{"paper":{"title":"The density of $k$-cacti via excluding minors","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Licheng Zhang, Yuanqiu Huang","submitted_at":"2026-06-04T15:36:57Z","abstract_excerpt":"A \\emph{$k$-cactus} generalizes forests and cacti by allowing each edge to lie on at most $k$ cycles. The maximum number of edges is classical for forests and cacti, but for $k$-cacti was known only for $k\\le 4$. In this note we treat general $k$. The key idea is that bounding the cycles through each edge forces a $k$-cactus to exclude a large complete minor; in particular, the class of $k$-cacti is minor-closed. From this we prove that every $n$-vertex $k$-cactus has $O\\!\\left(\\frac{\\log k}{\\sqrt{\\log\\log k}}\\,n\\right)$ edges for all sufficiently large $k$, and a construction shows this is op"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.06298","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/2606.06298/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"}