Bounds for the number of meeting edges in graph partitioning



Let G be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that G admits a bipartition such that each vertex class meets edges of total weight at least (w1−Δ1)/2+2w2/3, where wi is the total weight of edges of size i and Δ1 is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph G (i.e., multi-hypergraph), we show that there exists a bipartition of G such that each vertex class meets edges of total weight at least (w0−1)/6+(w1−Δ1)/3+2w2/3, where w0 is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with m edges, except for K2 and K1,3, admits a tripartition such that each vertex class meets at least [2m/5] edges, which establishes a special case of a more general conjecture of Bollobás and Scott.


graph weighted hypergraph partition judicious partition 

MSC 2010

05C35 05C75 


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  1. [1]
    N. Alon, B. Bollobás, M. Krivelevich, B. Sudakov: Maximum cuts and judicious partitions in graphs without short cycles. J. Comb. Theory, Ser. B 88 (2003), 329–346.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    B. Bollobás, A. D. Scott: Exact bounds for judicious partitions of graphs. Combinatorica 19 (1999), 473–486.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    B. Bollobás, A. D. Scott: Judicious partitions of 3-uniform hypergraphs. Eur. J. Comb. 21 (2000), 289–300.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    B. Bollobás, A. D. Scott: Better bounds for Max Cut. Contemporary Combinatorics (B. Bollobás, ed.). Bolyai Soc. Math. Stud. 10, János Bolyai Math. Soc., Budapest, 2002, pp. 185–246.Google Scholar
  5. [5]
    B. Bollobás, A. D. Scott: Problems and results on judicious partitions. Random Struct. Algorithms 21 (2002), 414–430.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    C. S. Edwards: Some extremal properties of bipartite subgraphs. Can. J. Math. 25 (1973), 475–485.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    C. S. Edwards: An improved lower bound for the number of edges in a largest bipartite subgraph. Recent Advances in Graph Theory. Proc. Symp., Praha 1974, Academia, Praha, 1975, 167–181.Google Scholar
  8. [8]
    G. Fan, J. Hou: Bounds for pairs in judicious partitions of graphs. Random Struct. Algorithms 50 (2017), 59–70.CrossRefMATHGoogle Scholar
  9. [9]
    G. Fan, J. Hou, Q. Zeng: A bound for judicious k-partitions of graphs. Discrete Appl. Math. 179 (2014), 86–99.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    J. Haslegrave: The Bollobás-Thomason conjecture for 3-uniform hypergraphs. Combinatorica 32 (2012), 451–471.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. Hou, S. Wu, G. Yan: On judicious partitions of uniform hypergraphs. J. Comb. Theory Ser. A 141 (2016), 16–32.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    J. Hou, Q. Zeng: Judicious partitioning of hypergraphs with edges of size at most 2. Comb. Probab. Comput. 26 (2017), 267–284.MathSciNetCrossRefGoogle Scholar
  13. [13]
    M. Liu, B. Xu: On judicious partitions of graphs. J. Comb. Optim. 31 (2016), 1383–1398.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    J. Ma, P.-L. Yen, X. Yu: On several partitioning problems of Bollobás and Scott. J. Comb. Theory, Ser. B 100 (2010), 631–649.CrossRefMATHGoogle Scholar
  15. [15]
    J. Ma, X. Yu: On judicious bipartitions of graphs. Combinatorica 36 (2016), 537–556.MathSciNetCrossRefGoogle Scholar
  16. [16]
    A. Scott: Judicious partitions and related problems. Surveys in Combinatorics (B. Webb, ed.). Conf. Proc., Durham, 2005, London Mathematical Society Lecture Note Series 327, Cambridge University Press, Cambridge, 2005, pp. 95–117.Google Scholar
  17. [17]
    X. Xu, G. Yan, Y. Zhang: Judicious partitions of weighted hypergraphs. Sci. China, Math. 59 (2016), 609–616.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    B. Xu, X. Yu: Judicious k-partitions of graphs. J. Comb. Theory, Ser. B 99 (2009), 324–337.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    B. Xu, X. Yu: Better bounds for k-partitions of graphs. Comb. Probab. Comput. 20 (2011), 631–640.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Center for Discrete MathematicsFuzhou UniversityFuzhou, FujianP.R. China

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