Distinguishing Cartesian Powers of Graphs
نویسندگان
چکیده
منابع مشابه
Distinguishing Cartesian powers of graphs
The distinguishing number D(G) of a graph is the least integer d such that there is a d-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. We show that the distinguishing number of the square and higher powers of a connected graph G 6= K2,K3 with respect to the Cartesian product is 2. This result strengthens results of Albertson [1] on powers of prime graph...
متن کاملDistinguishing Cartesian Powers of Graphs
Given a graph G, a labeling c : V (G) → {1, 2, . . . , d} is said to be d-distinguishing if the only element in Aut(G) that preserves the labels is the identity. The distinguishing number of G, denoted by D(G), is the minimum d such that G has a d-distinguishing labeling. If G2H denotes the Cartesian product of G and H, let G 2 = G2G and G r = G2G r−1 . A graph G is said to be prime with respec...
متن کاملThe Cost of 2-Distinguishing Cartesian Powers
A graph G is said to be 2-distinguishable if there is a labeling of the vertices with two labels so that only the trivial automorphism preserves the label classes. The minimum size of a label class in any such labeling of G is called the cost of 2-distinguishing G and is denoted by ρ(G). The determining number of a graph G, denoted Det(G), is the minimum size of a set of vertices whose pointwis...
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The distinguishing number (resp. index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (resp. edge labeling) with $d$ labels that is preserved only by a trivial automorphism. For any $n in mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$...
متن کاملDistinguishing colorings of Cartesian products of complete graphs
We determine the values of s and t for which there is a coloring of the edges of the complete bipartite graph Ks,t which admits only the identity automorphism. In particular this allows us to determine the distinguishing number of the Cartesian product of complete graphs. The distinguishing number of a graph is the minimum number of colors needed to label the vertices so that the only color pre...
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ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2005
ISSN: 1077-8926
DOI: 10.37236/1984