{"paper":{"title":"Strong binding numbers and factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guantao Chen, Hein van der Holst, Jennifer Vandenbussche, Mikhail Lavrov, Yuying Ma","submitted_at":"2025-08-25T23:12:48Z","abstract_excerpt":"Let $G$ be a simple graph. The $k$-th neighborhood of a vertex subset $S \\subseteq V(G)$, denoted $\\Lambda^k(S)$, is the set of vertices that are adjacent to at least $k$ vertices in $S$. The $k$-th binding number $\\beta^k(G)$ is defined as the minimum ratio $|\\Lambda^k(S)|/|S|$ over all subsets $S \\subseteq V(G)$ with $|S| \\ge k$ and $\\Lambda^k(S) \\ne V(G)$. This parameter generalizes the classical binding number introduced by Woodall. Andersen showed that the condition $\\beta^1(G) \\ge 1$ does not guarantee the existence of a $1$-factor in $G$, while Bar\\'at et al. proved that $\\beta^2(G) \\ge"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2508.18555","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2508.18555/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"}