{"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 $n+1$. As a corollary, we improv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.07132","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}