Tight bounds for judicious 3-partitions of graphs
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915523846995968 |
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| author | Kuang, Peiru Wang, Yan |
| author_facet | Kuang, Peiru Wang, Yan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_20994 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tight bounds for judicious 3-partitions of graphs Kuang, Peiru Wang, Yan Combinatorics 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. |
| title | Tight bounds for judicious 3-partitions of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.20994 |