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Cheng et al.\n  [Proper distance in edge-colored hypercubes, Applied Mathematics and Computation 313 (2017) 384-391.] determined the number of distinct shortest properly colored paths between a pair of vertices for the $(1)$-colored hypercubes. It is natural to consider the number for $(j)$-coloring, $j\\geq 2$. In this note, we determine the number o"},"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":"1708.02690","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-08-09T01:37:09Z","cross_cats_sorted":[],"title_canon_sha256":"87f7b8f4f07d605b3bb642f37778366efdd84bea7daf974a8376f19c10d2b02a","abstract_canon_sha256":"27651ddda1d23e6ccfb0ec60c87e7f7ef7f2f345f8c471a4c6c145fcc7350176"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:19.904919Z","signature_b64":"uN0wMf/AdEvNVYEMtmEVwWAwGmYHLNTFhFm8UEbxb+xut46gvASp5cEiH4OxIj2ggLT7NfO+o5epIPRPB8dcCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"010dc2b66b262526e6223e50fb657bde3bf37366c74065b1507f72eaa933f9c3","last_reissued_at":"2026-05-18T00:38:19.904142Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:19.904142Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the number of proper paths between vertices in edge-colored hypercubes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lina Xue, Shurong Zhang, Weihua Yang","submitted_at":"2017-08-09T01:37:09Z","abstract_excerpt":"Given an integer $1\\leq j <n$, define the $(j)$-coloring of a $n$-dimensional hypercube $H_{n}$ to be the $2$-coloring of the edges of $H_{n}$ in which all edges in dimension $i$, $1\\leq i \\leq j$, have color $1$ and all other edges have color $2$. Cheng et al.\n  [Proper distance in edge-colored hypercubes, Applied Mathematics and Computation 313 (2017) 384-391.] determined the number of distinct shortest properly colored paths between a pair of vertices for the $(1)$-colored hypercubes. It is natural to consider the number for $(j)$-coloring, $j\\geq 2$. 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