Cutsets in infinite graphs

TL;DR

This paper investigates the parameter $C_G$ related to minimal cutsets in Cayley graphs, establishing its finiteness across different graphs, providing bounds on minimal cutsets, and characterizing groups with infinite $C_G$.

Contribution

It proves the invariance of the finiteness of $C_G$ across Cayley graphs, offers bounds on minimal cutsets, and characterizes groups with infinite $C_G$.

Findings

Finiteness of $C_G$ is independent of the Cayley graph chosen.

Exponential bounds on minimal cutsets are also independent of the Cayley graph.

The lamplighter group provides an example where $C_G$ is infinite.

Abstract

We answer three questions posed in a paper by Babson and Benjamini. They introduced a parameter $C_G$ for Cayley graphs $G$ that has significant application to percolation. For a minimal cutset of $G$ and a partition of this cutset into two classes, take the minimal distance between the two classes. The supremum of this number over all minimal cutsets and all partitions is $C_G$. We show that if it is finite for some Cayley graph of the group then it is finite for any (finitely generated) Cayley graph. Having an exponential bound for the number of minimal cutsets of size $n$ separating $o$ from infinity also turns out to be independent of the Cayley graph chosen. We show a 1-ended example (the lamplighter group), where $C_G$ is infinite. Finally, we give a new proof for a question of de la Harpe, proving that the number of $n$-element cutsets separating $o$ from infinity is finite unless $G$ is a finite extension of $Z$.