End compactifications in non-locally-finite graphs

There are different definitions of ends in non-locally-finite graphs which are all equivalent in the locally finite case. We prove the compactness of the end-topology that is based on the principle of removing finite sets of vertices and give a proof of the compactness of the end-topology that is constructed by the principle of removing finite sets of edges. For the latter case there exists already a proof in \cite{cartwright93martin}, which only works on graphs with countably infinite vertex sets and in contrast to which we do not use the Theorem of Tychonoff. We also construct a new topology of ends that arises from the principle of removing sets of vertices with finite diameter and give applications that underline the advantages of this new definition.Comment: 17 pages, to appear in Math. Proc. Cambridge Philos. So

The language of self-avoiding walks

Let $X=(V\!X,E\!X)$ be an infinite, locally finite, connected graph without loops or multiple edges. We consider the edges to be oriented, and $E\!X$ is equipped with an involution which inverts the orientation. Each oriented edge is labelled by an element of a finite alphabet $\mathbf{\Sigma}$. The labelling is assumed to be deterministic: edges with the same initial (resp. terminal) vertex have distinct labels. Furthermore it is assumed that the group of label-preserving automorphisms of $X$ acts quasi-transitively. For any vertex $o$ of $X$, consider the language of all words over $\mathbf{\Sigma}$ which can be read along self-avoiding walks starting at $o$. We characterize under which conditions on the graph structure this language is regular or context-free. This is the case if and only if the graph has more than one end, and the size of all ends is $1$, or at most $2$, respectively.Comment: 24 pages, 3 figure