{"paper":{"title":"Partition of a Subset into Two Directed Cycles with Partial Degrees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Wang","submitted_at":"2019-07-26T16:33:57Z","abstract_excerpt":"Let $D=(V,A)$ be a directed graph of order $n\\geq 6$. Let $W$ be a subset of $V$ with $|W|\\geq 6$. Suppose that every vertex of $W$ has degree at least $(3n-3)/2$ in $D$. Then for any integer partition $|W|=n_1+n_2$ with $n_1\\geq 3$ and $n_2\\geq 3$, $D$ contains two disjoint directed cycles $C_1$ and $C_2$ such that $|V(C_1)\\cap W|=n_1$ and $|V(C_2)\\cap W|=n_2$. We conjecture that for any integer partition $|W|=n_1+n_2+\\cdots +n_k$ with $k\\geq 3$ and $n_i\\geq 3(1\\leq i\\leq k)$, $D$ contains $k$ disjoint directed cycles $C_1,C_2,\\ldots , C_k$ such that $|V(C_i)\\cap W|=n_i$ for all $1\\leq i\\leq "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.11668","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"}