Czechoslovak Mathematical Journal, Vol. 67, No. 3, pp. 741-752, 2017


Bounds for the number of meeting edges in graph partitioning

Qinghou Zeng, Jianfeng Hou

Received March 25, 2016.   First published July 14, 2017.

Abstract:  Let $G$ be a weighted hypergraph with edges of size at most 2. Bollobás and Scott conjectured that $G$ admits a bipartition such that each vertex class meets edges of total weight at least $(w_1-\Delta_1)/2+2w_2/3$, where $w_i$ is the total weight of edges of size $i$ and $\Delta_1$ is the maximum weight of an edge of size 1. In this paper, for positive integer weighted hypergraph $G$ (i.e., multi-hypergraph), we show that there exists a bipartition of $G$ such that each vertex class meets edges of total weight at least $(w_0-1)/6+(w_1-\Delta_1)/3+2w_2/3$, where $w_0$ is the number of edges of size 1. This generalizes a result of Haslegrave. Based on this result, we show that every graph with $m$ edges, except for $K_2$ and $K_{1,3}$, admits a tripartition such that each vertex class meets at least $\lceil{2m}/5\rceil$ edges, which establishes a special case of a more general conjecture of Bollobás and Scott.
Keywords:  graph; weighted hypergraph; partition; judicious partition
Classification MSC:  05C35, 05C75
DOI:  10.21136/CMJ.2017.0147-16


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Affiliations:   Qinghou Zeng, Jianfeng Hou, Center for Discrete Mathematics, Fuzhou University, Tongpan, Software Road 89-1, Fuzhou 350003, Fujian, P. R. China, e-mail: qinghouzeng@hotmail.com, jfhou@fzu.edu.cn

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