{"paper":{"title":"Strengthening a theorem of Meyniel","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Carl Feghali, Cl\\'ement Legrand-Duchesne, Franti\\v{s}ek Kardo\\v{s}, Quentin Deschamps, Th\\'eo Pierron","submitted_at":"2022-01-19T13:45:08Z","abstract_excerpt":"For an integer $k \\geq 1$ and a graph $G$, let $\\mathcal{K}_k(G)$ be the graph that has vertex set all proper $k$-colorings of $G$, and an edge between two vertices $\\alpha$ and~$\\beta$ whenever the coloring~$\\beta$ can be obtained from $\\alpha$ by a single Kempe change. A theorem of Meyniel from 1978 states that $\\mathcal{K}_5(G)$ is connected with diameter $O(5^{|V(G)|})$ for every planar graph $G$. We significantly strengthen this result, by showing that there is a positive constant $c$ such that $\\mathcal{K}_5(G)$ has diameter $O(|V(G)|^c)$ for every planar graph $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2201.07595","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2201.07595/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"}