{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:CAQANS3N5KLMB75WCM6WYOVOWI","short_pith_number":"pith:CAQANS3N","canonical_record":{"source":{"id":"1612.04069","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-13T09:16:03Z","cross_cats_sorted":[],"title_canon_sha256":"2220b22de507cbfb204897f1040fb6980052a74413d6d451a2db1ccacc3c5773","abstract_canon_sha256":"00196ad4015bbc216cd20ef710430b8366beffe639c2de9b5a4167094d455e37"},"schema_version":"1.0"},"canonical_sha256":"102006cb6dea96c0ffb6133d6c3aaeb22f5a637f293cc1975a555aedba52ceba","source":{"kind":"arxiv","id":"1612.04069","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.04069","created_at":"2026-05-18T00:55:05Z"},{"alias_kind":"arxiv_version","alias_value":"1612.04069v1","created_at":"2026-05-18T00:55:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04069","created_at":"2026-05-18T00:55:05Z"},{"alias_kind":"pith_short_12","alias_value":"CAQANS3N5KLM","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CAQANS3N5KLMB75W","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CAQANS3N","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:CAQANS3N5KLMB75WCM6WYOVOWI","target":"record","payload":{"canonical_record":{"source":{"id":"1612.04069","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-13T09:16:03Z","cross_cats_sorted":[],"title_canon_sha256":"2220b22de507cbfb204897f1040fb6980052a74413d6d451a2db1ccacc3c5773","abstract_canon_sha256":"00196ad4015bbc216cd20ef710430b8366beffe639c2de9b5a4167094d455e37"},"schema_version":"1.0"},"canonical_sha256":"102006cb6dea96c0ffb6133d6c3aaeb22f5a637f293cc1975a555aedba52ceba","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:55:05.632481Z","signature_b64":"qcFnCNcmzRFGGBxB97tTn+/muEGJPjnlEaQUSwD5p5u3FbTgD5z9dwr0aEh2ZuBzENcMllo01arepZr2qXPiCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"102006cb6dea96c0ffb6133d6c3aaeb22f5a637f293cc1975a555aedba52ceba","last_reissued_at":"2026-05-18T00:55:05.631810Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:55:05.631810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.04069","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":"2026-05-18T00:55:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cL8Tz+EPVLjgK+xU2XekiCS4FeiG3YsgQczm+LKtGn1tUxaFtS7mSZiLmuA1GFqcRKw+5WU9GAHAo2RlcpHCDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:22:23.015944Z"},"content_sha256":"54dfa2ee26fd889967a55696b717eb28d9ec7d5587bc8ea45366becb8d8861d0","schema_version":"1.0","event_id":"sha256:54dfa2ee26fd889967a55696b717eb28d9ec7d5587bc8ea45366becb8d8861d0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:CAQANS3N5KLMB75WCM6WYOVOWI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Triangulating Almost-Complete Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kayla Wright, Kim Nguyen Pham, Landon Settle, Padraic Bartlett","submitted_at":"2016-12-13T09:16:03Z","abstract_excerpt":"A triangle decomposition of a graph $G$ is a partition of the edges of $G$ into triangles. Two necessary conditions for $G$ to admit such a decomposition are that $|E(G)|$ is a multiple of three and that the degree of any vertex in $G$ is even; we call such graphs tridivisible.\n  Kirkman's work on Steiner triple systems established that for $G \\simeq K_n$, $G$ admits a triangle decomposition precisely when $G$ is tridivisible. In 1970, Nash-Williams conjectured that tridivisiblity is also sufficient for \"almost-complete\" graphs, which for this talk's purposes we interpret as any graph $G$ on $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04069","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":"2026-05-18T00:55:05Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2lCAywSuOhLjDukAQcu3WmLQib1yKBEbeWqb15ABdE55C9/an1ZyGMU3YfZ55pmdcTnExd7JTB4OaBUxcjI3DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T13:22:23.016595Z"},"content_sha256":"d9096422485cdf591eb1b5ac6b7bd4eca9f056b75be728d834d0d370ea21fc36","schema_version":"1.0","event_id":"sha256:d9096422485cdf591eb1b5ac6b7bd4eca9f056b75be728d834d0d370ea21fc36"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CAQANS3N5KLMB75WCM6WYOVOWI/bundle.json","state_url":"https://pith.science/pith/CAQANS3N5KLMB75WCM6WYOVOWI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CAQANS3N5KLMB75WCM6WYOVOWI/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":"2026-05-30T13:22:23Z","links":{"resolver":"https://pith.science/pith/CAQANS3N5KLMB75WCM6WYOVOWI","bundle":"https://pith.science/pith/CAQANS3N5KLMB75WCM6WYOVOWI/bundle.json","state":"https://pith.science/pith/CAQANS3N5KLMB75WCM6WYOVOWI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CAQANS3N5KLMB75WCM6WYOVOWI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:CAQANS3N5KLMB75WCM6WYOVOWI","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":"00196ad4015bbc216cd20ef710430b8366beffe639c2de9b5a4167094d455e37","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-13T09:16:03Z","title_canon_sha256":"2220b22de507cbfb204897f1040fb6980052a74413d6d451a2db1ccacc3c5773"},"schema_version":"1.0","source":{"id":"1612.04069","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.04069","created_at":"2026-05-18T00:55:05Z"},{"alias_kind":"arxiv_version","alias_value":"1612.04069v1","created_at":"2026-05-18T00:55:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.04069","created_at":"2026-05-18T00:55:05Z"},{"alias_kind":"pith_short_12","alias_value":"CAQANS3N5KLM","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"CAQANS3N5KLMB75W","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"CAQANS3N","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:d9096422485cdf591eb1b5ac6b7bd4eca9f056b75be728d834d0d370ea21fc36","target":"graph","created_at":"2026-05-18T00:55:05Z","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":"A triangle decomposition of a graph $G$ is a partition of the edges of $G$ into triangles. Two necessary conditions for $G$ to admit such a decomposition are that $|E(G)|$ is a multiple of three and that the degree of any vertex in $G$ is even; we call such graphs tridivisible.\n  Kirkman's work on Steiner triple systems established that for $G \\simeq K_n$, $G$ admits a triangle decomposition precisely when $G$ is tridivisible. In 1970, Nash-Williams conjectured that tridivisiblity is also sufficient for \"almost-complete\" graphs, which for this talk's purposes we interpret as any graph $G$ on $","authors_text":"Kayla Wright, Kim Nguyen Pham, Landon Settle, Padraic Bartlett","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-13T09:16:03Z","title":"Triangulating Almost-Complete Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.04069","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":"sha256:54dfa2ee26fd889967a55696b717eb28d9ec7d5587bc8ea45366becb8d8861d0","target":"record","created_at":"2026-05-18T00:55:05Z","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":"00196ad4015bbc216cd20ef710430b8366beffe639c2de9b5a4167094d455e37","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-12-13T09:16:03Z","title_canon_sha256":"2220b22de507cbfb204897f1040fb6980052a74413d6d451a2db1ccacc3c5773"},"schema_version":"1.0","source":{"id":"1612.04069","kind":"arxiv","version":1}},"canonical_sha256":"102006cb6dea96c0ffb6133d6c3aaeb22f5a637f293cc1975a555aedba52ceba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"102006cb6dea96c0ffb6133d6c3aaeb22f5a637f293cc1975a555aedba52ceba","first_computed_at":"2026-05-18T00:55:05.631810Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:05.631810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"qcFnCNcmzRFGGBxB97tTn+/muEGJPjnlEaQUSwD5p5u3FbTgD5z9dwr0aEh2ZuBzENcMllo01arepZr2qXPiCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:05.632481Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.04069","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:54dfa2ee26fd889967a55696b717eb28d9ec7d5587bc8ea45366becb8d8861d0","sha256:d9096422485cdf591eb1b5ac6b7bd4eca9f056b75be728d834d0d370ea21fc36"],"state_sha256":"3d789e8337b468e66f74513212af6b79d9ccc3ef909df09041aa35306f845261"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8kFNqVY7rfTzTxxUmeFo8+erHy+Ceo7ExpW/k9rAVO0RLvIC4CYk+QfgIW2YVBk/d56p1SoczauCI9X4+BE6CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T13:22:23.019803Z","bundle_sha256":"d5358f7ddcbff1c6282b71d096ed65ad3c4ae9dd88afcf4f3923a9f73189d2cb"}}