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On distinguishing and distinguishing chromatic numbers of hypercubes

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Klöckl W.

Discussiones Mathematicae Graph Theory

2008

tom 28

nr 3

419-429

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.

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1 Discussiones Mathematicae Graph Theory

1 Klöckl W.

1 2008

Wyniki wyszukiwania

On distinguishing and distinguishing chromatic numbers of hypercubes