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When $p=1$, this conjecture reduces to a theorem of Schrijver which says that a $d$--regular bipartite graph on $v(G)=2n$ vertices has at least $$\\left(\\frac{(d-1)^{d-1}}{d^{d-2}}\\right)^n$$ perfect matchings. L. Gurvits proved an asymptotic version of the Lower Matching Conjecture, namely he proved that $$\\frac{\\ln m_k(G"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1406.0766","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-06-03T16:09:09Z","cross_cats_sorted":[],"title_canon_sha256":"0e5605f34e878329da618d16883167475e7bfeb933482665e1422cfc436f9bc0","abstract_canon_sha256":"4a9f35b36aa333ea17f5ef6fe1e5a9c7f413ef64a5eb8a389a2c66dd94f6c6f5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:32.286984Z","signature_b64":"/Cy2qWIRNuJ+BovOopAOXV/UsPRLIcPYOjWFrkCtIz4Jn+exKXT0t63M+0HS+59JIJ3FyglXG50MLs+pDtEtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e0b3962e13b9ef64c0753de306752a262ad75da2e837f0e55d92d5e77ce70c75","last_reissued_at":"2026-05-18T00:46:32.286291Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:32.286291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lower matching conjecture, and a new proof of Schrijver's and Gurvits's theorems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"P\\'eter Csikv\\'ari","submitted_at":"2014-06-03T16:09:09Z","abstract_excerpt":"Friedland's Lower Matching Conjecture asserts that if $G$ is a $d$--regular bipartite graph on $v(G)=2n$ vertices, and $m_k(G)$ denotes the number of matchings of size $k$, then $$m_k(G)\\geq {n \\choose k}^2\\left(\\frac{d-p}{d}\\right)^{n(d-p)}(dp)^{np},$$ where $p=\\frac{k}{n}$. 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