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Judicious Partitioning of Hypergraphs with Edges of Size at Most 2†

Part of: Graph theory

Published online by Cambridge University Press:  16 August 2016

JIANFENG HOU  and
QINGHOU ZENG
JIANFENG HOU
Affiliation:
Center for Discrete Mathematics, Fuzhou University, Fujian, P. R. China350003 (e-mail: jfhou@fzu.edu.cn, qinghouzeng@hotmail.com)
QINGHOU ZENG*
Affiliation:
Center for Discrete Mathematics, Fuzhou University, Fujian, P. R. China350003 (e-mail: jfhou@fzu.edu.cn, qinghouzeng@hotmail.com)
*
‡Corresponding author.
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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 mi 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).

MSC classification

Primary: 05C35: Extremal problems
Secondary: 05C75: Structural characterization of families of graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 26 , Issue 2 , March 2017 , pp. 267 - 284
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

†

This work is supported by research grant NSFC.

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