The Structure of Locally Finite Two-Connected Graphs

The Structure of Locally Finite Two-Connected Graphs

We expand on Tutte’s theory of 3-blocks for 2-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the 3-block tree of a 2-connected graph. Mathematics Subject Classification: 05C40, 05C38, and 05C05.

Contractible Edges in 2-Connected Locally Finite Graphs

In this paper, we prove that every contraction-critical 2-connected infinite graph has no vertex of finite degree and contains uncountably many ends. Then, by investigating the distribution of contractible edges in a 2-connected locally finite infinite graph G, we show that the closure of the subgraph induced by all the contractible edges in the Freudenthal compactification of G is 2-arc-connec...

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

The topological approach to the study of infinite graphs of Diestel and K ̈hn has enabled several results on Hamilton cycles in finite graphs to be extended to locally finite graphs. We consider the result that the line graph of a finite 4-edge-connected graph is hamiltonian. We prove a weaker version of this result for infinite graphs: The line graph of locally finite, 6-edge-connected graph w...

Locally-finite connected-homogeneous digraphs

The classification of finite and locally finite connected-homogeneous digraphs

We classify the finite connected-homogeneous digraphs, as well as the infinite locally finite such digraphs with precisely one end. This completes the classification of all the locally finite connected-homogeneous digraphs.