## 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*
_{1}⋃*V*
_{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*.

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## 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.

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

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### Mathematics Subject Classification (2000)

- 05C35