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Judicious Partitioning of Hypergraphs with Edges of Size at Most 2

  • JIANFENG HOU (a1) and QINGHOU ZENG (a1)
Abstract

Judicious partitioning problems on graphs and hypergraphs ask for partitions that optimize several quantities simultaneously. Let k ≥ 2 be an integer and let G be a hypergraph with m i edges of size i for i=1,2. Bollobás and Scott conjectured that G has a partition into k classes, each of which contains at most $m_1/k+m_2/k^2+O(\sqrt{m_1+m_2})$ edges. In this paper, we confirm the conjecture affirmatively by showing that G has a partition into k classes, each of which contains at most $$m_1/k+m_2/k^2+\ffrac{k-1}{2k^2}\sqrt{2(km_1+m_2)}+O(1)$$. edges. This bound is tight up to O(1).

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This work is supported by research grant NSFC.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
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