In this paper we study the symmetry parameters determining number, distinguishing and cost of 2-distinguishing, for some variations on hypercubes, namely Hamming graphs, powers folded enhanced augmented hypercubes locally twisted hypercubes.

We consider infinite graphs. The distinguishing number D(G) of a graph G is the minimum number of colours in a vertex colouring of G that is preserved only by the trivial automorphism. An analogous invariant for edge colourings is called the distinguishing index, denoted by D′(G). We prove that D′(G) 6 D(G) + 1. For proper colourings, we study relevant invariants called the distinguishing chrom...

A group A acting faithfully on a set X has distinguishing number k, written D(A,X) = k, if there is a coloring of the elements of X with k colors such that no nonidentity element of A is color-preserving, and no such coloring with fewer than k colors exists. Given a map M with vertex set V and automorphism group Aut(M), let D(M) = D(Aut(M), V ). If M is orientable, let D+(M) = D(Aut+(M), V ), w...

We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper n-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph G satisfies the connected existentially closed property and admits a homomorphism to H, then it admits continuum-many distinguishing homomorphism...

Objetivo: focada na Hermenêutica Jurídica e adequada prestação jurisdicional, a pesquisa objetiva traçar elementos do precedente judicial, como um todo, distinguishing, objeto específico, sob o sistema de precedentes obrigatórios adotado pelo Código Processo Civil brasileiro. O argumento distinção impõe busca sua natureza jurídica, seus limites e, ainda, reforçou importância da atividade interp...

The distinguishing number of a group G acting faithfully on a set V is the least number of colors needed to color the elements of V so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph Γ is said to have connectivity 1 if there exists a vertex α ...

We examine the distinguishing number of the Cartesian product of an arbitrary number of complete graphs. We show that for u1 ≤ · · · ≤ ud the distinguishing number of the Cartesian product of complete graphs of these sizes is either du d e or du 1/s d e + 1 where s = Πd−1 i=1 ui. In most cases, which of these values it is can be explicitly determined.

We analyse several well-known key establishment protocols for mobile communications. The protocols are examined with respect to their security and suitability in mobile environments. In a number of cases weaknesses are pointed out, and in many cases reenements are suggested, either to improve the eeciency or to allow simpliied security analysis.

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...