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Cubic identity graphs and planar graphs derived from trees

Published online by Cambridge University Press:  09 April 2009

A. T. Balaban ,
Roy O. Davies ,
Frank Harary ,
Anthony Hill  and
Roy Westwick
A. T. Balaban
Affiliation:
Institute of Atomic Physics, Bucharest and International Atomic Energy Agency, Vienna
Roy O. Davies
Affiliation:
Leicester University and University College London
Frank Harary
Affiliation:
Research Center for Group Dynamics University of Michigan, Ann Arbor
Anthony Hill
Affiliation:
Chelsea School of Art, London
Roy Westwick
Affiliation:
University of British Columbia, Vancouver and University of Stockholm
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Abstract

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The smallest (nontrivial) identity graph is known to have six points and the smallest identity tree seven. It is now shown that the smallest cubic identity graphs have 12 points and that exactly two of them are planar, namely those constructed by Frucht in his proof that every finite group is isomorphic to the automorphism group of some cubic graph. Both of these graphs can be obtained from plane trees by joining consecutive endpoints; it is shown that when applied to identity trees this construction leads to identity graphs except in certain specified cases. In appendices all connected cubic graphs with 10 points or fewer, and all cubic nonseparable planar graphs with 12 points, are displayed.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 11 , Issue 2 , May 1970 , pp. 207 - 215
Copyright
Copyright © Australian Mathematical Society 1970

References

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