Tight bounds for judicious 3-partitions of graphs
Peiru Kuang, Yan Wang
TL;DR
The paper resolves the judicious 3-partition problem for graphs by proving tight bounds: every graph with $m$ edges admits a 3-partition with $\max_i e(V_i) \le \frac{m}{9}+\frac{1}{9}h(m)$ and $e(V_1,V_2,V_3) \ge \frac{2}{3}m+\frac{1}{3}h(m)$, answering Bollobás and Scott for $k=3$. The authors develop a two-pronged strategy: a stability result for judicious 2-partitions and a constructive reduction to 2-partitions on the complement, complemented by a mapping to weighted complete graphs to obtain tight bounds. They show the bounds are tight, with extremal graphs like $K_{3r+1}\cup pK_1$ and odd-order complete graphs achieving equality; they also extend the approach to graphs with few edges and connect to a general $k$-partition framework. The results advance the understanding of multiterminal partition problems by establishing precise thresholds for both within-part and cross-part edge counts, and by clarifying the role of graph structure in extremality.
Abstract
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 Bollobás and Scott affirmatively. We also solve several related problems of Bollobás and Scott. All of our results are tight.