We introduce the endomorphism distinguishing number De(G) of a graph G as the least cardinal d such that G has a vertex coloring with d colors that is only preserved by the trivial endomorphism. This generalizes the notion of the distinguishing number D(G) of a graph G, which is defined for automorphisms instead of endomorphisms. As the number of endomorphisms can vastly exceed the number of au...

A vertex k-labeling of graph G is distinguishing if the only automorphism that preserves the labels of G is the identity map. The distinguishing number of G, D(G), is the smallest integer k for which G has a distinguishing k-labeling. In this paper, we apply the principle of inclusion-exclusion and develop recursive formulas to count the number of inequivalent distinguishing k-labelings of a gr...

A graph G is distinguished if its vertices are labelled by a map φ : V (G) −→ {1, 2, . . . , k} so that no non-trivial graph automorphism preserves φ. The distinguishing number of G is the minimum number k necessary for φ to distinguish the graph. It measures the symmetry of the graph. We extend these definitions to an arbitrary group action of Γ on a set X. A labelling φ : X −→ {1, 2, . . . , ...

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...

The distinguishing number D(G) of a graph G is the least cardinal number א such that G has a labeling with א labels that is only preserved by the trivial automorphism. We show that the distinguishing number of the countable random graph is two, that tree-like graphs with not more than continuum many vertices have distinguishing number two, and determine the distinguishing number of many classes...

The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists vertex coloring $c:V(G)\to\{1,2,\dotsc,k\}$ whose induced edge labels $\{c(u),c(v)\}$ are distinct for all edges $uv$. Previous work has determined EDCN paths, cycles, and spider graphs with three legs. In this paper, we determine petal two petals loop, cycles one chord, fou...

Given a group Γ acting on a set X, a k-coloring φ : X → {1, . . . , k} of X is distinguishing with respect to Γ if the only γ ∈ Γ that fixes φ is the identity action. The distinguishing number of the action Γ, denoted DΓ(X), is then the smallest positive integer k such that there is a distinguishing k-coloring of X with respect to Γ. This notion has been studied in a number of settings, but by ...

Let G be a graph. A vertex labeling of G is distinguishing if the only label-preserving automorphism of G is the identity map. The distinguishing number of G, D(G), is the minimum number of labels needed so that G has a distinguishing labeling. In this paper, we present O(n log n)-time algorithms that compute the distinguishing numbers of trees and forests. Unlike most of the previous work in t...

A labeling of a graph G is distinguishing if it is only preserved by the trivial automorphism of G. The distinguishing chromatic number of G is the smallest integer k such that G has a distinguishing labeling that is at the same time a proper vertex coloring. The distinguishing chromatic number of the Cartesian product Kk Kn is determined for all k and n. In most of the cases it is equal to the...