{"paper":{"title":"Path-factors involving paths of order seven and nine","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michitaka Furuya, Yoshimi Egawa","submitted_at":"2015-07-25T04:43:56Z","abstract_excerpt":"In this paper, we show the following two theorems (here $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$): (i)~If a graph $G$ satisfies $c_{1}(G-X)+\\frac{1}{3}c_{3}(G-X)+\\frac{1}{3}c_{5}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{7}\\}$-factor. (ii)~If a graph $G$ satisfies $c_{1}(G-X)+c_{3}(G-X)+\\frac{2}{3}c_{5}(G-X)+\\frac{1}{3}c_{7}(G-X)\\leq \\frac{2}{3}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{9}\\}$-factor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.07066","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"}