Node bisectors of Cayley graphs

Abstract

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.