{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:3WUGWICVIFS3PJKZQKD5OPNVG4","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":"3dc1450ad910126d53a6e0fcb71d5029b5e3da3418d62593b50b12c603d6f729","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-11-15T05:34:46Z","title_canon_sha256":"f0d082e07986c610dfed8102a8e04c039da9214555ae1874624715f107b3bb0f"},"schema_version":"1.0","source":{"id":"2211.07913","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2211.07913","created_at":"2026-07-05T05:16:07Z"},{"alias_kind":"arxiv_version","alias_value":"2211.07913v1","created_at":"2026-07-05T05:16:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2211.07913","created_at":"2026-07-05T05:16:07Z"},{"alias_kind":"pith_short_12","alias_value":"3WUGWICVIFS3","created_at":"2026-07-05T05:16:07Z"},{"alias_kind":"pith_short_16","alias_value":"3WUGWICVIFS3PJKZ","created_at":"2026-07-05T05:16:07Z"},{"alias_kind":"pith_short_8","alias_value":"3WUGWICV","created_at":"2026-07-05T05:16:07Z"}],"graph_snapshots":[{"event_id":"sha256:a563b60983c151ccf344bae37bd2f5a8f528993bb2af86073f4ef3ad2c25a345","target":"graph","created_at":"2026-07-05T05:16:07Z","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/2211.07913/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Tur\\'{a}n number of a graph $H$, $\\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\\mathcal{F}$ of graphs, the suspension $\\mathcal{F}+v$ of $\\mathcal{F}$ is the graph obtained by connecting the vertex $v$ to all vertices of $F$ for each $F\\in \\mathcal{F}$. For two integers $k\\ge1$ and $r\\ge2$, let $H_i$ be a graph containing a critical edge with chromatic number $r$ for any $i\\in\\{1,\\ldots,k\\}$, and let $H=\\{H_1,\\ldots,H_k\\}+v$. In this paper, we determine $\\text{ex}(n, H)$ and characterize al","authors_text":"Heng Li, Jianfeng Hou, Qinghou Zeng","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-11-15T05:34:46Z","title":"Extremal graphs for the suspension of edge-critical graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.07913","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":"sha256:639628be186fdd2926a779ca580f74bc1b3c5b008eee29c02181c06ce9d4ef70","target":"record","created_at":"2026-07-05T05:16:07Z","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":"3dc1450ad910126d53a6e0fcb71d5029b5e3da3418d62593b50b12c603d6f729","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2022-11-15T05:34:46Z","title_canon_sha256":"f0d082e07986c610dfed8102a8e04c039da9214555ae1874624715f107b3bb0f"},"schema_version":"1.0","source":{"id":"2211.07913","kind":"arxiv","version":1}},"canonical_sha256":"dda86b20554165b7a5598287d73db5371310ad50d7a92f8b58df81c45800ad78","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dda86b20554165b7a5598287d73db5371310ad50d7a92f8b58df81c45800ad78","first_computed_at":"2026-07-05T05:16:07.796446Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T05:16:07.796446Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s9Pv8H9xFX5izUJempzBuRSYRRkBXmhEn/W9R2MTxXn0iXB32wimnw8YjLcDTtUpD0XE+hId3eZqPaRpLdiSAA==","signature_status":"signed_v1","signed_at":"2026-07-05T05:16:07.796897Z","signed_message":"canonical_sha256_bytes"},"source_id":"2211.07913","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:639628be186fdd2926a779ca580f74bc1b3c5b008eee29c02181c06ce9d4ef70","sha256:a563b60983c151ccf344bae37bd2f5a8f528993bb2af86073f4ef3ad2c25a345"],"state_sha256":"86ac70715876eff872c0373fcc2a64027be368250102fdf60feb5d20dc0e17bb"}