On distinguishing and distinguishing chromatic numbers of hypercubes

• Autorzy: Werner Klöckl
• 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.

Słowa kluczowe

EN

distinguishing number; distinguishing chromatic number; hypercube; weak Cartesian product

Kategorie tematyczne

• 05C12: Distance in graphs
• 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
• 05C15: Coloring of graphs and hypergraphs

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2008
• Tom: 28
• Numer: 3
• Strony: 419-429

Daty

wydano

2008

otrzymano

2006-10-18

poprawiono

2008-06-06

zaakceptowano

2008-06-06

Twórcy

autor

Werner Klöckl

• 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.

Identyfikatory

DOI

10.7151/dmgt.1416