Geodesic
Frucht’s Theorem for the Digraph Factorial
Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 329-336

Voir la notice de l'article provenant de la source Library of Science

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 
@article{DMGT_2013_33_2_a6,
     author = {Hammack, Richard H.},
     title = {Frucht{\textquoteright}s {Theorem} for the {Digraph} {Factorial}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {329--336},
     publisher = {mathdoc},
     volume = {33},
     number = {2},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a6/}
}
TY  - JOUR
AU  - Hammack, Richard H.
TI  - Frucht’s Theorem for the Digraph Factorial
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2013
SP  - 329
EP  - 336
VL  - 33
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a6/
LA  - en
ID  - DMGT_2013_33_2_a6
ER  - 
%0 Journal Article
%A Hammack, Richard H.
%T Frucht’s Theorem for the Digraph Factorial
%J Discussiones Mathematicae. Graph Theory
%D 2013
%P 329-336
%V 33
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a6/
%G en
%F DMGT_2013_33_2_a6
Hammack, Richard H. Frucht’s Theorem for the Digraph Factorial. Discussiones Mathematicae. Graph Theory, Tome 33 (2013) no. 2, pp. 329-336. http://geodesic.mathdoc.fr/item/DMGT_2013_33_2_a6/