{"paper":{"title":"Sufficient conditions for the existence of a path-factor which are related to odd components","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Kenta Ozeki, Michitaka Furuya, Yoshimi Egawa","submitted_at":"2017-05-24T03:24:06Z","abstract_excerpt":"In this paper, we are concerned with sufficient conditions for the existence of a $\\{P_{2},P_{2k+1}\\}$-factor. We prove that for $k\\geq 3$, there exists $\\varepsilon_{k}>0$ such that if a graph $G$ satisfies $\\sum_{0\\leq j\\leq k-1}c_{2j+1}(G-X)\\leq \\varepsilon_{k}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{2k+1}\\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$. On the other hand, we construct infinitely many graphs $G$ having no $\\{P_{2},P_{2k+1}\\}$-factor such that $\\sum_{0\\leq j\\leq k-1}c_{2j+1}(G-X)\\leq \\frac{32k+141}{72k-78}|X|$ for all "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.08592","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"}