{"paper":{"title":"Characterization of 1-Tough Graphs using Factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"H. Lu, M. Kano","submitted_at":"2017-02-20T06:30:18Z","abstract_excerpt":"For a graph $G$, let $odd(G)$ and $\\omega(G)$ denote the number of odd components and the number of components of $G$, respectively. Then it is well-known that $G$ has a 1-factor if and only if $odd(G-S)\\le |S|$ for all $S\\subset V(G)$. Also it is clear that $odd(G-S) \\le \\omega(G-S)$. In this paper we characterize a 1-tough graph $G$, which satisfies $\\omega(G-S) \\le |S|$ for all $\\emptyset \\ne S \\subset V(G)$, using an $H$-factor of a set-valued function $H:V(G) \\to \\{ \\{1\\}, \\{0,2\\} \\}$. Moreover, we generalize this characterization to a graph that satisfies $\\omega(G-S) \\le f(S)$ for all $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.05873","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":""},"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"}