Skip to main content
    • Aa
    • Aa

Judicious Partitioning of Hypergraphs with Edges of Size at Most 2


Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let k ≥ 2 be an integer and let G be a hypergraph with m i edges of size i for i=1,2. Bollobás and Scott conjectured that G has a partition into k classes, each of which contains at most $m_1/k+m_2/k^2+O(\sqrt{m_1+m_2})$ edges. In this paper, we confirm the conjecture affirmatively by showing that G has a partition into k classes, each of which contains at most $$m_1/k+m_2/k^2+\ffrac{k-1}{2k^2}\sqrt{2(km_1+m_2)}+O(1)$$. edges. This bound is tight up to O(1).

Corresponding author
Corresponding author.
Hide All

This work is supported by research grant NSFC.

Hide All
[1] AlonN. (1996) Bipartite subgraphs. Combinatorica 16 301311.
[2] AlonN., BollobásB., KrivelevichM. and SudakovB. (2003) Maximum cuts and judicious partitions in graphs without short cycles. J. Combin. Theory Ser. B 88 329346.
[3] BollobásB. and ScottA. D. (1997) Judicious partitions of hypergraphs. J. Combin. Theory Ser. A 78 1531.
[4] BollobásB. and ScottA. D. (1999) Exact bounds for judicious partitions of graphs. Combinatorica 19 473486.
[5] BollobásB. and ScottA. D. (2000) Judicious partitions of 3-uniform hypergraphs. European J. Combin. 21 289300.
[6] BollobásB. and ScottA. D. (2002) Better bounds for Max Cut. In Contemporary Combinatorics, Vol. 10 of Bolyai Society Mathematical Studies, pp. 185–246.
[7] BollobásB. and ScottA. D. (2002) Problems and results on judicious partitions. Random Struct. Alg. 21 414430.
[8] BollobásB. and ScottA. D. (2010) Max k-cut and judicious k-partitions. Discrete Math. 310 21262139.
[9] EdwardsC. S. (1973) Some extremal properties of bipartite graphs. Canad. J. Math. 3 475485.
[10] EdwardsC. S. (1975) An improved lower bound for the number of edges in a largest bipartite subgraph. In Proc. 2nd Czechoslovak Symposium on Graph Theory, pp. 167–181.
[11] FanG. and HouJ. Bounds for pairs in judicious partitions of graphs. Random Struct. Alg. doi:10.1002/rsa.20642
[12] FanG., HouJ. and ZengQ. (2014) A bound for judicious k-partitions of graphs. Discrete Appl. Math. 179 8699.
[13] HaslegraveJ. (2012) The Bollobás–Thomason conjecture for 3-uniform hypergraphs. Combinatorica 32 451471.
[14] HaslegraveJ. (2014) Judicious partitions of uniform hypergraphs. Combinatorica 34 561572.
[15] HouJ., WuS. and YanG. (2016) On judicious partitions of uniform hypergraphs. J. Combin. Theory Ser. A 141 1632.
[16] LeeC., LohP. and SudakovB. (2013) Bisections of graphs. J. Combin. Theory Ser. B 103 599629.
[17] MaJ., YenP. and YuX. (2010) On several partitioning problems of Bollobás and Scott. J. Combin. Theory Ser. B 100 631649.
[18] MaJ. and YuX. (2012) Partitioning 3-uniform hypergraphs. J. Combin. Theory Ser. B 102 212232.
[19] ScottA. D. (2005) Judicious partitions and related problems. In Surveys in Combinatorics, Vol. 327 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 95117.
[20] XuB. and YuX. (2008) Triangle-free subcubic graphs with minimum bipartite density. J. Combin. Theory Ser. B 98 516537.
[21] XuB. and YuX. (2014) On judicious bisections of graphs. J. Combin. Theory Ser. B 106 3069.
[22] YannakakisM. (1978) Node- and edge-deletion NP-complete problems. In STOC '78: Proc. 10th Annual ACM Symposium on Theory of Computing, pp. 253–264.
[23] ZhuX. (2009) Bipartite density of triangle-free subcubic graphs. Discrete Appl. Math. 157 710714.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 96 *
Loading metrics...

Abstract views

Total abstract views: 229 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.