Judicious bisection of hypergraphs

Abstract

Judicious bisection of hypergraphs asks for a balanced bipartition of the vertex set that optimizes several quantities simultaneously. In this paper, we prove that if G is a hypergraph with n vertices and m _i edges of size i for i = 1, 2, …, k, then G admits a bisection in which each vertex class spans at most $\frac{{m1}}{2} + \frac{1}{4}{m_2} + \cdots + \left( {\frac{1}{{{2^k}}}} \right){m_k} + o\left( {{m_1} + \cdots + {m_k}} \right)$ edges, where G is dense enough or Δ(G) = o(n) but has no isolated vertex, which turns out to be a bisection version of a conjecture proposed by Bollobás and Scott.