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The list Distinguishing Number Equals the Distinguishing Number for Interval Graphs

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Immel P., Wenger P. S.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
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nr 1
165-174
EN
A distinguishing coloring of a graph G is a coloring of the vertices so that every nontrivial automorphism of G maps some vertex to a vertex with a different color. The distinguishing number of G is the minimum k such that G has a distinguishing coloring where each vertex is assigned a color from {1, . . . , k}. A list assignment to G is an assignment L = {L(v)}v∈V (G) of lists of colors to the vertices of G. A distinguishing L-coloring of G is a distinguishing coloring of G where the color of each vertex v comes from L(v). The list distinguishing number of G is the minimum k such that every list assignment to G in which |L(v)| = k for all v ∈ V (G) yields a distinguishing L-coloring of G. We prove that if G is an interval graph, then its distinguishing number and list distinguishing number are equal.
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On distinguishing and distinguishing chromatic numbers of hypercubes

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Klöckl W.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 3
419-429
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.
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Distinguishing Cartesian Products of Countable Graphs

88%
Estaji E., Imrich W., Kalinowski R., Pilśniak M., Tucker T.
Discussiones Mathematicae Graph Theory
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2017
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tom 37
|
nr 1
155-164
EN
The distinguishing number D(G) of a graph G is the minimum number of colors needed to color the vertices of G such that the coloring is preserved only by the trivial automorphism. In this paper we improve results about the distinguishing number of Cartesian products of finite and infinite graphs by removing restrictions to prime or relatively prime factors.
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