{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2019:Y5W6SCMAQAFXFJQLWBXIJAEVL5","short_pith_number":"pith:Y5W6SCMA","canonical_record":{"source":{"id":"1906.10031","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-24T15:43:26Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"af98cf48443af7503c291d1d919b3b0d627ea413a7df18629ffcf811a510022a","abstract_canon_sha256":"16b10b90893ac1d2e90372a1f37709a568df443677e61ec7b61291f695deb74e"},"schema_version":"1.0"},"canonical_sha256":"c76de90980800b72a60bb06e8480955f4914fef54e57557aabf57136c3d8c920","source":{"kind":"arxiv","id":"1906.10031","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.10031","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"arxiv_version","alias_value":"1906.10031v1","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.10031","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"pith_short_12","alias_value":"Y5W6SCMAQAFX","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"Y5W6SCMAQAFXFJQL","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"Y5W6SCMA","created_at":"2026-05-18T12:33:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2019:Y5W6SCMAQAFXFJQLWBXIJAEVL5","target":"record","payload":{"canonical_record":{"source":{"id":"1906.10031","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-24T15:43:26Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"af98cf48443af7503c291d1d919b3b0d627ea413a7df18629ffcf811a510022a","abstract_canon_sha256":"16b10b90893ac1d2e90372a1f37709a568df443677e61ec7b61291f695deb74e"},"schema_version":"1.0"},"canonical_sha256":"c76de90980800b72a60bb06e8480955f4914fef54e57557aabf57136c3d8c920","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:42:35.670112Z","signature_b64":"6uvjx1Hxb1NPDnT/N27G2BICghWBX3VnNAaf9vKBpQ5F18Jqxz7RR3RF2t9L/ytm4TQ7oegP6FavKhX8d5T/Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c76de90980800b72a60bb06e8480955f4914fef54e57557aabf57136c3d8c920","last_reissued_at":"2026-05-17T23:42:35.669417Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:42:35.669417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1906.10031","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":"2026-05-17T23:42:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FOo3j1w9DnAHG9lYuYwAQyTefCpt+jc5vuSHxL7ugHGmHZh09In75+0QCvSv4yVdzG9fgCG2l+/Aqa3XxpzhCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T10:59:08.371197Z"},"content_sha256":"7eda5017de8bab3bf87b3623a1334f1e5bf8b12ab06a704dce11d0c648c970e3","schema_version":"1.0","event_id":"sha256:7eda5017de8bab3bf87b3623a1334f1e5bf8b12ab06a704dce11d0c648c970e3"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2019:Y5W6SCMAQAFXFJQLWBXIJAEVL5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hierarchical Colorings of Cographs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"D.I. Valdivia, M. Gei{\\ss}, M. Hellmuth, M. Hernandez Rosales, P.F. Stadler","submitted_at":"2019-06-24T15:43:26Z","abstract_excerpt":"Cographs are exactly hereditarily well-colored graphs, i.e., the graphs for which a greedy coloring of every induced subgraph uses only the minimally necessary number of colors $\\chi(G)$. In recent work on reciprocal best match graphs so-called hierarchically coloring play an important role. Here we show that greedy colorings are a special case of hierarchical coloring, which also require no more than $\\chi(G)$ colors."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.10031","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":"2026-05-17T23:42:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cMoUPSPeYujWSl84N1brMMBkOgKvvAHTgXEaRJbpvmo12ijgxgjR+r1pw8pcRN1c/cCsDiOrDToyqkIXSEK9Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T10:59:08.372009Z"},"content_sha256":"49ecf3a8ba5064dd4da61c83dfbc759d563ccfc597a6c3cfcecbed3434915197","schema_version":"1.0","event_id":"sha256:49ecf3a8ba5064dd4da61c83dfbc759d563ccfc597a6c3cfcecbed3434915197"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y5W6SCMAQAFXFJQLWBXIJAEVL5/bundle.json","state_url":"https://pith.science/pith/Y5W6SCMAQAFXFJQLWBXIJAEVL5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y5W6SCMAQAFXFJQLWBXIJAEVL5/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":"2026-05-25T10:59:08Z","links":{"resolver":"https://pith.science/pith/Y5W6SCMAQAFXFJQLWBXIJAEVL5","bundle":"https://pith.science/pith/Y5W6SCMAQAFXFJQLWBXIJAEVL5/bundle.json","state":"https://pith.science/pith/Y5W6SCMAQAFXFJQLWBXIJAEVL5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y5W6SCMAQAFXFJQLWBXIJAEVL5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:Y5W6SCMAQAFXFJQLWBXIJAEVL5","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":"16b10b90893ac1d2e90372a1f37709a568df443677e61ec7b61291f695deb74e","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-24T15:43:26Z","title_canon_sha256":"af98cf48443af7503c291d1d919b3b0d627ea413a7df18629ffcf811a510022a"},"schema_version":"1.0","source":{"id":"1906.10031","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1906.10031","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"arxiv_version","alias_value":"1906.10031v1","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1906.10031","created_at":"2026-05-17T23:42:35Z"},{"alias_kind":"pith_short_12","alias_value":"Y5W6SCMAQAFX","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_16","alias_value":"Y5W6SCMAQAFXFJQL","created_at":"2026-05-18T12:33:33Z"},{"alias_kind":"pith_short_8","alias_value":"Y5W6SCMA","created_at":"2026-05-18T12:33:33Z"}],"graph_snapshots":[{"event_id":"sha256:49ecf3a8ba5064dd4da61c83dfbc759d563ccfc597a6c3cfcecbed3434915197","target":"graph","created_at":"2026-05-17T23:42:35Z","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":"Cographs are exactly hereditarily well-colored graphs, i.e., the graphs for which a greedy coloring of every induced subgraph uses only the minimally necessary number of colors $\\chi(G)$. In recent work on reciprocal best match graphs so-called hierarchically coloring play an important role. Here we show that greedy colorings are a special case of hierarchical coloring, which also require no more than $\\chi(G)$ colors.","authors_text":"D.I. Valdivia, M. Gei{\\ss}, M. Hellmuth, M. Hernandez Rosales, P.F. Stadler","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-24T15:43:26Z","title":"Hierarchical Colorings of Cographs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1906.10031","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":"sha256:7eda5017de8bab3bf87b3623a1334f1e5bf8b12ab06a704dce11d0c648c970e3","target":"record","created_at":"2026-05-17T23:42:35Z","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":"16b10b90893ac1d2e90372a1f37709a568df443677e61ec7b61291f695deb74e","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-06-24T15:43:26Z","title_canon_sha256":"af98cf48443af7503c291d1d919b3b0d627ea413a7df18629ffcf811a510022a"},"schema_version":"1.0","source":{"id":"1906.10031","kind":"arxiv","version":1}},"canonical_sha256":"c76de90980800b72a60bb06e8480955f4914fef54e57557aabf57136c3d8c920","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c76de90980800b72a60bb06e8480955f4914fef54e57557aabf57136c3d8c920","first_computed_at":"2026-05-17T23:42:35.669417Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:42:35.669417Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6uvjx1Hxb1NPDnT/N27G2BICghWBX3VnNAaf9vKBpQ5F18Jqxz7RR3RF2t9L/ytm4TQ7oegP6FavKhX8d5T/Bg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:42:35.670112Z","signed_message":"canonical_sha256_bytes"},"source_id":"1906.10031","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7eda5017de8bab3bf87b3623a1334f1e5bf8b12ab06a704dce11d0c648c970e3","sha256:49ecf3a8ba5064dd4da61c83dfbc759d563ccfc597a6c3cfcecbed3434915197"],"state_sha256":"c3d4e7adc16b92f17026d00918fb563306a4bf3421d0fbed347c6d8a3f3f4322"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jRa2bE7Dmuip3PTccArXWVQtdxN7BqZOkOlJfSxBLpYNeJGG7+zsVK32sXBd0/msjF35Uu/VZy60RJEMPBriDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T10:59:08.374680Z","bundle_sha256":"635fa8b5d0821bdd4e17639abbc7cab60d7af9298ccd39ad88d4972d1acaba19"}}