{"paper":{"title":"On the Concentration of the Domination Number of the Random Graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Anita Liebenau, Roman Glebov, Tibor Szab\\'o","submitted_at":"2012-09-14T07:57:41Z","abstract_excerpt":"In this paper we study the behaviour of the domination number of the Erd\\H{o}s-R\\'enyi random graph $\\mathcal{G}(n,p)$. Extending a result of Wieland and Godbole we show that the domination number of $\\mathcal{G}(n,p)$ is equal to one of two values asymptotically almost surely whenever $p \\gg \\frac{\\ln^2n}{\\sqrt{n}}$. The explicit values are exactly at the first moment threshold, that is where the expected number of dominating sets starts to tend to infinity. For small $p$ we also provide various non-concentration results which indicate why some sort of lower bound on the probability $p$ is ne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3115","kind":"arxiv","version":4},"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"}