# On distinguishing and distinguishing chromatic numbers of hypercubes

Discussiones Mathematicae Graph Theory (2008)

- Volume: 28, Issue: 3, page 419-429
- ISSN: 2083-5892

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topWerner Klöckl. "On distinguishing and distinguishing chromatic numbers of hypercubes." Discussiones Mathematicae Graph Theory 28.3 (2008): 419-429. <http://eudml.org/doc/270608>.

@article{WernerKlöckl2008,

abstract = {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.},

author = {Werner Klöckl},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {distinguishing number; distinguishing chromatic number; hypercube; weak Cartesian product},

language = {eng},

number = {3},

pages = {419-429},

title = {On distinguishing and distinguishing chromatic numbers of hypercubes},

url = {http://eudml.org/doc/270608},

volume = {28},

year = {2008},

}

TY - JOUR

AU - Werner Klöckl

TI - On distinguishing and distinguishing chromatic numbers of hypercubes

JO - Discussiones Mathematicae Graph Theory

PY - 2008

VL - 28

IS - 3

SP - 419

EP - 429

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/270608

ER -

## References

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- [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. Zbl1205.05198
- [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.
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