Partitioning dense uniform hypergraphs

Shufei Wu and Jianfeng Hou ()
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Shufei Wu: Henan Polytechnic University
Jianfeng Hou: Fuzhou University

Journal of Combinatorial Optimization, 2018, vol. 35, issue 1, No 5, 48-63

Abstract: Abstract Let $$r\ge 3$$ r ≥ 3 and $$k\ge 2$$ k ≥ 2 be fixed integers, and let H be an r-uniform hypergraph with n vertices and m edges. In 1997, Bollobás and Scott conjectured that H has a vertex-partition into k sets with at most $$m/k^r+o(m)$$ m / k r + o ( m ) edges in each set. So far, this conjecture was confirmed when $$r=3$$ r = 3 or $$m=\Omega (n^{r-1+o(1)})$$ m = Ω ( n r - 1 + o ( 1 ) ) . In this paper, we show that it holds for $$m=\Omega (n^{r-3+\epsilon })$$ m = Ω ( n r - 3 + ϵ ) for any $$\epsilon >0$$ ϵ > 0 .

Keywords: Hypergraph; Partition; Max-Cut; Azuma–Hoeffding inequality; 05C35; 05C65 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10878-017-0153-x

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