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Maximum independent sets on random regular graphs. (English) Zbl 1371.05261

Summary: We determine the asymptotics of the independence number of the random \(d\)-regular graph for all \({d\geq d_0}\). It is highly concentrated, with constant-order fluctuations around \({n\alpha_\ast-c_\ast\log n}\) for explicit constants \({\alpha_\ast(d)}\) and \({c_\ast(d)}\). Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs.

MSC:

05C80 Random graphs (graph-theoretic aspects)
05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
05C35 Extremal problems in graph theory
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