{"paper":{"title":"Extremal Cuts of Sparse Random Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"math.PR","authors_text":"Amir Dembo, Andrea Montanari, Subhabrata Sen","submitted_at":"2015-03-13T01:22:57Z","abstract_excerpt":"For Erd\\H{o}s-R\\'enyi random graphs with average degree $\\gamma$, and uniformly random $\\gamma$-regular graph on $n$ vertices, we prove that with high probability the size of both the Max-Cut and maximum bisection are $n\\Big(\\frac{\\gamma}{4} + {{\\sf P}}_* \\sqrt{\\frac{\\gamma}{4}} + o(\\sqrt{\\gamma})\\Big) + o(n)$ while the size of the minimum bisection is $n\\Big(\\frac{\\gamma}{4}-{{\\sf P}}_*\\sqrt{\\frac{\\gamma}{4}} + o(\\sqrt{\\gamma})\\Big) + o(n)$. Our derivation relates the free energy of the anti-ferromagnetic Ising model on such graphs to that of the Sherrington-Kirkpatrick model, with ${{\\sf P}}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03923","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"}