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A $k$-ranking is a relaxation in which all nontrivial paths of length at most $k$ are well-ranked. The $k$-ranking number of a graph $G$ is the minimum $t$ such that there is a $k$-ranking of $G$ using ranks in $\\{1,\\ldots,t\\}$.\n  We prove that the $2$-ranking number of the $n$-dimensional hypercube $Q_n$ is $n+1$. As a corollary, we improv"},"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":"1607.07132","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-25T03:15:28Z","cross_cats_sorted":[],"title_canon_sha256":"b888d14132b93a620ea7f4a0e40367d35ea4e98ae50e98244374955afaf9bd46","abstract_canon_sha256":"f59c0a71ab5a10ff90efc05eb226e869a9903340625adff2079090e3415596f9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:10:34.670066Z","signature_b64":"Hoz7XNWGogE4XameqEBGheigFPf6sxAeVmtlK5CawpC1YmZKIWG4WdVuzcNKjXMuqEaA6jyK5PQNj5llR5zXDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10fb2ec17969440d6aafe9d4a388e586300094e0dd259602554140e5dae6430f","last_reissued_at":"2026-05-18T01:10:34.669345Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:10:34.669345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The 2-Ranking Numbers of Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrew Kallmeyer, Jordan Almeter, Kevin G. Milans, Robert Winslow, Samet Demircan","submitted_at":"2016-07-25T03:15:28Z","abstract_excerpt":"In a graph whose vertices are assigned integer ranks, a path is well-ranked if the endpoints have distinct ranks or some interior point has a higher rank than the endpoints. A ranking is an assignment of ranks such that all nontrivial paths are well-ranked. A $k$-ranking is a relaxation in which all nontrivial paths of length at most $k$ are well-ranked. The $k$-ranking number of a graph $G$ is the minimum $t$ such that there is a $k$-ranking of $G$ using ranks in $\\{1,\\ldots,t\\}$.\n  We prove that the $2$-ranking number of the $n$-dimensional hypercube $Q_n$ is $n+1$. 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