On weak cop numbers of transitive graphs

TL;DR

This paper investigates the weak cop number in infinite vertex transitive graphs, establishing that such graphs either have a weak cop number of 1 or infinity, thus answering a previously posed open question.

Contribution

It proves that infinite vertex transitive graphs with at least one thick end have infinite weak cop number, and classifies all connected vertex transitive graphs as having weak cop number 1 or infinity.

Findings

Graphs with thick ends have infinite weak cop number.

Connected vertex transitive graphs have weak cop number 1 or infinity.

The result answers an open question in the field.

Abstract

The weak cop number of infinite graphs can be seen as a coarse-geometric analogue to the cop number of finite graphs. We show that every vertex transitive graph with at least one thick end has infinite weak cop number. It follows that every connected, vertex transitive graph has weak cop number $1$ or $\infty$, answering a question posed by Lee, Mart\'inez-Pedroza, and Rodr\'iguez-Quinche, and reiterated in recent preprints by Appenzeller and Klinge, and by Esperet, Gahlawat, and Giocanti.