{"paper":{"title":"On the Hamilton-Waterloo Problem with cycle lengths of distinct parities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Andrea Burgess, Peter Danziger, Tommaso Traetta","submitted_at":"2018-01-23T16:22:56Z","abstract_excerpt":"Let $K_v^*$ denote the complete graph $K_v$ if $v$ is odd and $K_v-I$, the complete graph with the edges of a 1-factor removed, if $v$ is even. Given non-negative integers $v, M, N, \\alpha, \\beta$, the Hamilton-Waterloo problem asks for a $2$-factorization of $K^*_v$ into $\\alpha$ $C_M$-factors and $\\beta$ $C_N$-factors. Clearly, $M,N\\geq 3$, $M\\mid v$, $N\\mid v$ and $\\alpha+\\beta = \\lfloor\\frac{v-1}{2}\\rfloor$ are necessary conditions.\n  Very little is known on the case where $M$ and $N$ have different parities. In this paper, we make some progress on this case by showing, among other things,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07638","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":""},"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"}