Judicious partitions of uniform hypergraphs

Abstract

The vertices of any graph with m edges may be partitioned into two parts so that each part meets at least \(\tfrac{{2m}} {3}\) edges. Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges may likewise be partitioned into r classes such that each part meets at least \(\tfrac{r} {{2r - 1}}\) edges. In this paper we prove the weaker statement that, for each r ≥ 4, a partition into r classes may be found in which each class meets at least \(\tfrac{r} {{3r - 4}}\) edges, a substantial improvement on previous bounds.

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References

  1. [1]

    N. Alon and E. Halperin: Bipartite subgraphs of integer weighted graphs, Discrete Math. 181 (1998), 19–29.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    B. Bollobás: Modern Graph Theory, Graduate Texts in Mathematics 184, Springer-Verlag, New York, 1998. xiv+394 pp.

    Google Scholar 

  3. [3]

    B. Bollobás, B. Reed and A. Thomason: An extremal function for the achromatic number, in: Graph Structure Theory (Seattle, WA, 1991), 161–165, Contemp. Math. 147, Amer. Math. Soc., Providence, RI, 1993.

    Google Scholar 

  4. [4]

    B. Bollobás and A.D. Scott: On judicious partitions of graphs, Period. Math. Hungar. 26 (1993), 127–139.

    Article  Google Scholar 

  5. [5]

    B. Bollobás and A.D. Scott: Judicious partitions of hypergraphs, J. Combin. Theory Ser. A 78 (1997), 15–31.

    Article  MATH  MathSciNet  Google Scholar 

  6. [6]

    B. Bollobás and A. D. Scott: Exact bounds for judicious partitions of graphs, Combinatorica 19 (1999), 473–486.

    Article  MATH  MathSciNet  Google Scholar 

  7. [7]

    B. Bollobás and A.D. Scott: Judicious partitions of 3-uniform hypergraphs, European J. Combin. 21 (2000), 289–300.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    B. Bollobás and A.D. Scott: Problems and results on judicious partitions, Random Structures Algorithms 21 (2002), 414–430.

    Article  MATH  MathSciNet  Google Scholar 

  9. [9]

    B. Bollobás and A.D. Scott: Better bounds for Max Cut, in: Contemporary Combinatorics, Bolyai Soc. Math. Stud. 10 (2002), 185–246.

    Google Scholar 

  10. [10]

    J. Haslegrave: The Bollobás-Thomason conjecture for 3-uniform hypergraphs, Combinatorica 32 (2012), 451–471.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    T. D. Porter: On a bottleneck bipartition conjecture of Erdős, Combinatorica 12 (1992), 317–321.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    T. D. Porter and Bing Yang: Graph partitions II, J. Combin. Math. Combin. Comput. 37 (2001), 149–158.

    MATH  MathSciNet  Google Scholar 

  13. [13]

    A. D. Scott: Judicious partitions and related problems, in: Surveys in Combinatorics 2005, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005, 95–117.

    Google Scholar 

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Correspondence to John Haslegrave.

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Research supported by the Engineering and Physical Sciences Research Council.

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Haslegrave, J. Judicious partitions of uniform hypergraphs. Combinatorica 34, 561–572 (2014). https://doi.org/10.1007/s00493-014-2916-7

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Mathematics Subject Classification (2000)

  • 05C65