Bounds for the number of meeting edges in graph partitioning

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?Δ₁)/2+2w ₂/3, 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 (w ₀?1)/6+(w ₁?Δ₁)/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 2m/5] edges, which establishes a special case of a more general conjecture of Bollobás and Scott.