On judicious bipartitions of graphs

Abstract

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)=V 1V 2 such that G has at most s edges with both incident vertices in V i . 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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon: Bipartite subgraphs, Combinatorica 16 (1996), 301–311.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

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

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  Google Scholar 

  5. [5]

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

    MathSciNet  Article  MATH  Google Scholar 

  6. [6]

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

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

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

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  10. [10]

    B. Bollobás and A. D. Scott: Max k-cut and judicious k-partitions, Discrete Math. 310 (2010), 2126–2139.

    MathSciNet  Article  MATH  Google Scholar 

  11. [11]

    C. S. Edwards: Some extremal properties of bipartite graphs, Canadian J. math. 25 (1973), 475–485.

    Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  19. [19]

    M. Yannakakis: Node- and Edge-Deletion NP-Complete Problems, STOC (1978), 253–264.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jie Ma.

Additional information

Partially supported by NSFC project 11501539.

Partially supported by NSF grants DMS-1265564 and AST-1247545 and NSA grant H98230-13-1-0255.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Ma, J., Yu, X. On judicious bipartitions of graphs. Combinatorica 36, 537–556 (2016). https://doi.org/10.1007/s00493-015-2944-y

Download citation

Mathematics Subject Classification (2000)

  • 05C35