{"paper":{"title":"On the geometric $k$-colored crossing number of $K_n$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.CG","authors_text":"Benedikt Hahn, Bettina Klinz, Birgit Vogtenhuber","submitted_at":"2025-05-23T15:15:27Z","abstract_excerpt":"We study the \\emph{geometric $k$-colored crossing number} of complete graphs $\\overline{\\overline{\\text{cr}}}_k(K_n)$, which is the smallest number of monochromatic crossings in any $k$-edge colored straight-line drawing of $K_n$.\n  We substantially improve asymptotic upper bounds on $\\overline{\\overline{\\text{cr}}}_k(K_n)$ for $k=2,\\ldots, 10$ by developing a procedure for general $k$ that derives $k$-edge colored drawings of $K_n$ for arbitrarily large $n$ from initial drawings with a low number of monochromatic crossings.\n  We obtain the latter by heuristic search, employing a \\textsc{MAX-$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2505.18014","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/2505.18014/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"}