For an assignment of numbers to the vertices of a graph, let S[u] be the sum of the labels of all the vertices in the closed neighborhood of u, for a vertex u. Such an assignment is called closed distinguishing if S[u] 6= S[v] for any two adjacent vertices u and v unless the closed neighborhoods of u and v coincide. In this note we investigate dis[G], the smallest integer k such that there is a...

The distinguishing number of a graph G is a symmetry related graph invariant whose study started two decades ago. The distinguishing number D(G) is the least integer d such that G has a distinguishing d-coloring. A distinguishing d-coloring is a coloring c : V (G) → {1, · · · , d} invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing num...

The local chromatic number of a graph G is the number of colors appearing in the most colorful closed neighborhood of a vertex minimized over all proper colorings of G. We show that two specific topological obstructions that have the same implications for the chromatic number have different implications for the local chromatic number. These two obstructions can be formulated in terms of the hom...

The distinguishing number of a graph G, denoted D(G), is the minimum number of colors such that there exists a coloring of the vertices of G where no nontrivial graph automorphism is color-preserving. In this paper, we answer an open question posed in [3] by showing that the distinguishing number of Qpn, the p th graph power of the n-dimensional hypercube, is 2 whenever 2 < p < n − 1. This comp...

Let G be a group acting faithfully on a set X . The distinguishing number of the action of G on X , denoted DG(X ), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a colorpreserving permutation of X . In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. ...

The i-local distinguishing number of G, denoted by LD i (G), was deened in 3]. Let T be a tree on n > 2 vertices with maximum degree bounded by some constant k. It is shown that LD 1 (T) = O(p n) and that for some trees this bound is tight. The result is extended to show that LD i (T) = O(n 1=(i+1)).

A group of permutations G of a set V is k-distinguishable if there exists a partition of V into k cells such that only the identity permutation in G fixes setwise all of the cells of the partition. The least cardinal number k such that (G,V ) is k-distinguishable is its distinguishing number, D(G,V ). In particular, a graph Γ is k-distinguishable if its automorphism group Aut(Γ) satisfies D(Aut...

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 stream ciphers Py, Py6 were designed by Biham and Seberry for the ECRYPT-eSTREAM project in 2005. However, due to several recent cryptanalytic attacks on them, a strengthened version Pypy was proposed to rule out those attacks. The ciphers have been promoted to the ‘Focus’ ciphers of the Phase II of the eSTREAM project. The impressive speed of the ciphers make them the forerunners in the co...

The Distinguishing Chromatic Number of a graph G, denoted χD(G), was first defined in [5] as the minimum number of colors needed to properly color G such that no non-trivial automorphism φ of the graph G fixes each color class of G. In this paper, we consider certain ‘natural’ families of bipartite graphs that have reasonably large automorphism groups and we show that in all those cases, the di...