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Mathematics > Combinatorics
Title: Topological remarks on end and edge-end spaces
(Submitted on 26 Apr 2024 (v1), last revised 2 May 2024 (this version, v2))
Abstract: The notion of ends in an infinite graph $G$ might be modified if we consider them as equivalence classes of infinitely edge-connected rays, rather than equivalence classes of infinitely (vertex-)connected ones. This alternative definition yields to the edge-end space $\Omega_E(G)$ of $G$, in which we can endow a natural (edge-)end topology. For every graph $G$, this paper proves that $\Omega_E(G)$ is homeomorphic to $\Omega(H)$ for some possibly another graph $H$, where $\Omega(H)$ denotes its usual end space. However, we also show that the converse statement does not hold: there is a graph $H$ such that $\Omega(H)$ is not homeomorphic to $\Omega_E(G)$ for any other graph $G$. In other words, as a main result, we conclude that the class of topological spaces $\Omega_E = \{\Omega_E(G) : G \text{ graph}\}$ is strictly contained in $\Omega = \{\Omega(H) : H \text{ graph}\}$.
Submission history
From: Lucas Real [view email][v1] Fri, 26 Apr 2024 02:15:18 GMT (33kb)
[v2] Thu, 2 May 2024 01:56:29 GMT (33kb)
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