{"paper":{"title":"On decomposition thresholds for odd-length cycles and other tripartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Barbara Maenhaut, Daniel Horsley, Darryn Bryant, Peter Dukes, Richard Montgomery","submitted_at":"2024-11-26T08:54:32Z","abstract_excerpt":"An (edge) decomposition of a graph $G$ is a set of subgraphs of $G$ whose edge sets partition the edge set of $G$. Here we show, for each odd $\\ell \\geq 5$, that any graph $G$ of sufficiently large order $n$ with minimum degree at least $(\\frac{1}{2}+\\frac{1}{2\\ell-4}+o(1))n$ has a decomposition into $\\ell$-cycles if and only if $\\ell$ divides $|E(G)|$ and each vertex of $G$ has even degree. This threshold cannot be improved beyond $\\frac{1}{2}+\\frac{1}{2\\ell-2}$. It was previously shown that the thresholds approach $\\frac{1}{2}$ as $\\ell$ becomes large, but our thresholds do so significantly "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17232","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/2411.17232/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"}