{"paper":{"title":"On the Hamilton-Waterloo Problem with odd orders","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Burgess, P. Danziger, T. Traetta","submitted_at":"2015-10-23T22:27:41Z","abstract_excerpt":"Given non-negative integers $v, m, n, \\alpha, \\beta$, the Hamilton-Waterloo problem asks for a factorization of the complete graph $K_v$ into $\\alpha$ $C_m$-factors and $\\beta$ $C_n$-factors. Clearly, $v$ odd, $n,m\\geq 3$, $m\\mid v$, $n\\mid v$ and $\\alpha+\\beta = (v-1)/2$ are necessary conditions. To date results have only been found for specific values of $m$ and $n$. In this paper we show that for any $m$ and $n$ the necessary conditions are sufficient when $v$ is a multiple of $mn$ and $v>mn$, except possibly when $\\beta=1$ or 3, with five additional possible exceptions in $(m,n,\\beta)$. Fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07079","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":""},"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"}