Relationships between topology and combinatorics
This thesis addresses certain problems concerning infinite graphs and is divided into two parts: Finite-Infinite Type Extensions and End Theory. In the Finite-Infinite Type Extensions part, we study the case of infinite graphs in relation to two conjectures that were originally considered in the con...
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| Formato: | tesis doctoral |
| Estado: | Versión publicada |
| Fecha de publicación: | 2025 |
| País: | Brasil |
| Recursos: | Universidade de São Paulo (USP) |
| Repositorio: | Biblioteca Digital de Teses e Dissertações da USP |
| Idioma: | inglés |
| OAI Identifier: | oai:teses.usp.br:tde-01092025-112137 |
| Acesso em linha: | https://www.teses.usp.br/teses/disponiveis/55/55135/tde-01092025-112137/ |
| Access Level: | acceso abierto |
| Palavra-chave: | Edge-end space End space Espaço de aresta-extremidade Espaço de extremidades F-limit F-limite Grafos infinitos Infinite Graphs Ultrafilters Ultrafiltros |
| Resumo: | This thesis addresses certain problems concerning infinite graphs and is divided into two parts: Finite-Infinite Type Extensions and End Theory. In the Finite-Infinite Type Extensions part, we study the case of infinite graphs in relation to two conjectures that were originally considered in the context of finite graphs. These conjectures concern cycle covers of a graph. To deal with them, we use the concept of infinite cycles, already established in the literature, and introduce a new concept: the F-limit of cycles. Intuitively, an F-limit of cycles is an approximation of infinite cycles by finite ones. The main focus of the chapters in this part is to establish the relationship between the finite and infinite cases of the conjectures, as well as to study properties of the F-limit of cycles. In the second part, we address problems related to the end theory of infinite graphs. These investigations are conducted primarily from topological, set-theoretical, and combinatorial perspectives. We prove topological characterization results for end spaces, present a new counterexample to Halins End Degree Conjecture in the case ℵ1, and show the connection between the topological classes of end spaces and edge-end spaces. |
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