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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory


Discussiones Mathematicae Graph Theory 33(2) (2013) 329-336
DOI: 10.7151/dmgt.1661

Frucht's Theorem for the Digraph Factorial

Richard H. Hammack

Department of Mathematics and Applied Mathematics
Virginia Commonwealth University
Richmond, VA 23284 USA


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.


[1]G. Chartrand, L. Lesniak and P. Zhang, Graphs and Digraphs, 5th edition (CRC Press, Boca Raton, FL, 2011).
[2]R. Hammack, Direct product cancellation of digraphs, European J. Combin. 34 (2013) 846--858, doi: 10.1016/j.ejc.2012.11.003.
[3]R. Hammack, On direct product cancellation of graphs, Discrete Math. 309 (2009) 2538--2543, doi: 10.1016/j.disc.2008.06.004.
[4]R. Hammack and H. Smith, Zero divisors among digraphs, Graphs Combin, doi: 10.1007/s00373-012-1248-x.
[5]R. Hammack and K. Toman, Cancellation of direct products of digraphs, Discuss. Math. Graph Theory 30 (2010) 575--590, doi: 10.7151/dmgt.1515.
[6]R. Hammack, W. Imrich, and S. Klavžar, Handbook of Product Graphs, 2nd edition, Series: Discrete Mathematics and its Applications (CRC Press, Boca Raton, FL, 2011).
[7]P. Hell and J. Nešetřil, Graphs and Homomorphisms, Oxford Lecture Series in Mathematics (Oxford Univ. Press, 2004), doi: 10.1093/acprof:oso/9780198528173.001.0001.
[8]L. Lovász, On the cancellation law among finite relational structures, Period. Math. Hungar. 1 (1971) 145--156, doi: 10.1007/BF02029172.

Received 10 September 2011
Revised 19 March 2012
Accepted 22 March 2012