Exact bounds for judicious partitions of graphs
Edwards showed that every graph of size m ≥ 1 has a bipartite subgraph of size at least m/2 + √ m/8 + 1/64 - 1/8. We show that every graph of size m ≥ 1 has a bipartition in which the Edwards bound holds, and in addition each vertex class contains at most m/4+ √ m/32+1/256-1/16 edges. This is exact for complete graphs of odd order, which we show are the only extremal graphs without isolated vertices. We also give results for partitions into more than two classes.
Bollobás, B., & Scott, A. (1999). Exact bounds for judicious partitions of graphs. Combinatorica, 19 (4), 473-486. https://doi.org/10.1007/s004939970002