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2013 | 33 | 2 | 329-336
Tytuł artykułu

Frucht’s Theorem for the Digraph Factorial

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
33
Numer
2
Strony
329-336
Opis fizyczny
Daty
wydano
2013-05-01
online
2013-04-13
Twórcy
  • Department of Mathematics and Applied Mathematics Virginia Commonwealth University Richmond, VA 23284 USA, rhammack@vcu.edu
Bibliografia
  • [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[Crossref]
  • [3] R. Hammack, On direct product cancellation of graphs, Discrete Math. 309 (2009) 2538-2543. doi:10.1016/j.disc.2008.06.004[Crossref][WoS]
  • [4] R. Hammack and H. Smith, Zero divisors among digraphs, Graphs Combin. doi:10.1007/s00373-012-1248-x[Crossref]
  • [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[Crossref]
  • [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[Crossref]
  • [8] L. Lovász, On the cancellation law among finite relational structures, Period. Math. Hungar. 1 (1971) 145-156. doi:10.1007/BF02029172[Crossref]
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1661
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