{"paper":{"title":"Sufficient conditions for the existence of path-factors with given properties","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guowei Dai, Hui Qin, Ting Jin, Yuan Chen, Yuan Yuan","submitted_at":"2023-05-08T13:55:36Z","abstract_excerpt":"A spanning subgraph $H$ of a graph $G$ is called a $P_{\\geq k}$-factor of $G$ if every component of $H$ is isomorphic to a path of order at least $k$, where $k\\geq2$ is an integer. A graph $G$ is called a $(P_{\\geq k},l)$-factor critical graph if $G-V'$ contains a $P_{\\geq k}$-factor for any $V'\\subseteq V(G)$ with $|V'|=l$. A graph $G$ is called a $(P_{\\geq k},m)$-factor deleted graph if $G-E'$ has a $P_{\\geq k}$-factor for any $E'\\subseteq E(G)$ with $|E'|=m$. Intuitively, if a graph is dense enough, it will have a $P_{\\geq 3}$-factor. In this paper, we give some sufficient conditions for a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.04713","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2305.04713/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}