On end degrees and infinite cycles in locally finite graphs

  title={On end degrees and infinite cycles in locally finite graphs},
  author={Henning Bruhn and Maya Jakobine Stein},
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. 

Eulerian edge sets in locally finite graphs

In a finite graph, an edge set Z is an element of the cycle space if and only if every vertex has even degree in Z. We extend this basic result to the topological cycle space, which allows infinite

On the hamiltonicity of line graphs of locally finite, 6‐edge‐connected graphs

A weaker version of this result for infinite graphs is proved: the line graph of locally finite, 6-edge-connected graph with a finite number of ends, each of which is thin, is hamiltonian.

The relative degree and large complete minors in infinite graphs

Extending Cycles Locally to Hamilton Cycles

It is proved that every connected, locallyconnected, locally finite, claw-free graph has a Hamilton circle in an infinite graph, and it is shown that such graphs are Hamilton-connected if and only if they are $3$-connected.

Hamilton circles in infinite planar graphs

Forcing Hamiltonicity in locally finite graphs via forbidden induced subgraphs I: Nets and bulls

In a series of papers, of which this is the first, we study sufficient conditions for Hamiltonicity in terms of forbidden induced subgraphs and extend such results to locally finite infinite graphs.

The Line Graph of Every Locally Finite 6-Edge-Connected Graph with Finitely Many Ends is Hamiltonian

Homeomorphic images of the unit circle in the Freudenthal compactification of a graph can be seen as an infinite analogue of finite graph theoretic circles. In this sense the main result of the

A sufficient condition for Hamiltonicity in locally finite graphs

Connected but not path-connected subspaces of infinite graphs

It is shown that the Freudenthal compactification of a locally finite graph can have connected subsets that are not path-connected, and it is proved that connectedness and path- connectedness to coincide for all but a few sets, which have a complicated structure.



On Infinite Cycles I

We adapt the cycle space of a finite graph to locally finite infinite graphs, using as infinite cycles the homeomorphic images of the unit circle S1 in the graph compactified by its ends. We prove

On Infinite Cycles II

The spanning trees whose fundamental cycles generate this cycle space are characterized, and infinite analogues to the standard characterizations of finite cycle spaces in terms of edge-decomposition into single cycles and orthogonality to cuts are proved.

Topological paths, cycles and spanning trees in infinite graphs

A note on Menger's theorem for infinite locally finite graphs

Introduction. In [1] the ends of an infinite graph G were introduced; they are the equivalence classes induced by the following equivalence relation on the set ~1 (G) of all one way infinite paths in

Decomposition of Graphs into Two-Way Infinite Paths

We consider undirected graphs in which two vertices may be joined by more than one edge and in which a vertex may be joined to itself by one or more edges. G will always denote a graph. The set of

The Cycle Space of an Infinite Graph

  • R. Diestel
  • Mathematics
    Combinatorics, Probability and Computing
  • 2005
A new ‘singular’ approach is presented that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of $S^1$ in the space formed by the graph together with its ends.

Decompositions of infinite graphs: Part II circuit decompositions

Graph Theory

Gaph Teory Fourth Edition is standard textbook of modern graph theory which covers the core material of the subject with concise yet reliably complete proofs, while offering glimpses of more advanced methods in each chapter by one or two deeper results.

Graph Theory (3rd edition)

  • Springer-Verlag
  • 2005