Bounds for the number of meeting edges in graph partitioning
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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 (w1−Δ1)/2+2w2/3, where wi is the total weight of edges of size i and Δ1 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 (w0−1)/6+(w1−Δ1)/3+2w2/3, where w0 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 K2 and K1,3, admits a tripartition such that each vertex class meets at least [2m/5] edges, which establishes a special case of a more general conjecture of Bollobás and Scott.
Keywordsgraph weighted hypergraph partition judicious partition
MSC 201005C35 05C75
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