{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:4R6ZQZ6ENNQFAPAR6LYCB6JHU6","short_pith_number":"pith:4R6ZQZ6E","canonical_record":{"source":{"id":"1911.06275","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-11-14T17:58:40Z","cross_cats_sorted":[],"title_canon_sha256":"2ce43d0233f199552f5edca4bd0e78cfac4c128c2bde78b171162e13c0995e08","abstract_canon_sha256":"1168c4939a755f77943a744d8e25713affd893eab715faad7a9e296bbc7df69d"},"schema_version":"1.0"},"canonical_sha256":"e47d9867c46b60503c11f2f020f927a792582407a8d49def9668b0edf71e3350","source":{"kind":"arxiv","id":"1911.06275","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1911.06275","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"arxiv_version","alias_value":"1911.06275v1","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.06275","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"pith_short_12","alias_value":"4R6ZQZ6ENNQF","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"pith_short_16","alias_value":"4R6ZQZ6ENNQFAPAR","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"pith_short_8","alias_value":"4R6ZQZ6E","created_at":"2026-07-05T00:19:16Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:4R6ZQZ6ENNQFAPAR6LYCB6JHU6","target":"record","payload":{"canonical_record":{"source":{"id":"1911.06275","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-11-14T17:58:40Z","cross_cats_sorted":[],"title_canon_sha256":"2ce43d0233f199552f5edca4bd0e78cfac4c128c2bde78b171162e13c0995e08","abstract_canon_sha256":"1168c4939a755f77943a744d8e25713affd893eab715faad7a9e296bbc7df69d"},"schema_version":"1.0"},"canonical_sha256":"e47d9867c46b60503c11f2f020f927a792582407a8d49def9668b0edf71e3350","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:19:16.501339Z","signature_b64":"c+JeAc6PtP7RHYbrxJD4RrCRSGLYJs2cfNbf1P88Kg9X8kcm+oqua3JAU+WyAPjZynxCcmt27UcKO9bg9JzPDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e47d9867c46b60503c11f2f020f927a792582407a8d49def9668b0edf71e3350","last_reissued_at":"2026-07-05T00:19:16.500963Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:19:16.500963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1911.06275","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":"2026-07-05T00:19:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gzTidaJAa0rNJGvhiWy23ZH4ZRruRq1zQXxcrTHcL8cSrOd6msvEk3DLuS7mVevlvcuRL9DTeuqSlMqnKlV3Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T08:08:31.021898Z"},"content_sha256":"2e336a3a73158a91a8a4f7a4c6e730105be0c5e0b2769c09eb377248bf528b9a","schema_version":"1.0","event_id":"sha256:2e336a3a73158a91a8a4f7a4c6e730105be0c5e0b2769c09eb377248bf528b9a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:4R6ZQZ6ENNQFAPAR6LYCB6JHU6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Colourings of star systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Pike, Iren Darijani","submitted_at":"2019-11-14T17:58:40Z","abstract_excerpt":"An $e$-star is a complete bipartite graph $K_{1,e}$. An $e$-star system of order $n>1$, $S_e(n)$, is a partition of the edges of the complete graph $K_n$ into $e$-stars. An $e$-star system is said to be $k$-colourable if its vertex set can be partitioned into $k$ sets (called colour classes) such that no $e$-star is monochromatic. The system $S_e(n)$ is $k$-chromatic if $S_e(n)$ is $k$-colourable but is not $(k-1)$-colourable. If every $k$-colouring of an $e$-star system can be obtained from some $k$-colouring $\\phi$ by a permutation of the colours, we say that the system is uniquely $k$-colou"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.06275","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1911.06275/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T00:19:16Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HxFHbsifaFrsQyDokwWSawE4tL/A56Rjn1sF9tPinX7nDdwqFw9vXSX0K2roRGEAVZPx2yurWshpHm0cL+BoCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T08:08:31.022516Z"},"content_sha256":"d7006535eff16a84693660559f76e4d30aefd9a8c928c642e3945888b17a990b","schema_version":"1.0","event_id":"sha256:d7006535eff16a84693660559f76e4d30aefd9a8c928c642e3945888b17a990b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4R6ZQZ6ENNQFAPAR6LYCB6JHU6/bundle.json","state_url":"https://pith.science/pith/4R6ZQZ6ENNQFAPAR6LYCB6JHU6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4R6ZQZ6ENNQFAPAR6LYCB6JHU6/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":"2026-08-06T08:08:31Z","links":{"resolver":"https://pith.science/pith/4R6ZQZ6ENNQFAPAR6LYCB6JHU6","bundle":"https://pith.science/pith/4R6ZQZ6ENNQFAPAR6LYCB6JHU6/bundle.json","state":"https://pith.science/pith/4R6ZQZ6ENNQFAPAR6LYCB6JHU6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4R6ZQZ6ENNQFAPAR6LYCB6JHU6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:4R6ZQZ6ENNQFAPAR6LYCB6JHU6","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":"1168c4939a755f77943a744d8e25713affd893eab715faad7a9e296bbc7df69d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-11-14T17:58:40Z","title_canon_sha256":"2ce43d0233f199552f5edca4bd0e78cfac4c128c2bde78b171162e13c0995e08"},"schema_version":"1.0","source":{"id":"1911.06275","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1911.06275","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"arxiv_version","alias_value":"1911.06275v1","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1911.06275","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"pith_short_12","alias_value":"4R6ZQZ6ENNQF","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"pith_short_16","alias_value":"4R6ZQZ6ENNQFAPAR","created_at":"2026-07-05T00:19:16Z"},{"alias_kind":"pith_short_8","alias_value":"4R6ZQZ6E","created_at":"2026-07-05T00:19:16Z"}],"graph_snapshots":[{"event_id":"sha256:d7006535eff16a84693660559f76e4d30aefd9a8c928c642e3945888b17a990b","target":"graph","created_at":"2026-07-05T00:19:16Z","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/1911.06275/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"An $e$-star is a complete bipartite graph $K_{1,e}$. An $e$-star system of order $n>1$, $S_e(n)$, is a partition of the edges of the complete graph $K_n$ into $e$-stars. An $e$-star system is said to be $k$-colourable if its vertex set can be partitioned into $k$ sets (called colour classes) such that no $e$-star is monochromatic. The system $S_e(n)$ is $k$-chromatic if $S_e(n)$ is $k$-colourable but is not $(k-1)$-colourable. If every $k$-colouring of an $e$-star system can be obtained from some $k$-colouring $\\phi$ by a permutation of the colours, we say that the system is uniquely $k$-colou","authors_text":"David Pike, Iren Darijani","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-11-14T17:58:40Z","title":"Colourings of star systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.06275","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":"sha256:2e336a3a73158a91a8a4f7a4c6e730105be0c5e0b2769c09eb377248bf528b9a","target":"record","created_at":"2026-07-05T00:19:16Z","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":"1168c4939a755f77943a744d8e25713affd893eab715faad7a9e296bbc7df69d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-11-14T17:58:40Z","title_canon_sha256":"2ce43d0233f199552f5edca4bd0e78cfac4c128c2bde78b171162e13c0995e08"},"schema_version":"1.0","source":{"id":"1911.06275","kind":"arxiv","version":1}},"canonical_sha256":"e47d9867c46b60503c11f2f020f927a792582407a8d49def9668b0edf71e3350","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e47d9867c46b60503c11f2f020f927a792582407a8d49def9668b0edf71e3350","first_computed_at":"2026-07-05T00:19:16.500963Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:19:16.500963Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"c+JeAc6PtP7RHYbrxJD4RrCRSGLYJs2cfNbf1P88Kg9X8kcm+oqua3JAU+WyAPjZynxCcmt27UcKO9bg9JzPDw==","signature_status":"signed_v1","signed_at":"2026-07-05T00:19:16.501339Z","signed_message":"canonical_sha256_bytes"},"source_id":"1911.06275","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e336a3a73158a91a8a4f7a4c6e730105be0c5e0b2769c09eb377248bf528b9a","sha256:d7006535eff16a84693660559f76e4d30aefd9a8c928c642e3945888b17a990b"],"state_sha256":"9ccc62509039061efadd6349209961c2e807087f8bf51d25e690c24c874e524b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s+K6v8zEbTElpUuLr0vI0r1hFAFpRIlc+TGgWShwr15HONMxAqGNCH1EklvXh+Cw+9OZUv3qVobhWyq884mhAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T08:08:31.027641Z","bundle_sha256":"2e2be799fa4e1a20c1356c9bbbdce35310d7452156315491fad4b976a521c0d9"}}