On graphs with a metric end space

Waas, Kerstin

doi:10.1090/S0002-9947-99-02255-2

On graphs with a metric end space
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Trans. Amer. Math. Soc. 351 (1999), 1043-1062 Request permission

Abstract:

R. Diestel conjectured that an infinite graph contains a topologically end-faithful forest if and only if its end space is metrizable. We prove this conjecture for uniform end spaces.

References

J.-M. Brochet and R. Diestel, Normal tree orders for infinite graphs, Trans. Amer. Math. Soc. 345 (1994), no. 2, 871–895. MR 1260198, DOI 10.1090/S0002-9947-1994-1260198-4
Reinhard Diestel, The end structure of a graph: recent results and open problems, Discrete Math. 100 (1992), no. 1-3, 313–327. Special volume to mark the centennial of Julius Petersen’s “Die Theorie der regulären Graphs”, Part I. MR 1172358, DOI 10.1016/0012-365X(92)90650-5
Reinhard Diestel, On spanning trees and $k$-connectedness in infinite graphs, J. Combin. Theory Ser. B 56 (1992), no. 2, 263–277. MR 1186759, DOI 10.1016/0095-8956(92)90022-P
R. Halin, Über unendliche Wege in Graphen, Math. Ann. 157 (1964), 125–137 (German). MR 170340, DOI 10.1007/BF01362670
R. Halin, Simplicial decompositions of infinite graphs, Ann. Discrete Math. 3 (1978), 93–109. MR 499113
Rudolf Halin, Graphentheorie. I, Erträge der Forschung [Research Results], vol. 138, Wissenschaftliche Buchgesellschaft, Darmstadt, 1980 (German). MR 586234
I. M. James, Topological and uniform spaces, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1987. MR 884154, DOI 10.1007/978-1-4612-4716-6
H. A. Jung, Wurzelbäume und unendliche Wege in Graphen, Math. Nachr. 41 (1969), 1–22 (German). MR 266807, DOI 10.1002/mana.19690410102
Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
Horst Schubert, Topologie. Eine Einführung, Mathematische Leitfäden, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1964 (German). MR 0170313
P.D. Seymour and R. Thomas, An end-faithful counterexample. Directions in Infinite Graph Theory and Combinatorics (R. Diestel, Ed.), Topics in Discrete Mathematics 3, North-Holland Publ. Co., Amsterdam / London, 1992.
C. Thomassen, Infinitely connected graphs with no end-preserving spanning trees. preprint, 1990.
K. Waas, Topologisch endentreue Bäume in unendlichen Graphen. Diplomarbeit, Universität Bielefeld, 1994.