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2008 | 28 | 3 | 419-429
Tytuł artykułu

On distinguishing and distinguishing chromatic numbers of hypercubes

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The distinguishing number D(G) of a graph G is the least integer d such that G has a labeling with d colors that is not preserved by any nontrivial automorphism. The restriction to proper labelings leads to the definition of the distinguishing chromatic number $χ_D(G)$ of G.
Extending these concepts to infinite graphs we prove that $D(Q_ℵ₀) = 2$ and $χ_D(Q_ℵ₀) = 3$, where $Q_ℵ₀$ denotes the hypercube of countable dimension. We also show that $χ_D(Q₄) = 4$, thereby completing the investigation of finite hypercubes with respect to $χ_D$.
Our results extend work on finite graphs by Bogstad and Cowen on the distinguishing number and Choi, Hartke and Kaul on the distinguishing chromatic number.
Wydawca
Rocznik
Tom
28
Numer
3
Strony
419-429
Opis fizyczny
Daty
wydano
2008
otrzymano
2006-10-18
poprawiono
2008-06-06
zaakceptowano
2008-06-06
Twórcy
  • Chair of Applied Mathematics, Montanuniversität Leoben, 8700 Leoben, Austria
Bibliografia
  • [1] M.O. Albertson, Distinguishing Cartesian powers of graphs, Electron. J. Combin. 12 (2005) N17.
  • [2] M.O. Albertson and K.L. Collins, Symmetry breaking in graphs, Electron. J. Combin. 3 (1996) R18.
  • [3] B. Bogstad and L.J. Cowen, The distinguishing number of hypercubes, Discrete Math. 283 (2004) 29-35, doi: 10.1016/j.disc.2003.11.018.
  • [4] M. Chan, The distinguishing number of the augmented cube and hypercube powers, Discrete Math. 308 (2008) 2330-2336, doi: 10.1016/j.disc.2006.09.056.
  • [5] J.O. Choi, S.G. Hartke and H. Kaul, Distinguishing chromatic number of Cartesian products of graphs, submitted.
  • [6] K.T. Collins and A.N. Trenk, The distinguishing chromatic number, Electr. J. Combin. 13 (2006) R16.
  • [7] W. Imrich, J. Jerebic and S. Klavžar, The distinguishing number of Cartesian products of complete graphs, Eur. J. Combin. 29 (2008) 922-927, doi: 10.1016/j.ejc.2007.11.018.
  • [8] W. Imrich and S. Klavžar, Product Graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization (Wiley-Interscience, New York, 2000). Structure and recognition, With a foreword by Peter Winkler.
  • [9] W. Imrich and S. Klavžar, Distinguishing Cartesian powers of graphs, J. Graph Theory 53 (2006) 250-260, doi: 10.1002/jgt.20190.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1416
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