2025/07/22 by Max Pitz, Pitz, Max
Mathematics · #05C40 #05C63 #54E35 #Advanced Banach Space Theory #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2507.16625
openalex publication_date 2025/07/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that the edge-end space of an infinite graph is metrizable if and only if it is first-countable. This strengthens a recent result by Aurichi, Magalhaes Jr. and Real (2024). Our central graph-theoretic tool is the use of tree-cut decompositions, introduced by Wollan (2015) as a variation of tree decompositions that is based on edge cuts instead of vertex separations. In particular, we give a new, elementary proof for Kurkofka's result (2022) that every infinite graph has a tree-cut decomposition of finite adhesion into its ω-edge blocks. Along the way, we also give a new, short proof for a classic result by Halin (1984) on Kk,κ-subdivisions in k-connected graphs, making this paper self-contained.