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


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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  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.


  1. 1.

    Alon, N.: On the edge-expansion of graphs. Comb. Probab. Comput. 6(2), 145–152 (1997)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ban, A., Linial, N.: Internal partitions of regular graphs. J. Graph Theory 83(1), 5–18 (2016)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Ban, A.: Decomposing weighted graphs. J. Graph Theory 86(2), 250–254 (2017)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bazgan, C., Tuza, Z., Vanderpooten, D.: On the existence and determination of satisfactory partitions in a graph. In: ISAAC, pp. 444–453 (2003)

  5. 5.

    Bazgan, C., Tuza, Z., Vanderpooten, D.: Complexity and approximation of satisfactory partition problems. Lect. Notes Comput. Sci. 3595, 829 (2005)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bazgan, C., Tuza, Z., Vanderpooten, D.: The satisfactory partition problem. Discrete Appl. Math. 154(8), 1236–1245 (2006)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bazgan, C., Tuza, Z., Vanderpooten, D.: Satisfactory graph partition, variants, and generalizations. Eur. J. Oper. Res. 206(2), 271–280 (2010)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chvátal, V.: Recognizing decomposable graphs. J. Graph Theory 8(1), 51–53 (1984)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Cizma, D., Linial, N.: Private Communication (2019)

  10. 10.

    Diwan, A.A.: Decomposing graphs with girth at least five under degree constraints. J. Graph Theory 33(4), 237–239 (2000)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Eroh, L., Gera, R.: Global Alliance Partitions in Graphs, Paper presented at the annual meeting of the Mathematical Association of America, The Fairmont Hotel, San Jose, CA, Aug 03, 2007 (2008)

  12. 12.

    Fernau, H., Rodriguez-Velazquez, J.A.: A survey on alliances and related parameters in graphs. Electron. J. Graph Theory Appl. (EJGTA) 2(1), 70–86 (2014)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Gerber, M.U., Kobler, D.: Algorithmic approach to the satisfactory graph partitioning problem. Eur. J. Oper. Res. 125(2), 283–291 (2000)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Gerber, M.U., Kobler, D.: Algorithms for vertex-partitioning problems on graphs with fixed clique-width. Theor. Comput. Sci. 299(1–3), 719–734 (2003)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Gerber, M.U., Kobler, D.: Classes of graphs that can be partitioned to satisfy all their vertices. Australas. J. Comb. 29, 201–214 (2004)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Kaneko, A.: On decomposition of triangle-free graphs under degree constraints. J. Graph Theory 27(1), 7–9 (1998)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Liu, M., Xu, B.: On partitions of graphs under degree constraints. Discrete Appl. Math. 226, 87–93 (2017)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Ma, J., Yang, T.: Decomposing \(C_4\)-free graphs under degree constraints. J. Graph Theory 90, 13–23 (2017)

    Article  Google Scholar 

  19. 19.

    Morris, S.: Contagion. Rev. Econ. Stud. 67(1), 57–78 (2000)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Shafique, K.H., Dutton, R.D.: On satisfactory partitioning of graphs. Congressus Numerantium 154, 183–194 (2002)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Stiebitz, M.: Decomposing graphs under degree constraints. J. Graph Theory 23(3), 321–324 (1996)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Schweser, T., Stiebitz, M.: Partitions of multigraphs under degree constraints. (2017) arXiv preprint arXiv:1703.08502

  23. 23.

    Thomassen, C.: Graph decomposition with constraints on the connectivity and minimum degree. J. Graph Theory 7(2), 165–167 (1983)

    MathSciNet  Article  Google Scholar 

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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).

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