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 $$\underset{1\le i\le 3}{\max }e(V_i)\le \frac{m}{9}+\frac{1}{9}h(m)\text{and}e(V_1,V_2,V_3)\ge \frac{2}{3}m+\frac{1}{3}h(m),$$ where $h(m)=\sqrt{2m+1\mathrm{/}4}-1\mathrm{/}2$. This answers a problem of Bollobás and Scott affirmatively. We also solve several related problems of Bollobás and Scott. All of our results are tight.