{"paper":{"title":"The Euclidean MST-ratio for Bi-colored Lattices","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.CG","authors_text":"Herbert Edelsbrunner, Morteza Saghafian, Ond\\v{r}ej Draganov, Sebastiano Cultrera di Montesano","submitted_at":"2024-03-15T11:10:50Z","abstract_excerpt":"Given a finite set, $A \\subseteq \\mathbb{R}^2$, and a subset, $B \\subseteq A$, the \\emph{MST-ratio} is the combined length of the minimum spanning trees of $B$ and $A \\setminus B$ divided by the length of the minimum spanning tree of $A$.\n  The question of the supremum, over all sets $A$, of the maximum, over all subsets $B$, is related to the Steiner ratio, and we prove this sup-max is between $2.154$ and $2.427$.\n  Restricting ourselves to $2$-dimensional lattices, we prove that the sup-max is $2.0$, while the inf-max is $1.25$.\n  By some margin the most difficult of these results is the upp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.10204","kind":"arxiv","version":2},"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/2403.10204/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"}