{"paper":{"title":"The Tutte's condition in terms of graph factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David G.L. Wang, Hongliang Lu","submitted_at":"2018-06-25T10:08:35Z","abstract_excerpt":"Let $G$ be a connected general graph of even order, with a function $f\\colon V(G)\\to\\Z^+$. We obtain that $G$ satisfies the Tutte's condition \\[ o(G-S)\\le \\sum_{v\\in S}f(v)\\qquad\\text{for any nonempty set $S\\subset V(G)$}, \\] with respect to $f$ if and only if $G$ contains an $H$-factor for any function $H\\colon V(G)\\to 2^\\N$ such that $H(v)\\in \\{J_f(v),\\,J_f^+(v)\\}$ for each $v\\in V(G)$, where the set $J_f(v)$ consists of the integer $f(v)$ and all positive odd integers less than $f(v)$, and the set $J^+_f(v)$ consists of positive odd integers less than or equal to $f(v)+1$. We also obtain a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.09357","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"}