Discussiones Mathematicae Graph Theory 28(3) (2008)
419-429
doi: 10.7151/dmgt.1416
Werner Klöckl
Chair of Applied Mathematics
Montanuniversität Leoben, 8700 Leoben, Austria
e-mail: werner.kloeckl@mu-leoben.at
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}(Q_{4}) = 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.
Keywords: distinguishing number, distinguishing chromatic number, hypercube, weak Cartesian product.
2000 Mathematics Subject Classification: Primary: 05C25, 05C15; Secondary: 05C12.
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Received 18 October 2006
Revised 6 June 2008
Accepted 6 June 2008