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(ii)~If a graph $G$ satisfies $c_{1}(G-X)+c_{3}(G-X)+\\frac{2}{3}c_{5}(G-X)+\\frac{1}{3}c_{7}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{9}\\}$-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":"1507.07066","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-25T04:43:56Z","cross_cats_sorted":[],"title_canon_sha256":"8503f1e62c94c967067b1affd01195eb6cf4b07ed251f2502be81d60fa5c8074","abstract_canon_sha256":"eda4d21ff33132add0e216f0da2d7f9507540d2fa3e1ea5695f2c3ddcb12be47"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:17.344564Z","signature_b64":"rFrwNWSG3bNt2AXVG9RCeByWbSk2KZWH8hWFWPwWLdrpICA0aJwBebKZncU2MBElRU4dsz62DrzmplEyMAYhDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45b540eacb5a1e7cbe1278a0609d50b611253c8453a358f64406d30f59fa2148","last_reissued_at":"2026-05-18T01:36:17.343951Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:17.343951Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Path-factors involving paths of order seven and nine","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michitaka Furuya, Yoshimi Egawa","submitted_at":"2015-07-25T04:43:56Z","abstract_excerpt":"In this paper, we show the following two theorems (here $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$): (i)~If a graph $G$ satisfies $c_{1}(G-X)+\\frac{1}{3}c_{3}(G-X)+\\frac{1}{3}c_{5}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{7}\\}$-factor. 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