{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:OXZVLH7BXN43BAZDHUNVO5VZX5","short_pith_number":"pith:OXZVLH7B","canonical_record":{"source":{"id":"1109.4344","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-20T16:45:58Z","cross_cats_sorted":[],"title_canon_sha256":"8e9c841e27bae354644b3358642ae0b2a9d179784eee7943d088ae13334d9c67","abstract_canon_sha256":"3a2a9e541ac4dd5ea59bf952a948796220798c087c1e5302f3d82a9c8d7fc213"},"schema_version":"1.0"},"canonical_sha256":"75f3559fe1bb79b083233d1b5776b9bf498650e7f4eaa0f2d39861b81f63a602","source":{"kind":"arxiv","id":"1109.4344","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4344","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4344v2","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4344","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"pith_short_12","alias_value":"OXZVLH7BXN43","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"OXZVLH7BXN43BAZD","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"OXZVLH7B","created_at":"2026-05-18T12:26:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:OXZVLH7BXN43BAZDHUNVO5VZX5","target":"record","payload":{"canonical_record":{"source":{"id":"1109.4344","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-20T16:45:58Z","cross_cats_sorted":[],"title_canon_sha256":"8e9c841e27bae354644b3358642ae0b2a9d179784eee7943d088ae13334d9c67","abstract_canon_sha256":"3a2a9e541ac4dd5ea59bf952a948796220798c087c1e5302f3d82a9c8d7fc213"},"schema_version":"1.0"},"canonical_sha256":"75f3559fe1bb79b083233d1b5776b9bf498650e7f4eaa0f2d39861b81f63a602","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:13.860979Z","signature_b64":"gBaRYCSof5ZbPs9mbMhAb3LLjQs+XN26zAAN6LkPcfFo8QGQ+NvRywt4r79IgCk7U70LvZQpKPl0vcDSC2aLDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"75f3559fe1bb79b083233d1b5776b9bf498650e7f4eaa0f2d39861b81f63a602","last_reissued_at":"2026-05-18T00:17:13.860372Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:13.860372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.4344","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:17:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l+B9E584uHGpoTnCemKpWSsRvgHAxIilqMBA5mpSDeRZ9sFCGsKpDy9f7U3xrQByvXnCKL1M0JYxOyjhpAxUAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T18:29:00.217222Z"},"content_sha256":"ec5ebd12c316d333c210dd852b24b90381342cd709e811751613b93c82adfd6a","schema_version":"1.0","event_id":"sha256:ec5ebd12c316d333c210dd852b24b90381342cd709e811751613b93c82adfd6a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:OXZVLH7BXN43BAZDHUNVO5VZX5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Long paths and cycles passing through specified vertices under the average degree condition","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Binlong Li, Bo Ning, Shenggui Zhang","submitted_at":"2011-09-20T16:45:58Z","abstract_excerpt":"Let $G$ be a $k$-connected graph with $k\\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erd\\\"os and Gallai on the existence "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4344","kind":"arxiv","version":2},"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":"2026-05-18T00:17:13Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qKXVkK4kqT9Dz/DGhulkfedS1LopuIuM+UbCOaf8Z3UJF83K7U8SQADlBhVffk80Lp7eNw7miLQR2dCstDluAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T18:29:00.217582Z"},"content_sha256":"a0c5107de57294fb07118a739604533579119cd24731a5e68c9a3790e4bd9550","schema_version":"1.0","event_id":"sha256:a0c5107de57294fb07118a739604533579119cd24731a5e68c9a3790e4bd9550"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OXZVLH7BXN43BAZDHUNVO5VZX5/bundle.json","state_url":"https://pith.science/pith/OXZVLH7BXN43BAZDHUNVO5VZX5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OXZVLH7BXN43BAZDHUNVO5VZX5/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":"2026-06-04T18:29:00Z","links":{"resolver":"https://pith.science/pith/OXZVLH7BXN43BAZDHUNVO5VZX5","bundle":"https://pith.science/pith/OXZVLH7BXN43BAZDHUNVO5VZX5/bundle.json","state":"https://pith.science/pith/OXZVLH7BXN43BAZDHUNVO5VZX5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OXZVLH7BXN43BAZDHUNVO5VZX5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:OXZVLH7BXN43BAZDHUNVO5VZX5","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":"3a2a9e541ac4dd5ea59bf952a948796220798c087c1e5302f3d82a9c8d7fc213","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-20T16:45:58Z","title_canon_sha256":"8e9c841e27bae354644b3358642ae0b2a9d179784eee7943d088ae13334d9c67"},"schema_version":"1.0","source":{"id":"1109.4344","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.4344","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"arxiv_version","alias_value":"1109.4344v2","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.4344","created_at":"2026-05-18T00:17:13Z"},{"alias_kind":"pith_short_12","alias_value":"OXZVLH7BXN43","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_16","alias_value":"OXZVLH7BXN43BAZD","created_at":"2026-05-18T12:26:37Z"},{"alias_kind":"pith_short_8","alias_value":"OXZVLH7B","created_at":"2026-05-18T12:26:37Z"}],"graph_snapshots":[{"event_id":"sha256:a0c5107de57294fb07118a739604533579119cd24731a5e68c9a3790e4bd9550","target":"graph","created_at":"2026-05-18T00:17:13Z","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 $G$ be a $k$-connected graph with $k\\geq 2$. In this paper we first prove that: For two distinct vertices $x$ and $z$ in $G$, it contains a path passing through its any $k-2$ {specified} vertices with length at least the average degree of the vertices other than $x$ and $z$. Further, with this result, we prove that: If $G$ has $n$ vertices and $m$ edges, then it contains a cycle of length at least $2m/(n-1)$ passing through its any $k-1$ specified vertices. Our results generalize a theorem of Fan on the existence of long paths and a classical theorem of Erd\\\"os and Gallai on the existence ","authors_text":"Binlong Li, Bo Ning, Shenggui Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-20T16:45:58Z","title":"Long paths and cycles passing through specified vertices under the average degree condition"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.4344","kind":"arxiv","version":2},"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:ec5ebd12c316d333c210dd852b24b90381342cd709e811751613b93c82adfd6a","target":"record","created_at":"2026-05-18T00:17:13Z","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":"3a2a9e541ac4dd5ea59bf952a948796220798c087c1e5302f3d82a9c8d7fc213","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2011-09-20T16:45:58Z","title_canon_sha256":"8e9c841e27bae354644b3358642ae0b2a9d179784eee7943d088ae13334d9c67"},"schema_version":"1.0","source":{"id":"1109.4344","kind":"arxiv","version":2}},"canonical_sha256":"75f3559fe1bb79b083233d1b5776b9bf498650e7f4eaa0f2d39861b81f63a602","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"75f3559fe1bb79b083233d1b5776b9bf498650e7f4eaa0f2d39861b81f63a602","first_computed_at":"2026-05-18T00:17:13.860372Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:13.860372Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gBaRYCSof5ZbPs9mbMhAb3LLjQs+XN26zAAN6LkPcfFo8QGQ+NvRywt4r79IgCk7U70LvZQpKPl0vcDSC2aLDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:13.860979Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.4344","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ec5ebd12c316d333c210dd852b24b90381342cd709e811751613b93c82adfd6a","sha256:a0c5107de57294fb07118a739604533579119cd24731a5e68c9a3790e4bd9550"],"state_sha256":"0dc7a523a012d871e9c4313aee744d70873384bab25d8575396accfbce836826"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eTtkX01XlxiybMbvFetRirbA8jtOIcv4boYkMwBn4Nv9WtXc2Cuhn63xIrF3pgVN7L6ZrPcMVb+34yE5CETFAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T18:29:00.219635Z","bundle_sha256":"2ec8890e2995ca8435cb306a0e1c06c112ff5871f5d0717858fdc84e7da68314"}}