{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:YXJJJLC6V7J2EUAPSL4ZXBKJPY","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":"d0049b68243f6e9985b084d2176d78f6019568d2f84342a1a2820b011dad34e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T18:06:47Z","title_canon_sha256":"740db9cc7d94bfa2e5e25ac9c9d7e65c5dbab39bf77375919d863495a4f20e39"},"schema_version":"1.0","source":{"id":"1704.06667","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.06667","created_at":"2026-05-18T00:45:57Z"},{"alias_kind":"arxiv_version","alias_value":"1704.06667v1","created_at":"2026-05-18T00:45:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.06667","created_at":"2026-05-18T00:45:57Z"},{"alias_kind":"pith_short_12","alias_value":"YXJJJLC6V7J2","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_16","alias_value":"YXJJJLC6V7J2EUAP","created_at":"2026-05-18T12:31:56Z"},{"alias_kind":"pith_short_8","alias_value":"YXJJJLC6","created_at":"2026-05-18T12:31:56Z"}],"graph_snapshots":[{"event_id":"sha256:b57c9bc3c8ed66d77782ac8aec3c4c9e37a0f81b3e84bf14b8073a98f3d84c37","target":"graph","created_at":"2026-05-18T00:45:57Z","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 graph $G$ is said to be $2$-divisible if for all (nonempty) induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $\\omega(A) < \\omega(H)$ and $\\omega(B) < \\omega(H)$. A graph $G$ is said to be perfectly divisible if for all induced subgraphs $H$ of $G$, $V(H)$ can be partitioned into two sets $A,B$ such that $H[A]$ is perfect and $\\omega(B) < \\omega(H)$. We prove that if a graph is $(P_5,C_5)$-free, then it is $2$-divisible. We also prove that if a graph is bull-free and either odd-hole-free or $P_5$-free, then it is perfectly divisible.","authors_text":"Maria Chudnovsky, Vaidy Sivaraman","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T18:06:47Z","title":"Perfect divisibility and 2-divisibility"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.06667","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":"sha256:14b3754e283a5ea403037a4c6f4fe628b49345ce40fe748faacb9ee98a7b6b0d","target":"record","created_at":"2026-05-18T00:45:57Z","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":"d0049b68243f6e9985b084d2176d78f6019568d2f84342a1a2820b011dad34e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-04-21T18:06:47Z","title_canon_sha256":"740db9cc7d94bfa2e5e25ac9c9d7e65c5dbab39bf77375919d863495a4f20e39"},"schema_version":"1.0","source":{"id":"1704.06667","kind":"arxiv","version":1}},"canonical_sha256":"c5d294ac5eafd3a2500f92f99b85497e38bfb90781d8dc80df264ed8c927f854","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c5d294ac5eafd3a2500f92f99b85497e38bfb90781d8dc80df264ed8c927f854","first_computed_at":"2026-05-18T00:45:57.946498Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:45:57.946498Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ArW6Y+WSe6O3zORigCqMcEI6dSMeXBdl57x/+6tlkRVwgIDatXYWSfL4GCKAP3jfmtfkv/OTH6DkyCJYk5inDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:45:57.946984Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.06667","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14b3754e283a5ea403037a4c6f4fe628b49345ce40fe748faacb9ee98a7b6b0d","sha256:b57c9bc3c8ed66d77782ac8aec3c4c9e37a0f81b3e84bf14b8073a98f3d84c37"],"state_sha256":"dfd1ff0833896c175a2fa533725900616972f961474c40b627d7abb9f3f76d44"}