Asymptotically Almost Every 2r-Regular Graph Has an Internal Partition

Abstract

An internal partition of a graph \(G=(V,E)\) is a partitioning of V into two parts such that every vertex has at least a half of its neighbors on its own side. We prove that for every positive integer r, asymptotically almost every 2r-regular graph has an internal partition. Whereas previous results in this area apply only to a small fraction of all 2r-regular graphs, ours applies to almost all of them.

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Notes

  1. 1.

    In Sect. 3 we present a polynomial time algorithm that finds a partition whose existence is stated in Theorem 2.2. For algorithmic purposes we cannot consider a globally optimal \(A\in {\mathcal {F}}\) as described here, but as we show below, a properly chosen locally optimal version of such A will do.

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Acknowledgements

We thank Amir Ban, David Eisenberg, David Louis, Zur Luria, Jonathan Mosheiff, and Yuval Peled for careful reading and useful comments.

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Correspondence to Sria Louis.

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N. Linial: Supported by ISF Grant 1169/14, Local and Global Combinatorics.

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Linial, N., Louis, S. Asymptotically Almost Every 2r-Regular Graph Has an Internal Partition. Graphs and Combinatorics 36, 41–50 (2020). https://doi.org/10.1007/s00373-019-02116-0

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Keywords

  • Graph partitions
  • Internal partition
  • Satisfactory partition
  • Asymptotic
  • Vertex degree
  • Optimization