On end degrees and infinite cycles in locally finite graphs

Authors:
Henning Bruhn;Maya Stein
Affiliations:
Universität Hamburg, Mathematisches Seminar, Bundesstraße 55, 20146, Hamburg, Germany;Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão, 1010, CEP 05508-090, São Paulo, SP, Brasil
Venue:
Combinatorica
Year:
2007

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Abstract

We introduce a natural extension of the vertex degree to ends. For the cycle space C(G) as proposed by Diestel and Kühn [4, 5], which allows for infinite cycles, we prove that the edge set of a locally finite graph G lies in C(G) if and only if every vertex and every end has even degree. In the same way we generalise to locally finite graphs the characterisation of the cycles in a finite graph as its 2-regular connected subgraphs.