbdQVJrLlm56LDYd0sW5qzxEcCsd+aLYitctS6tqEQhCpgUVPKFT5ICJLDLx6TRCLXLNMgiPH2AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:56.425827Z","signed_message":"canonical_sha256_bytes"},"source_id":"1706.06945","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c608a3efd2690abff241b114fae202341355cf8ad034658ef3f51fb81e61a28f","sha256:3856daf9c2dc875c715f2ad930d2a6daa1b9a2e54df159555f3ed25ce259b25b"],"state_sha256":"7370d8e1b3ccbc6b2baf185a2ba5ea92f0ffe009da66da858c7b830efeffbfc8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LA6Va+pk3oxTLsvNghbuFrlH1Uhnf71j7bS9vEqqDpk+wi2+0KSlxuR2zZwfdtcOQmMq6G59ivxLJp73iogkCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T15:58:08.535520Z","bundle_sha256":"f1dee78e924577c81faec49dd7f759a40be2cda04effcc654b2eca96543af050"}}