{"paper":{"title":"On maximum matchings in almost regular graphs","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Petros A. Petrosyan","submitted_at":"2012-02-03T12:39:14Z","abstract_excerpt":"In 2010, Mkrtchyan, Petrosyan and Vardanyan proved that every graph $G$ with $2\\leq \\delta(G)\\leq \\Delta(G)\\leq 3$ contains a maximum matching whose unsaturated vertices do not have a common neighbor, where $\\Delta(G)$ and $\\delta(G)$ denote the maximum and minimum degrees of vertices in $G$, respectively. In the same paper they suggested the following conjecture: every graph $G$ with $\\Delta(G)-\\delta(G)\\leq 1$ contains a maximum matching whose unsaturated vertices do not have a common neighbor. Recently, Picouleau disproved this conjecture by constructing a bipartite counterexample $G$ with "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0681","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"}