{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:F55J2GEAB2CDN34IQZE2OQD4MQ","short_pith_number":"pith:F55J2GEA","canonical_record":{"source":{"id":"1804.10359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-27T06:49:23Z","cross_cats_sorted":[],"title_canon_sha256":"ff1c7fedd7e23caa7ac64a660e8807c3a303b92bd09dcb21f8f8036412ad0108","abstract_canon_sha256":"30ae186ac55a701156c397d01d7529b1a0a7e3b38869671d488317054fc570a8"},"schema_version":"1.0"},"canonical_sha256":"2f7a9d18800e8436ef888649a7407c6408ca97c806a9e4254916bee0c7b06950","source":{"kind":"arxiv","id":"1804.10359","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10359","created_at":"2026-05-18T00:17:20Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10359v1","created_at":"2026-05-18T00:17:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10359","created_at":"2026-05-18T00:17:20Z"},{"alias_kind":"pith_short_12","alias_value":"F55J2GEAB2CD","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"F55J2GEAB2CDN34I","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"F55J2GEA","created_at":"2026-05-18T12:32:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:F55J2GEAB2CDN34IQZE2OQD4MQ","target":"record","payload":{"canonical_record":{"source":{"id":"1804.10359","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-27T06:49:23Z","cross_cats_sorted":[],"title_canon_sha256":"ff1c7fedd7e23caa7ac64a660e8807c3a303b92bd09dcb21f8f8036412ad0108","abstract_canon_sha256":"30ae186ac55a701156c397d01d7529b1a0a7e3b38869671d488317054fc570a8"},"schema_version":"1.0"},"canonical_sha256":"2f7a9d18800e8436ef888649a7407c6408ca97c806a9e4254916bee0c7b06950","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:20.983157Z","signature_b64":"RlKGjEXWw7japcW1FcENYTk9R/F4G5wDG7ED26SMLlGE3N4hXeq7eD3afpCJgega2vh2EgMraAlfdxGe+eTJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f7a9d18800e8436ef888649a7407c6408ca97c806a9e4254916bee0c7b06950","last_reissued_at":"2026-05-18T00:17:20.982450Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:20.982450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.10359","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":"2026-05-18T00:17:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qau4FUz69lZLH73ZPGJzYyEU2RIQkJ99x1nQAhfP9lzu9ZqLL9N4cfAzFtLh7SD2jZ5Lr+wvKLH/zkKS3PsgDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:57:41.375551Z"},"content_sha256":"71487093ec58a06910699bf9728d4ac73aa9d1ee52d7dd32da1a3e85851d8225","schema_version":"1.0","event_id":"sha256:71487093ec58a06910699bf9728d4ac73aa9d1ee52d7dd32da1a3e85851d8225"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:F55J2GEAB2CDN34IQZE2OQD4MQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The edge spectrum of $K_4^-$-saturated graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun Gao, Xinmin Hou, Yue Ma","submitted_at":"2018-04-27T06:49:23Z","abstract_excerpt":"Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on $n$ vertices. Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\\lfloor \\frac{n-1}{2} \\rfloor \\lceil \\frac{n-1}{2} \\rceil +2$, then $G$ must be a bipartite graph; (b) there exists a $K_4^-$-saturated non-bipartite graph on $n\\ge 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10359","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":"2026-05-18T00:17:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"it7e68nbG/YXLIPW2p9L+xVd2+zgcJzwfKa/dRsegZfli9I5+ulua5grXAyrO+/bk6s0Dd0iMuyYx9xL5++xAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T21:57:41.375905Z"},"content_sha256":"e40afd36cf037209a499cb83be672bad9f8cc6711d3dc277f5aa9bda6a8b78ac","schema_version":"1.0","event_id":"sha256:e40afd36cf037209a499cb83be672bad9f8cc6711d3dc277f5aa9bda6a8b78ac"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/bundle.json","state_url":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/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":"2026-06-02T21:57:41Z","links":{"resolver":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ","bundle":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/bundle.json","state":"https://pith.science/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/F55J2GEAB2CDN34IQZE2OQD4MQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:F55J2GEAB2CDN34IQZE2OQD4MQ","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":"30ae186ac55a701156c397d01d7529b1a0a7e3b38869671d488317054fc570a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-27T06:49:23Z","title_canon_sha256":"ff1c7fedd7e23caa7ac64a660e8807c3a303b92bd09dcb21f8f8036412ad0108"},"schema_version":"1.0","source":{"id":"1804.10359","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.10359","created_at":"2026-05-18T00:17:20Z"},{"alias_kind":"arxiv_version","alias_value":"1804.10359v1","created_at":"2026-05-18T00:17:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.10359","created_at":"2026-05-18T00:17:20Z"},{"alias_kind":"pith_short_12","alias_value":"F55J2GEAB2CD","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_16","alias_value":"F55J2GEAB2CDN34I","created_at":"2026-05-18T12:32:22Z"},{"alias_kind":"pith_short_8","alias_value":"F55J2GEA","created_at":"2026-05-18T12:32:22Z"}],"graph_snapshots":[{"event_id":"sha256:e40afd36cf037209a499cb83be672bad9f8cc6711d3dc277f5aa9bda6a8b78ac","target":"graph","created_at":"2026-05-18T00:17:20Z","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":"Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on $n$ vertices. Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\\lfloor \\frac{n-1}{2} \\rfloor \\lceil \\frac{n-1}{2} \\rceil +2$, then $G$ must be a bipartite graph; (b) there exists a $K_4^-$-saturated non-bipartite graph on $n\\ge 1","authors_text":"Jun Gao, Xinmin Hou, Yue Ma","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-27T06:49:23Z","title":"The edge spectrum of $K_4^-$-saturated graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10359","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":"sha256:71487093ec58a06910699bf9728d4ac73aa9d1ee52d7dd32da1a3e85851d8225","target":"record","created_at":"2026-05-18T00:17:20Z","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":"30ae186ac55a701156c397d01d7529b1a0a7e3b38869671d488317054fc570a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-04-27T06:49:23Z","title_canon_sha256":"ff1c7fedd7e23caa7ac64a660e8807c3a303b92bd09dcb21f8f8036412ad0108"},"schema_version":"1.0","source":{"id":"1804.10359","kind":"arxiv","version":1}},"canonical_sha256":"2f7a9d18800e8436ef888649a7407c6408ca97c806a9e4254916bee0c7b06950","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f7a9d18800e8436ef888649a7407c6408ca97c806a9e4254916bee0c7b06950","first_computed_at":"2026-05-18T00:17:20.982450Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:20.982450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RlKGjEXWw7japcW1FcENYTk9R/F4G5wDG7ED26SMLlGE3N4hXeq7eD3afpCJgega2vh2EgMraAlfdxGe+eTJBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:20.983157Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.10359","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71487093ec58a06910699bf9728d4ac73aa9d1ee52d7dd32da1a3e85851d8225","sha256:e40afd36cf037209a499cb83be672bad9f8cc6711d3dc277f5aa9bda6a8b78ac"],"state_sha256":"dde55c34633340179cf8b4b5e9a9d8c2f9c370f79c53f38ada1cc6c3b1307576"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qy0nx3gUMypp+NvDIUPBRnAleehpjiZfLnkeZUatAObyIsOVIAWKp/HvZWH4yOloyT1z/X6RxX+h5lxmDAYbDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T21:57:41.377877Z","bundle_sha256":"dabfb9093762ca0169ac3da307dc389812653e0d3b6cb8752a741b6fe067832f"}}