Discussiones Mathematicae Graph Theory 33(2) (2013)
329-336
DOI: https://doi.org/10.7151/dmgt.1661
Frucht's Theorem for the Digraph Factorial
| Richard H. Hammack
Department of Mathematics and Applied Mathematics |
Abstract
To every graph (or digraph) A, there is an associated automorphism group Aut(A). Frucht's theorem asserts the converse association; that for any finite group G there is a graph (or digraph) A for which Aut(A) ≅ G.A new operation on digraphs was introduced recently as an aid in solving certain questions regarding cancellation over the direct product of digraphs. Given a digraph A, its factorial A! is certain digraph whose vertex set is the permutations of V(A). The arc set E(A!) forms a group, and the loops form a subgroup that is isomorphic to Aut(A). (So E(A!) can be regarded as an extension of Aut(A).)
This note proves an analogue of Frucht's theorem in which Aut(A) is replaced by the group E(A!). Given any finite group G, we show that there is a graph A for which E(A!) ≅ G.
Keywords: Frucht's theorem, digraphs, graph automorphisms, digraph factorial
2010 Mathematics Subject Classification: 05C25.
References
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Received 10 September 2011
Revised 19 March 2012
Accepted 22 March 2012
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