Related papers: The distinguishing number of the augmented cube an…

An adjacent vertex distinguishing coloring of a graph G is a proper edge coloring of G such that any pair of adjacent vertices are incident with distinct sets of colors. The minimum number of colors needed for an adjacent vertex…

Given a subgraph G of the hypercube Q_n, a coloring of the edges of Q_n such that every embedding of G contains an edge of every color is called a G-polychromatic coloring. The maximum number of colors with which it is possible to…

The distinguishing index of a graph $G$, denoted by $D'(G)$, is the least number of labels in an edge labeling of $G$ not preserved by any non-trivial automorphism. The distinguishing chromatic index $\chi'_D (G)$ of a graph $G$ is the…

The distinguishing number $D(G)$ of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling with $d$ labels that is preserved only by a trivial automorphism. The minimum size of a label class in such a labeling of $G$ with…

A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The distinguishing threshold of a graph $G$, denoted by $\theta(G)$, is the minimum number of colors $k$ so that every $k$-coloring of $G$ is…

The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $\alpha$-regular graph for some…

A packing $k$-coloring of a graph $G$ is a partition of $V(G)$ into sets $V_1,\ldots,V_k$ such that for each $1\leq i\leq k$ the distance between any two distinct $x,y\in V_i$ is at least $i+1$. The packing chromatic number, $\chi_p(G)$, of…

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 a graph $G$ consists in finding an assignment of colors $c: V(G)\to\{1,\ldots,p\}$ such that any pair of adjacent vertices receives different colors. The minimum integer $p$ such that a coloring exists is called the chromatic…

We prove analogs of Brooks' Theorem for the list-distinguishing chromatic number of different classes of simple finite connected graphs. Moreover, we determine two upper bounds for the list-distinguishing chromatic number of a graph G in…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. In this paper, we investigate…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…

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…

The distinguishing number $D(\Gamma)$ of a graph $\Gamma$ is the least size of a partition of the vertices of $\Gamma$ such that no non-trivial automorphism of $\Gamma$ preserves this partition. We show that if the automorphism group of a…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. A graphoidal cover of $G$ is a…

For a graph $G$, a $k$-coloring $c:V(G)\to \{1,2,\ldots, k\}$ is called distinguishing, if the only automorphism $f$ of $G$ with the property $c(v)=c(f(v))$ for every vertex $v\in G$ (color-preserving automorphism), is the identity. In this…

The independence number $\alpha(G)$ and the dissociation number ${\rm diss}(G)$ of a graph $G$ are the largest orders of induced subgraphs of $G$ of maximum degree at most $0$ and at most $1$, respectively. We consider possible improvements…

Given a graph $G$ and an integer $r\ge 1$, the $r$th power $G^r$ of $G$ is the graph obtained from $G$ by adding edges for all pairs of distinct vertices at distance at most $r$ from each other. We focus on two basic structural properties…

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…