# Cutsets in infinite graphs

Adam Timar

Arxiv ID: 0711.1711•Last updated: 5/11/2020

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 $Z$.

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