Edge-ends versus topological ends of graphs
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908854562848768 |
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| author | Aurichi, Leandro Júnior, Paulo Magalhães Pinto, Guilherme Eduardo |
| author_facet | Aurichi, Leandro Júnior, Paulo Magalhães Pinto, Guilherme Eduardo |
| contents | Diestel and Kühn proved that the topological ends of an infinite graph are precisely its undominated graph ends, yielding a canonical embedding of the space of topological ends into the space of graph ends. For edge-ends, introduced by Hahn, Laviolette and Širáň, such an embedding does not exist in general.
In this note, we characterize the class of infinite graphs for which the topological ends admit a natural injective map into the space of edge-ends that is compatible with the canonical maps between end spaces. Our characterization is purely combinatorial and is expressed in terms of edge-equivalence classes of vertices.
Moreover, when such an embedding exists, we identify precisely which edge-ends arise from topological ends, showing that they are exactly the edge-ends containing a non-dominated ray. This establishes a parallel result to the theorem of Diestel and Kühn for edge-end spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14943 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Edge-ends versus topological ends of graphs Aurichi, Leandro Júnior, Paulo Magalhães Pinto, Guilherme Eduardo Combinatorics General Topology 05C63 05C40 Diestel and Kühn proved that the topological ends of an infinite graph are precisely its undominated graph ends, yielding a canonical embedding of the space of topological ends into the space of graph ends. For edge-ends, introduced by Hahn, Laviolette and Širáň, such an embedding does not exist in general. In this note, we characterize the class of infinite graphs for which the topological ends admit a natural injective map into the space of edge-ends that is compatible with the canonical maps between end spaces. Our characterization is purely combinatorial and is expressed in terms of edge-equivalence classes of vertices. Moreover, when such an embedding exists, we identify precisely which edge-ends arise from topological ends, showing that they are exactly the edge-ends containing a non-dominated ray. This establishes a parallel result to the theorem of Diestel and Kühn for edge-end spaces. |
| title | Edge-ends versus topological ends of graphs |
| topic | Combinatorics General Topology 05C63 05C40 |
| url | https://arxiv.org/abs/2602.14943 |