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Let $o(G)$ denote the number of odd components of $G$.\n  We show that if one of the following conditions holds, then $G$ contain"},"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":"1301.4657","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-20T13:35:02Z","cross_cats_sorted":[],"title_canon_sha256":"c1bd97c25c5b4d7dc2a49528641e8192598054de0eb967998869c06ad8231007","abstract_canon_sha256":"15f5931421aac95c35b43d7931b5c66799392295c87829ed575fc1debe9ece72"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:35:25.030576Z","signature_b64":"CoQbMqJ+xdrKm5RPUBC0GnMiv4YgQqBbUgtGSiw81AbHO+ux9PiL2BIqwypWlV1cHrSU6ZyYH/CkgzSAIudUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f41893f742a12c9d25ecb5a5c3f6d3210c3c58a59270ea10e6a08cf270a6ff5","last_reissued_at":"2026-05-18T03:35:25.029904Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:35:25.029904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Extension of Cui-Kano's Characterization Problem on Graph Factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hongliang Lu","submitted_at":"2013-01-20T13:35:02Z","abstract_excerpt":"Let $G$ be a graph with vertex set $V(G)$ and let $H:V(G)\\rightarrow 2^N$ be a set function associating with $G$. 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