{"paper":{"title":"Decomposition of a complete bipartite multigraph into arbitrary cycle sizes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"James Hammer, Joe Chaffee, John Asplund","submitted_at":"2016-08-19T21:52:01Z","abstract_excerpt":"In a graph $G$, let $\\mu_G(xy)$ denote the number of edges between $x$ and $y$ in $G$. Let $\\lambda K_{v,u}$ be the graph $(V\\cup U,E)$ with $|V|=v$, $|U|=u$, and \\[ \\mu_G(xy)=\\begin{cases} \\lambda &\\mbox{if $x\\in U$ and $y\\in V$ or if $x\\in V$ and $y\\in U$}\\\\ 0 &\\mbox{otherwise.} \\\\ \\end{cases} \\] Let $M$ be a sequence of non-negative integers $m_1,m_2,\\ldots,m_n$. An $(M)$-cycle decomposition of a graph $G$ is a partition of the edge set into cycles of lengths $m_1,m_2,\\ldots,m_n$. In this paper, we establish necessary and sufficient conditions for the existence of an $(M)$-cycle decompositi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05744","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"}