Cop-number dichotomy for locally finite vertex-transitive graphs

Determine whether every connected locally finite vertex-transitive graph has weak cop number and strong cop number each belonging to the set {1,∞}.

Background

The question generalizes the group dichotomy to locally finite vertex-transitive graphs. The paper obtains related results for locally finite quasi-transitive graphs under consequences of the Georgakopoulos–Papasoglu conjecture, but does not establish the stated dichotomy in full generality.

References

More generally, they asked whether for any connected locally finite vertex-transitive graph $G$ we always have $\wco(G)=1\text{ or }\infty$, and $\sco(G)=1\text{ or }\infty$ Question L.

— Coarse cops and robber in graphs and groups  (2502.15571 - Esperet et al., 21 Feb 2025) in Immediately following Question 1.3, Section 1, subsection “Weak cop number 1”