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Percolation with Small Clusters on Random Graphs

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Abstract

Consider the problem of determining the maximal induced subgraph in a random d-regular graph such that its components remain bounded as the size of the graph becomes arbitrarily large. We show, for asymptotically large d, that any such induced subgraph has size density at most \(2(\log d)/d\) with high probability. A matching lower bound is known for independent sets. We also prove the analogous result for sparse Erdős–Rényi graphs.

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Acknowledgments

The author thanks Bálint Virág for suggesting the problem.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Mustazee Rahman

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  1. Mustazee Rahman
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Correspondence to Mustazee Rahman.

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The author’s research was supported by an NSERC CGS grant.

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Rahman, M. Percolation with Small Clusters on Random Graphs. Graphs and Combinatorics 32, 1167–1185 (2016). https://doi.org/10.1007/s00373-015-1628-0

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  • DOI: https://doi.org/10.1007/s00373-015-1628-0

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