Cartesian powers of 3-manifolds

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...

The 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...

On the metric dimension of Cartesian powers of a graph

A set of vertices S resolves a graph if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of a graph is the minimum cardinality of a resolving set of the graph. Fix a connected graph G on q ≥ 2 vertices, and let M be the distance matrix of G. We prove that if there exists w ∈ Z such that ∑ i wi = 0 and the vector Mw, after sorting its coor...

Cartesian Products of Contractible Open Manifolds