{"paper":{"title":"Tight lower bounds on the matching number in a graph with given maximum degree","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anders Yeo, Michael A. Henning","submitted_at":"2016-04-18T07:37:27Z","abstract_excerpt":"Let $k \\geq 3$. We prove the following three bounds for the matching number, $\\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$.\n If $k$ is odd, then $\\alpha'(G) \\ge \\left( \\frac{k-1}{k(k^2 - 3)} \\right) n \\, + \\, \\left( \\frac{k^2 - k - 2}{k(k^2 - 3)} \\right) m \\, - \\, \\frac{k-1}{k(k^2 - 3)}$. If $k$ is even, then $\\alpha'(G) \\ge \\frac{n}{k(k+1)} \\, + \\, \\frac{m}{k+1} - \\frac{1}{k}$. If $k$ is even, then $\\alpha'(G) \\ge \\left( \\frac{k+2}{k^2+k+2} \\right) m \\, - \\, \\left( \\frac{k-2}{k^2+k+2} \\right) n \\, - \\frac{k+2}{k^2+k+2}$.\n  In this paper we actually prove "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.05020","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"}