Tight bounds for judicious 3-partitions of graphs

In this paper, we show that every graph with $m$ edges admits a 3-partition such that $$\max_{1 \leq i \leq 3} e(V_i) \leq \frac{m}{9} + \frac{1}{9}h(m) \quad \text{and} \quad e(V_1, V_2, V_3) \geq \frac{2}{3}m + \frac{1}{3}h(m),$$ where $h(m) = \sqrt{2m + 1/4} - 1/2$. This answers a problem of Bollob\'as and Scott affirmatively. We also solve several related problems of Bollob\'as and Scott. All of our results are tight.