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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, ("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0910.4681","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2009-10-24T21:07:20Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"12420557c80181c90a900b1d3ff6542c231c7b3c762f1c4f4f2d83926eed3d93","abstract_canon_sha256":"b4427ec176776e785f70b0484dd99a502ac75a43f10ffe9c50c1673396fbe753"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:17:03.446776Z","signature_b64":"eQtGcogNOTwzMnbZbHFTnogj8ApMq15u7fd+sk/13hECfy8DIzlQ5HJ224V3KKpot1gnICexUXs3no+ugLFxAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"db68555e3f2462b00c5dc4b14a3a474a0f5c2f07a9f7d2689d1988f8e24f1979","last_reissued_at":"2026-05-18T04:17:03.446347Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:17:03.446347Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Packing 3-vertex paths in claw-free graphs and related topics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Alexander Kelmans","submitted_at":"2009-10-24T21:07:20Z","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, ("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.4681","kind":"arxiv","version":2},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0910.4681","created_at":"2026-05-18T04:17:03.446406+00:00"},{"alias_kind":"arxiv_version","alias_value":"0910.4681v2","created_at":"2026-05-18T04:17:03.446406+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0910.4681","created_at":"2026-05-18T04:17:03.446406+00:00"},{"alias_kind":"pith_short_12","alias_value":"3NUFKXR7ERRL","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_16","alias_value":"3NUFKXR7ERRLADC5","created_at":"2026-05-18T12:25:58.018023+00:00"},{"alias_kind":"pith_short_8","alias_value":"3NUFKXR7","created_at":"2026-05-18T12:25:58.018023+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI","json":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI.json","graph_json":"https://pith.science/api/pith-number/3NUFKXR7ERRLADC5YSYUUOSHJI/graph.json","events_json":"https://pith.science/api/pith-number/3NUFKXR7ERRLADC5YSYUUOSHJI/events.json","paper":"https://pith.science/paper/3NUFKXR7"},"agent_actions":{"view_html":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI","download_json":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI.json","view_paper":"https://pith.science/paper/3NUFKXR7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0910.4681&json=true","fetch_graph":"https://pith.science/api/pith-number/3NUFKXR7ERRLADC5YSYUUOSHJI/graph.json","fetch_events":"https://pith.science/api/pith-number/3NUFKXR7ERRLADC5YSYUUOSHJI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI/action/storage_attestation","attest_author":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI/action/author_attestation","sign_citation":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI/action/citation_signature","submit_replication":"https://pith.science/pith/3NUFKXR7ERRLADC5YSYUUOSHJI/action/replication_record"}},"created_at":"2026-05-18T04:17:03.446406+00:00","updated_at":"2026-05-18T04:17:03.446406+00:00"}