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Maximum independent sets on random regular graphs

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Acta Mathematica

Abstract

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 \({n\alpha_*-c_*\log n}\) for explicit constants \({\alpha_*(d)}\) and \({c_*(d)}\). 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.

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Authors and Affiliations

  1. Statistics Department, University of Chicago, 5747 South Ellis Avenue, Chicago, IL, 60637, U.S.A.

    Jian Ding

  2. Department of Statistics, University of California, Berkeley, 367 Evans Hall, Berkeley, CA, 94720, U.S.A.

    Allan Sly

  3. Mathematical Sciences Institute, Australian National University, John Dedman Building 27, Union Lane, Canberra, ACT, 0200, Australia

    Allan Sly

  4. Department of Statistics, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA, 94305, U.S.A.

    Nike Sun

Authors
  1. Jian Ding
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  2. Allan Sly
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  3. Nike Sun
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Corresponding author

Correspondence to Jian Ding.

Additional information

Research supported by NSF grant DMS-1313596 (J. D.), Sloan Research Fellowship (A. S.), NDSEG and NSF GRF (N. S.).

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Ding, J., Sly, A. & Sun, N. Maximum independent sets on random regular graphs. Acta Math 217, 263–340 (2016). https://doi.org/10.1007/s11511-017-0145-9

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  • DOI: https://doi.org/10.1007/s11511-017-0145-9

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