A bound on judicious bipartitions of directed graphs

  • Jianfeng Hou
  • Huawen Ma
  • Xingxing YuEmail author
  • Xia Zhang


Judicious partitioning problems on graphs ask for partitions that bound several quantities simultaneously, which have received much attention lately. Scott (2005) asked the following natural question: What is the maximum constant cd such that every directed graph D with m arcs and minimum outdegree d admits a bipartition V(D) = V1V2 satisfying min{e(V1, V2), e(V2, V1)} ⩾ cdm? Here, for i = 1, 2, e(Vi, V3-i) denotes the number of arcs in D from Vi to V3-i. Lee et al. (2016) conjectured that every directed graph D with m arcs and minimum outdegree at least d ⩾ 2 admits a bipartition V(D) = V1V2 such that
$$\min \left\{ {e\left( {V_1 ,V_2 } \right),e\left( {V_2 ,V_1 } \right)} \right\} \geqslant \left( {\frac{{d - 1}}{{2\left( {2d - 1} \right)}} + o\left( 1 \right)} \right)m$$
. In this paper, we show that this conjecture holds under the additional natural condition that the minimum indegree is also at least d.


directed graph partition outdegree indegree tight component 


05C20 05C35 


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The first author was supported by National Natural Science Foundation of China (Grant No. 11671087). The third author was supported by National Science Foundation of USA (Grant No. DMS-1600738) and the Hundred Talents Program of Fujian Province. The fourth author was supported by the Shandong Provincial Natural Science Foundation of China (Grant No. ZR2014JL001) and the Excellent Young Scholars Research Fund of Shandong Normal University of China. The authors thank the anonymous referees for their helpful comments and suggestions.


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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Jianfeng Hou
    • 1
  • Huawen Ma
    • 1
  • Xingxing Yu
    • 2
    Email author
  • Xia Zhang
    • 3
  1. 1.Center for Discrete MathematicsFuzhou UniversityFuzhouChina
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  3. 3.School of Mathematics and StatisticsShandong Normal UniversityJinanChina

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