The Bollobás-Thomason conjecture for 3-uniform hypergraphs

Abstract

The vertices of any graph with m edges can be partitioned into two parts so that each part meets at least \(\frac{{2m}} {3}\) edges. Bollobás and Thomason conjectured that the vertices of any r-uniform graph may be likewise partitioned into r classes such that each part meets at least cm edges, with \(\frac{r} {{2r - 1}}\). In this paper, we prove this conjecture for the case r=3. In the course of the proof we shall also prove an extension of the graph case which was conjectured by Bollobás and Scott.

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References

  1. [1]

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

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

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

    MATH  Book  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.

  4. [4]

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

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

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

    MathSciNet  MATH  Article  Google Scholar 

  6. [6]

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

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

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

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

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

    Google Scholar 

  9. [9]

    J. Haslegrave: Judicious partitions of uniform hypergraphs, preprint.

  10. [10]

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

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

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

    MathSciNet  MATH  Google Scholar 

  12. [12]

    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. The Bollobás-Thomason conjecture for 3-uniform hypergraphs. Combinatorica 32, 451–471 (2012). https://doi.org/10.1007/s00493-012-2696-x

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Mathematics Subject Classification (2000)

  • 05C35
  • 05C65