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The parity of an $i$-th dimension edge $\\edg{v_1}{v_2}$ is the number of 1's modulus 2 of any of its vertex ignoring the $i$-th entry. We prove that the number of $i$-th dimension edges appearing in a given Hamiltonian cycle of $Q_n$ with parity zero coincides with the number of edges with parity one. As an application of this result it is introduced and explored the conjecture of the inscribed"},"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":"1009.3304","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-09-17T00:43:02Z","cross_cats_sorted":[],"title_canon_sha256":"7687f75906f918a35c6539a1cc70772dbbe56079d3f761ed076705c7ab7de86b","abstract_canon_sha256":"c5907a018fa982a7b55c5ffcf7f2a663e8652d0168b08a26962a3f898439c07f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:40:46.001183Z","signature_b64":"4OrlF9UR8hfbEjbYEbINh+WjZl3hwkn7eP0TyWxAu0JCD7/NJmSsNywq52k2iLn8Pj8Vk/lJdXLC73xv6Is3AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7a94671ef1d4a64e415e15724ddc19be4f91198892abb554283f003b1c31bfd3","last_reissued_at":"2026-05-18T04:40:46.000651Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:40:46.000651Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Parity balance of the $i$-th dimension edges in Hamiltonian cycles of the hypercube","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Feli\\'u Sagols, Guillermo Morales-Luna","submitted_at":"2010-09-17T00:43:02Z","abstract_excerpt":"Let $n\\geq 2$ be an integer, and let $i\\in\\{0,...,n-1\\}$. 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