{"paper":{"title":"The Balanced Four-Color Theorem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DS","authors_text":"Hirotaka Yoneda, Ken-ichi Kawarabayashi, Masataka Yoneda","submitted_at":"2026-07-14T17:58:34Z","abstract_excerpt":"We show that every planar graph with $n \\geq 3$ vertices admits a 4-coloring in which each color is used on fewer than $n/2$ vertices. This bound is the best possible. Moreover, such a coloring can be found in $O(n \\log n)$ time. We also extend these results to five or more colors and to graphs on general surfaces."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.13025","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/2607.13025/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"}