A coloring of the vertices of a graph G is said to be distinguishing provided no nontrivial automorphism of G preserves all of the vertex colors. The distinguishing number of G, D(G), is the minimum number of colors in a distinguishing coloring of G. The distinguishing chromatic number of G, χD(G), is the minimum number of colors in a distinguishing coloring of G that is also a proper coloring....

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...

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has a vertex labeling   with $d$ labels  that is preserved only by a trivial automorphism. The distinguishing chromatic number $chi_{D}(G)$ of $G$ is defined similarly, where, in addition, $f$ is assumed to be a proper labeling. We prove that if $G$ is a bipartite graph of girth at least six with the maximum ...

In this paper we define and study the distinguishing chromatic number, χD(G), of a graph G, building on the work of Albertson and Collins who studied the distinguishing number. We find χD(G) for various families of graphs and characterize those graphs with χD(G) = |V (G)|, and those trees with the maximum chromatic distingushing number for trees. We prove analogs of Brooks’ Theorem for both the...

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...

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 6= idX , there exists an element x ∈ X such that c(x) 6= c(g(x)). The distinguishing number, DΓ(X), of the action of Γ on X is the minimum k for which there is a k-labeling which is distinguishing. This paper...

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.