Node bisectors of Cayley graphs

A node bisector of a graph Γ is a subset Ω of the nodes of Γ such that Γ may be expressed as the disjoint union $$Gamma = Omega _1 dot cup Omega dot cup Omega _2 ,$$ , where $left| {Omega _1 } right| geqslant frac{1}{3}left| Gamma right|,Omega _2 geqslant frac{1}{3}left| Gamma right|$ , and where any path from Ω₁ to Ω₂ must pass through Ω. Suppose Γ is the Cayley graph of an abelian groupG with respect to a generating set of cardinalityr, regarded as an undirected graph. Then we show that Γ has a node bisector of order at mostc¦G ^1-1/r wherec is a constant depending only onr. We show that the exponent in this result is the best possible.