Henning Bruhn

The cycle space of a 3-connected locally finite graph is generated by its finite and infinite peripheral circuits

We extend Tuttes result that in a finite 3-connected graph the cycle space is generated by the peripheral circuits to locally finite graphs. Such a generalization becomes possible by the admission of infinite circuits in the graph compactified by its ends.

Duality in Infinite Graphs

The adaption of combinatorial duality to infinite graphs has been hampered by the fact that while cuts (or cocycles) can be infinite, cycles are finite. We show that these obstructions fall away when duality is reinterpreted on the basis of a ‘singular’ approach to graph homology, whose cycles are defined topologically in a space formed by the graph together with its ends and can be infinite. Our approach enables us to complete Thomassens results about ‘finitary’ duality for infinite graphs to full duality, including his extensions of Whitneys theorem.

On end degrees and infinite cycles in locally finite graphs

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.

Infinite matroids in graphs

It has recently been shown that infinite matroids can be axiomatized in a way that is very similar to finite matroids and permits duality. This was previously thought impossible, since finitary infinite matroids must have non-finitary duals. In this paper we illustrate the new theory by exhibiting its implications for the cycle and bond matroids of infinite graphs. We also describe their algebraic cycle matroids, those whose circuits are the finite cycles and double rays, and determine their duals. Finally, we give a sufficient condition for a matroid to be representable in a sense adapted to infinite matroids. Which graphic matroids are representable in this sense remains an open question.

Finite connectivity in infinite matroids

We introduce a connectivity function for infinite matroids with properties similar to the connectivity function of a finite matroid, such as submodularity and invariance under duality. As an application we use it to extend Tuttes Linking Theorem to finitary and cofinitary matroids.

Duality of ends

We investigate the end spaces of infinite dual graphs. We show that there exists a natural homeomorphism * between the end spaces of a graph and its dual, and that * preserves the ‘end degree’. In particular, * maps thick ends to thick ends. Along the way, we prove that Tutte-connectivity is invariant under taking (infinite) duals.

Hamilton Cycles in Planar Locally Finite Graphs

A classical theorem by Tutte ensures the existence of a Hamilton cycle in every finite

Eulerian edge sets in locally finite graphs

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Single Source Multiroute Flows and Cuts on Uniform Capacity Networks

-connected planar graph. Extensions of this result to infinite graphs require a suitable concept of an infinite cycle. Such a concept was provided by Diestel and Kuhn, who defined circles to be homeomorphic images of the unit circle in the Freudenthal compactification of the (locally finite) graph. With this definition we prove a partial extension of Tuttes result to locally finite graphs.

Bicycles and left-right tours 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 circuits, of locally finite graphs. In order to do so, it becomes necessary to attribute a parity to the ends of the graph.