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...

Albertson and Collins defined the distinguishing number of a graph to be the smallest number of colors needed to color its vertices so that the coloring is preserved only by the identity automorphism. Collins and Trenk followed by defining the distinguishing chromatic number of a graph to be the smallest size of a coloring that is both proper and distinguishing. We show that, with two exception...

The Mycielskian construction, denoted ?(G), takes a finite simple graph G to larger with of the same clique number but chromatic number. generalized ?t(G), odd girth. In this chapter we look at symmetry parameters ?(G) and ?t(G) in terms G. These include determining number, distinguishing cost distinguishing.

The most recent results and some of the open key questions on the evolution of early-type galaxies are reviewed in the general cosmological context of massive galaxy formation.

A distinguishing partition for an action of a group Γ on a set X is a partition of X that is preserved by no nontrivial element of Γ. As a special case, a distinguishing partition of a graph is a partition of the vertex set that is preserved by no nontrivial automorphism. Not all graphs admit a distinguishing partition – for example, the complete graph Kn for n ≥ 2 does not admit a distinguishi...

4 A graph is said to be d-distinguishable if there exists a d-labeling of its vertices which is 5 only preserved by the identity map. The distinguishing number of a graph G is the smallest 6 number d for which G is d-distinguishable. We show that the distinguishing number of trees 7 and forests can be computed in linear time, improving the previously known O(n log n) time 8 algorithm. 9

Chi-square tests are generally used for distinguishing purposes; however when they are combined to simultaneously test several independent variables, extra notation is required. In this study, the chi-square statistics in some previous works is revealed to be computed half of its real value. Therefore, the notion of Multi _ Chi-square tests is formulated to avoid possible future confusions. In ...

Suppose Γ is a group acting on a set X. A k-labeling of X is a mapping c :X → {1,2, . . . , k}. A labeling c of X is distinguishing (with respect to the action of Γ ) if for any g ∈ Γ , g = idX , there exists an element x ∈X such that c(x) = c(g(x)). The distinguishing number, DΓ (X), of the action of Γ onX is the minimum k for which there is a k-labeling which is distinguishing. This paper stu...

Definition 1.2 (Distinguishing number of a graph). Let Γ = (V,E) be a graph and let f : V → [r] be a coloring of the set of vertices by r colors. The map f need not be surjective, in fact when r > |V | it cannot be surjective. We say that f is r-distinguishing if the only automorphism of Γ that fixes the coloring f is the trivial automorphism. The distinguishing number of Γ is denoted by D(Γ) a...

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 χD(G) of G. Extending these concepts to infinite graphs we prove that D(Qא0) = 2 and χD(Qא0) = 3, where Qא0 denotes the hypercube of cou...