On weak cop numbers of transitive graphs
Florian Lehner
TL;DR
This work studies the coarse-geometric analogue of the cop number on infinite graphs, focusing on the weak cop number $wCop(G)$ for vertex transitive graphs. The authors fix cop speed $s_c$ and capture radius $\rho$, construct a radius sequence $R_0<R_1<\dots$ with $s_r=b(R_{k\cdot b(\rho)+1})$, and exploit thick ends via havens to route the robber along safe vertices, preventing capture against any finite number of cops. They prove that if a vertex transitive graph has a thick end, then $wCop(G)=\infty$, which implies $wCop(G)\in\{1,\infty\}$ for all connected vertex transitive graphs, resolving the open question in the negative for $wCop$. The result connects thick-end geometry to pursuit-evasion dynamics and complements existing results linking $sCop(G)=1$ to Gromov-hyperbolicity; the thin-end case reduces to a tree-like quasi-isometry giving $wCop(G)=1$.
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, Martínez-Pedroza, and Rodríguez-Quinche, and reiterated in recent preprints by Appenzeller and Klinge, and by Esperet, Gahlawat, and Giocanti.