Locally finite graphs with ends: a topological approach

TL;DR

This survey introduces a topological framework for studying locally finite graphs with ends, extending classical finite graph theorems and highlighting new research directions in the field.

Contribution

It provides an accessible overview of a new topological approach to infinite graphs, connecting classical results with modern techniques and open problems.

Findings

Extension of finite graph theorems to locally finite graphs

Introduction of topological arcs and circles through ends

Identification of key open problems in the field

Abstract

This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open problems, and offers pointers to the literature for more detail.