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Graphs of Degree Three with a Given Abstract Group

Published online by Cambridge University Press:  20 November 2018

Robert Frucht
Robert Frucht*
Affiliation:
Universidad Técnica F. Santa Maria Valparaiso, Chile
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1. Introduction. In his well-known book on graphs [1] König proposed the following problem: “When can a given abstract group be represented as the group of the automorphisms of a (finite) graph, and if possible how can the graph be constructed?”

To understand this problem well let us begin with the precise definition of a graph and its group (of automorphisms).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 1 , Issue 4 , 01 August 1949 , pp. 365 - 378
Copyright
Copyright © Canadian Mathematical Society 1949

References

[1] König, Denés, Theorie der endlichen und unendlichen Graphen. Kombinatorische Topologie der Streckenkomplexe. (Leipzig, 1936). See page 5.Google Scholar
[2] Frucht, R., “Herstellung von Graphen mit vorgegebener abstrakter Gruppe,” Compositio Math., vol. 6 (1938), 239–250.Google Scholar
[3] Kagno, I. N., “Desargues’ and Pappus’ Graphs and Their Groups,” Amer. J. Math., vol. 69 (1947), 859-862; see footnote 2 on page 859.Google Scholar
[4] Tutte, W. T., “A Family of Cubical Graphs,” Proc. Cambridge Philos. Soc, vol. 43 (1948), 459-474.Google Scholar
[5] Frucht, R., “Die Gruppe des Petersenschen Graphen und der Kantensysteme der regulären Polyeder,” Comment. Math. Helv., vol. 9 (1937), 217-223.Google Scholar
[6] Kagno, I. N., “Linear Graphs of Degree ≦ 6 and Their Groups,” Amer. J. Math., vol. 68 (1946), 505-520, and vol. 69 (1947), 872.Google Scholar