Cartesian powers of graphs can be distinguished by two labels

Cartesian powers of graphs can be distinguished by two labels

1 The distinguishing number D(G) of a graph G is the least integer d such that there is a d-labeling 2 of the vertices of G which is not preserved by any nontrivial automorphism. For a graph G let Gr 3 be the r th power of G with respect to the Cartesian product. It is proved that D(Gr ) = 2 for any 4 connected graph G with at least 3 vertices and for any r ≥ 3. This confirms and strengthens a ...

Cartesian powers of graphs can be distinguished with two labels

The distinguishing number D(G) of a graph G is the least integer d such that there is a d-labeling of the vertices of G which is not preserved by any nontrivial automorphism. For a graph G let G be the rth power of G with respect to the Cartesian product. It is proved that D(G) = 2 for any connected graph G with at least 3 vertices and for any r ≥ 3. This confirms and strengthens a conjecture o...

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...

Distinguishing 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...

Edge-Isoperimetri Problems for Cartesian Powers of Regular Graphs