{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:3NUFKXR7ERRLADC5YSYUUOSHJI","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":"b4427ec176776e785f70b0484dd99a502ac75a43f10ffe9c50c1673396fbe753","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-24T21:07:20Z","title_canon_sha256":"12420557c80181c90a900b1d3ff6542c231c7b3c762f1c4f4f2d83926eed3d93"},"schema_version":"1.0","source":{"id":"0910.4681","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0910.4681","created_at":"2026-05-18T04:17:03Z"},{"alias_kind":"arxiv_version","alias_value":"0910.4681v2","created_at":"2026-05-18T04:17:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.4681","created_at":"2026-05-18T04:17:03Z"},{"alias_kind":"pith_short_12","alias_value":"3NUFKXR7ERRL","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"3NUFKXR7ERRLADC5","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"3NUFKXR7","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:826d3c0f2efbbc1eecb9bc37b82855be5bc5af73d15f3f5d7d7d82c68696207c","target":"graph","created_at":"2026-05-18T04:17:03Z","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":"An L-factor of a graph G is a spanning subgraph of G whose every component is a 3-vertex path. Let v(G) be the number of vertices of G and d(G) the domination number of G. A claw is a graph with four vertices and three edges incident to the same vertex. A graph is claw-free if it has no induced subgraph isomorphic to a claw. Our results include the following. Let G be a 3-connected claw-free graph, x a vertex in G, e = xy an edge in G, and P a 3-vertex path in G. Then\n  (a1) if v(G) = 0 mod 3, then G has an L-factor containing (avoiding) e, (a2) if v(G) = 1 mod 3, then G - x has an L-factor, (","authors_text":"Alexander Kelmans","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-24T21:07:20Z","title":"Packing 3-vertex paths in claw-free graphs and related topics"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4681","kind":"arxiv","version":2},"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":"sha256:33f78de96df2f4d82f1853955f05bba4e2d20b326649801bd47a2a3ef28d8291","target":"record","created_at":"2026-05-18T04:17:03Z","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":"b4427ec176776e785f70b0484dd99a502ac75a43f10ffe9c50c1673396fbe753","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-24T21:07:20Z","title_canon_sha256":"12420557c80181c90a900b1d3ff6542c231c7b3c762f1c4f4f2d83926eed3d93"},"schema_version":"1.0","source":{"id":"0910.4681","kind":"arxiv","version":2}},"canonical_sha256":"db68555e3f2462b00c5dc4b14a3a474a0f5c2f07a9f7d2689d1988f8e24f1979","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db68555e3f2462b00c5dc4b14a3a474a0f5c2f07a9f7d2689d1988f8e24f1979","first_computed_at":"2026-05-18T04:17:03.446347Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:17:03.446347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"eQtGcogNOTwzMnbZbHFTnogj8ApMq15u7fd+sk/13hECfy8DIzlQ5HJ224V3KKpot1gnICexUXs3no+ugLFxAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:17:03.446776Z","signed_message":"canonical_sha256_bytes"},"source_id":"0910.4681","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:33f78de96df2f4d82f1853955f05bba4e2d20b326649801bd47a2a3ef28d8291","sha256:826d3c0f2efbbc1eecb9bc37b82855be5bc5af73d15f3f5d7d7d82c68696207c"],"state_sha256":"80e1db7787ff4721541c515cc72e6d70388e69fd3ab56b5b4ab71aa5669b6160"}