{"paper":{"title":"Improved Decomposition Bounds for Partition Polytopes and Odd-Covers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Abigail Nix, Bryce Frederickson, Steffen Borgwardt, Youngho Yoo, Zden\\v{e}k Dvo\\v{r}\\'ak","submitted_at":"2025-07-17T03:05:23Z","abstract_excerpt":"The assignments of a set of $m$ items into $n$ clusters of prescribed sizes $k_1,\\dots,k_n$ can be encoded as the vertices of the partition polytope $\\mathrm{PP}(k_1,\\dots,k_n)$. We prove that, if $K = \\max\\{k_1,\\dots,k_n\\}$, then the combinatorial diameter of $\\mathrm{PP}(k_1,\\dots,k_n)$ is at most $\\lceil 3K/2\\rceil$. This improves the previously known upper bound of $2K$.\n  A cycle (or path) odd-cover of a graph $G$ is a set of cycles (or paths) with symmetric difference $G$. We prove that every Eulerian graph $G$ with maximum degree $\\Delta$ admits a cycle odd-cover and a path odd-cover, e"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.12748","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/2507.12748/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"}