{"paper":{"title":"On the maximum number of maximum independent sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dieter Rautenbach, Elena Mohr","submitted_at":"2018-05-07T13:30:11Z","abstract_excerpt":"We give a very short and simple proof of Zykov's generalization of Tur\\'{a}n's theorem, which implies that the number of maximum independent sets of a graph of order $n$ and independence number $\\alpha$ with $\\alpha<n$ is at most $\\left\\lceil\\frac{n}{\\alpha}\\right\\rceil^{n\\,{\\rm mod}\\,\\alpha} \\left\\lfloor\\frac{n}{\\alpha}\\right\\rfloor^{\\alpha-(n\\,{\\rm mod}\\,\\alpha)}$. Generalizing a result of Zito, we show that the number of maximum independent sets of a tree of order $n$ and independence number $\\alpha$ is at most $2^{n-\\alpha-1}+1$, if $2\\alpha=n$, and, $2^{n-\\alpha-1}$, if $2\\alpha>n$, and w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.02519","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"}