{"paper":{"title":"Asymptotically Optimal Vertex Ranking of Planar Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"math.CO","authors_text":"Mehrnoosh Javarsineh, Pat Morin, Prosenjit Bose, Vida Dujmovi\\'c","submitted_at":"2020-07-13T15:40:57Z","abstract_excerpt":"A (vertex) $\\ell$-ranking is a colouring $\\varphi:V(G)\\to\\mathbb{N}$ of the vertices of a graph $G$ with integer colours so that for any path $u_0,\\ldots,u_p$ of length at most $\\ell$, $\\varphi(u_0)\\neq\\varphi(u_p)$ or $\\varphi(u_0)<\\max\\{\\varphi(u_0),\\ldots,\\varphi(u_p)\\}$. We show that, for any fixed integer $\\ell\\ge 2$, every $n$-vertex planar graph has an $\\ell$-ranking using $O(\\log n/\\log\\log\\log n)$ colours and this is tight even when $\\ell=2$; for infinitely many values of $n$, there are $n$-vertex planar graphs, for which any 2-ranking requires $\\Omega(\\log n/\\log\\log\\log n)$ colours."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.06455","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2007.06455/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}