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arXiv · 2509.20994

Tight bounds for judicious 3-partitions of graphs

Peiru Kuang·Yan Wang
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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.

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Peiru Kuang, Yan Wang. 2025-09-30. Tight bounds for judicious 3-partitions of graphs. https://arxiv.org/abs/2509.20994

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