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Hamiltonian Cubic Graphs and Centralizers of Involutions

Published online by Cambridge University Press:  20 November 2018

László Babai ,
Péter Frankl ,
János Kollár  and
Gert Sabidussi
László Babai
Affiliation:
Eötvös L. University, Budapest, Hungary
Péter Frankl
Affiliation:
Hungarian Academy of Science, Budapest, Hungary
János Kollár
Affiliation:
Université de Montréal, Montréal, Québec
Gert Sabidussi
Affiliation:
Université de Montréal, Montréal, Québec
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In 1948, R. Frucht [5] proved that, given a finite group G, there are infinitely many connected cubic graphs X such that the automorphism group Aut X is isomorphic to G. In a letter, Professor Frucht has proposed the problem, whether in addition X can be required to be hamiltonian. One of the aims of the present note is to answer this question affirmatively.

THEOREM 1.1. Given a finite group G there are infinitely many finite hamiltonian cubic graphs Y such that Aut Y ≌ G.

In fact, we prove the following:

THEOREM 1.2. Given a finite cubic graph X having no component isomorphic to K4, there exists a hamiltonian cubic graph Y such that Aut Y ≌ Aut X and |V(Y)| = 6|V(X)|.

This implies 1.1 by the theorem of Frucht [5] mentioned above.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 3 , 01 June 1979 , pp. 458 - 464
Copyright
Copyright © Canadian Mathematical Society 1979

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