, Volume 36, Issue 5, pp 537–556 | Cite as

On judicious bipartitions of graphs



For a positive integer m, let f(m) be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let g(m) be the minimum value s such that for any graph G with m edges there exists a bipartition V (G)=V1V2 such that G has at most s edges with both incident vertices in Vi. Alon proved that the limsup of \(f\left( m \right) - \left( {m/2 + \sqrt {m/8} } \right)\) tends to infinity as m tends to infinity, establishing a conjecture of Erdős. Bollobás and Scott proposed the following judicious version of Erdős' conjecture: the limsup of \(m/4 + \left( {\sqrt {m/32} - g(m)} \right)\) tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to k-partitions for all even integers k. On the other hand, we generalize Alon's result to multi-partitions, which should be useful for generalizing the above Bollobás-Scott conjecture to k-partitions for odd integers k.

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Alon: Bipartite subgraphs, Combinatorica 16 (1996), 301–311.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    N. Alon, B. Bollobás, M. Krivelevich, and B. Sudakov: Maximum cuts and judicious partitions in graphs without short cycles, J. Combin. Theory Ser. B 88 (2003), 329–346.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    N. Alon and E. Halperin: Bipartite subgraphs of integer weighted graphs, Discrete Math. 181 (1998), 19–29.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    P. Berman and M. Karpinski: On some tighter inapproximability results, (extended abstract) Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 1644 (1999), 200–209.MathSciNetCrossRefGoogle Scholar
  5. [5]
    B. Bollobás and A. D. Scott: On judicious partitions, Period. Math. Hungar. 26 (1993), 127–139.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    B. Bollobás and A. D. Scott: Exact bounds for judicious partitions of graphs, Combinatorica 19 (1999), 473–486.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    B. Bollobás and A. D. Scott: Problems and results on judicious partitions, Random Structures and Algorithms 21 (2002), 414–430.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    B. Bollobás and A. D. Scott: Better bounds for Max Cut, in Contemporary Comb, Bolyai Soc Math Stud 10, Janos Bolyai Math Soc, Budapest (2002), 185–246.Google Scholar
  9. [9]
    B. Bollobás and A. D. Scott: Judicious partitions of bounded-degree graphs, J. Graph Theory 46 (2004), 131–143.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    B. Bollobás and A. D. Scott: Max k-cut and judicious k-partitions, Discrete Math. 310 (2010), 2126–2139.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    C. S. Edwards: Some extremal properties of bipartite graphs, Canadian J. math. 25 (1973), 475–485.CrossRefMATHGoogle Scholar
  12. [12]
    C. S. Edwards: An improved lower bound for the number of edges in a largest bipartite subgraph, In: Proc. 2nd Czechoslovak Symposium on Graph Theory, Prague (1975), 167–181.Google Scholar
  13. [13]
    P. Erdős: Some recent problems in Combinatorics and Graph Theory, Proc. 26 th Southeastern International Conference on Graph Theory, Combinatorics and Com- puting, Boca Raton, 1995, Congressus Numerantium.Google Scholar
  14. [14]
    U. Feige, M. Karpinski, and M. Langberg: Improved approximation of max-cut on graphs of bounded degree, J. Algorithms 43 (2002), 201–219.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    M. X. Goemans and D. P. Williamson: Improved approximation algorithms for maximum cut and satisfiability using semidefinite programming, J. ACM 42 (1995), 1115–1145.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    R. M. Karp: Reducibility among combinatorial problems, in Complexity of Computer Computations, (R. Miller and J. Thatcher, eds) Plenum Press, New York, (1972), 85–103.CrossRefGoogle Scholar
  17. [17]
    A. Scott: Judicious partitions and related problems, in Surveys in Combinatorics (2005), 95–117, London Math. Soc. Lecture Note Ser., 327, Cambridge Univ. Press, Cambridge, 2005.Google Scholar
  18. [18]
    B. Xu and X. Yu: Better bounds for k-partitions of graphs, Combinatorics, Probability and Computing 20 (2011), 631–640.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    M. Yannakakis: Node- and Edge-Deletion NP-Complete Problems, STOC (1978), 253–264.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefei, AnhuiChina
  2. 2.School of MathematicsGeorgia Institute of Technology AtlantaGeorgiaUSA

Personalised recommendations