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It is called $k$-semi-perfect if the union of any pair of 1-factors $M_i, M_j$ with $1 \\le i \\le k$ and $k+1 \\le j \\le n$ is a Hamilton cycle.\n  We consider 1-factorizations of the discrete cube $Q_d$. There is no perfect 1-factorization of $Q_d$, but it was previously shown that there is a 1-semi-perfect 1-factorization of $Q_d$ for all $d$. Our main result is to prove that there is a $k$-semi-perfect 1-factorization of "},"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":"1811.06389","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-11-15T14:29:34Z","cross_cats_sorted":[],"title_canon_sha256":"965ec0e3e20a6c7c488ce3f3ed6b36b6a7f7ae9079256d68c8a4fc04322383c7","abstract_canon_sha256":"8e4ff289174328a22b0d3e0ab73ff26f0a7c4d1732695390489f5600788e6e23"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:30:48.179208Z","signature_b64":"+PmVosfY7P0t/iZ4w2ygRRMTnJIi3MNl/Il6fjVn1TZgrfQK8FCZUs8A63A7qGMNGP5VNJ0d7YU7P4NRGPbAAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"67ebbb16d29216b7c857db8240f6ef99c6a56098c74112d6444295f50fa102b9","last_reissued_at":"2026-07-05T01:30:48.178747Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:30:48.178747Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Semi-perfect 1-Factorizations of the Hypercube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Natalie C. 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