On weak cop numbers of transitive graphs

2025/02/27 by Florian Lehner, Lehner, Florian · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2502.19724

openalex publication_date 2025/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∞, 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.

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• A coarse Halin Grid Theorem with applications to quasi-transitive, locally finite graphs

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