The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study starte... more 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 $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending...
Introduced by Albertson and al. [1], the distinguishing number D(G) of a graph G is the least int... more Introduced by Albertson and al. [1], the distinguishing number D(G) of a graph G is the least integer r such that there is a r-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs not depend on n. In this paper, we study circulant graphs of order n where the adjacency is defined using a symmetric subset A of Zn, called generator. We show that we can built a family of circulant graphs of order n having distinct distinguishing numbers not depend on n.
Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ ... more Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator. We give a construction of a family of circulant graphs of order $n$ and we show that this class has distinct distinguishing numbers and these lasters are not depending on $n$.
The distinguishing number of a graph $H$ is a symmetry related graph invariant whose study starte... more The distinguishing number of a graph $H$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(H)$ is the least integer $d$ such that $H$ has a $d$-distinguishing coloring. A $d$-distinguishing coloring is a coloring $c:V(H)\rightarrow\{1,\dots,d\}$ invariant only under the trivial automorphism. In this paper, we continue the study of a game variant of this parameter, recently introduced. The distinguishing game is a game with two players, Gentle and Rascal, with antagonist goals. This game is played on a graph $H$ with a fixed set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $H$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then Gentle wins if the coloring is $d$-distinguishing and Rascal wins otherwise. This game defines two new invariants, which are the minimum numbers of colors needed to ensure that Gentle has a winning strategy, depending who start...
The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study starte... more 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 $d$-distinguishing coloring. A distinguishing $d$-coloring is a coloring $c:V(G)\rightarrow\{1,...,d\}$ invariant only under the trivial automorphism. In this paper, we introduce a game variant of the distinguishing number. The distinguishing game is a game with two players, the Gentle and the Rascal, with antagonist goals. This game is played on a graph $G$ with a set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $G$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then the Gentle wins if the coloring is distinguishing and the Rascal wins otherwise. This game leads to define two new invariants for a graph $G$, which are the minimum numbers of colors needed to ensure that the Gentle has a winning strategy, depending...
Introduced by Albertson and al. [1], the distinguishing number D(G) of a graph G is the least int... more Introduced by Albertson and al. [1], the distinguishing number D(G) of a graph G is the least integer r such that there is a r-labeling of the vertices of G that is not preserved by any nontrivial automorphism of G. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs not depend on n. In this paper, we study circulant graphs of order n where the adjacency is defined using a symmetric subset A of Zn, called generator. We show that we can built a family of circulant graphs of order n having distinct distinguishing numbers not depend on n.
Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ ... more Introduced by Albertson et al. \cite{albertson}, the distinguishing number $D(G)$ of a graph $G$ is the least integer $r$ such that there is a $r$-labeling of the vertices of $G$ that is not preserved by any nontrivial automorphism of $G$. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on $n$. In this paper, we study circulant graphs of order $n$ where the adjacency is defined using a symmetric subset $A$ of $\mathbb{Z}_n$, called generator. We give a construction of a family of circulant graphs of order $n$ and we show that this class has distinct distinguishing numbers and these lasters are not depending on $n$.
The distinguishing number of a graph $H$ is a symmetry related graph invariant whose study starte... more The distinguishing number of a graph $H$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(H)$ is the least integer $d$ such that $H$ has a $d$-distinguishing coloring. A $d$-distinguishing coloring is a coloring $c:V(H)\rightarrow\{1,\dots,d\}$ invariant only under the trivial automorphism. In this paper, we continue the study of a game variant of this parameter, recently introduced. The distinguishing game is a game with two players, Gentle and Rascal, with antagonist goals. This game is played on a graph $H$ with a fixed set of $d\in\mathbb N^*$ colors. Alternately, the two players choose a vertex of $H$ and color it with one of the $d$ colors. The game ends when all the vertices have been colored. Then Gentle wins if the coloring is $d$-distinguishing and Rascal wins otherwise. This game defines two new invariants, which are the minimum numbers of colors needed to ensure that Gentle has a winning strategy, depending who start...
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