Abstract
A graphX is said to beprimitive if its automorphism groupG acts primitively on the vertex setVX; that is, the onlyG-invariant equivalence relations onVX are the one where all the classes have size one and the equivalence relation which has only one class, the whole ofVX. We investigate the end structure of locally finite primitive graphs. Our main result shows that it has a very simple description; in particular, locally finite primitive graphs are accessible in the sense of Thomassen and Woess.
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References
W. Dicks, andM. J. Dunwoody:Groups acting on graphs, Cambridge University Press, Cambridge 1989.
M. J. Dunwoody: Cutting up graphs,Combinatorica 2 (1982), 15–23.
M. J. Dunwoody: An inaccessible group, in: (ed. G. A. Niblo and M. Roller)Proceedings of Geometric Group Theory 1991, London Mathematical Society Lecture note series 181, Cambridge University Press 1993, 75–78.
H. A. Jung, andM. E. Watkins: On the structure of infinite vertex-transitive graphs,Discrete Math. 18 (1977), 45–53.
H. A. Jung, andM. E. Watkins: The connectivities of locally finite primitive graphs,Combinatorica 9 (1989), 261–267.
R. G. Möller: Ends of graphs,Math. Proc. Camb. Phil. Soc. 111 (1992), 255–266.
R. G. Möller: Ends of graphs II,Math. Proc. Camb. Phil. Soc. 111 (1992), 455–460.
J.-P. Serre:Trees, Springer 1980.
C. Thomassen, andW. Woess: Vertex-transitive graphs and accessibility,J. Combin. Theory, Ser. B 58 (1993), 248–268.