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We show that $G$ has an $(a,b)$-parity factor, if $max\\{d_G(u),d_G(v)\\}\\geq \\frac{an}{a+b}$ for any two nonadjacent vertices $u,v$ of $G$. It is an extension of Nishimura's results for the existence of $k$-factors (\\emph{J. Graph Theory}, \\textbf{16} (1992), 141--151) and generalizes Li and Cai's result in some senses (\\emph{J. Graph Theory}, \\textbf{27} (1998), 1--6). These conditions are tight."},"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":"1606.04608","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-15T01:23:52Z","cross_cats_sorted":[],"title_canon_sha256":"37be039ae9c20ff3d6ed19ab7066708afabea3eadf591866365630822119c97b","abstract_canon_sha256":"19ff7531704c4805186c377b5d9b095afaa74c7b2b14be99505bb7c2bfd16e9a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:24.703220Z","signature_b64":"s5I0A2wvEKehfr1lZuHFzRjXIUPRe7MEojlH/d/nOFkU/n04ohKOLts5wVOaDQQPvWKq9SW9AjpU8/Pix1ERBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2d69be383a86437f602a00c93e991f0944052a27ea0df876f1948805effaa2d2","last_reissued_at":"2026-05-18T01:12:24.702895Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:24.702895Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Degree Condition for a Graph to have $(a,b)$-Parity Factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Haodong Liu, Hongliang Lu","submitted_at":"2016-06-15T01:23:52Z","abstract_excerpt":"Let $a,b,n$ be three positive integers such that $a\\equiv b\\pmod 2$ and $n\\geq b(a+b)(a+b+2)/(2a)$. Let $G$ be a graph of order $n$ with minimum degree at least $a+b/a-1$. We show that $G$ has an $(a,b)$-parity factor, if $max\\{d_G(u),d_G(v)\\}\\geq \\frac{an}{a+b}$ for any two nonadjacent vertices $u,v$ of $G$. It is an extension of Nishimura's results for the existence of $k$-factors (\\emph{J. Graph Theory}, \\textbf{16} (1992), 141--151) and generalizes Li and Cai's result in some senses (\\emph{J. Graph Theory}, \\textbf{27} (1998), 1--6). 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