# A bound on judicious bipartitions of directed graphs

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## Abstract

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 . In this paper, we show that this conjecture holds under the additional natural condition that the minimum indegree is also at least

*c*_{d}such that every directed graph*D*with*m*arcs and minimum outdegree*d*admits a bipartition*V*(*D*) =*V*_{1}∪*V*_{2}satisfying min{*e*(*V*_{1},*V*_{2}),*e*(*V*_{2},*V**1*)} ⩾*c*_{d}*m*? Here, for*i*= 1, 2,*e*(*V*_{i},*V*_{3-i}) denotes the number of arcs in*D*from*V*_{i}to*V*_{3-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*) =*V*_{1}∪*V*_{2}such that$$\min \left\{ {e\left( {V_1 ,V_2 } \right),e\left( {V_2 ,V_1 } \right)} \right\} \geqslant \left( {\frac{{d - 1}}{{2\left( {2d - 1} \right)}} + o\left( 1 \right)} \right)m$$

*d*.## Keywords

directed graph partition outdegree indegree tight component## MSC(2010)

05C20 05C35## Preview

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## Notes

### Acknowledgements

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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