Symmetry breaking in planar and maximal outerplanar graphs

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has a vertex (edge) labeling with $d$ labels that is preserved only by a trivial automorphism. In this paper we consider the maximal outerplanar graphs (MOP graphs) and show that MOP graphs, except $K_3$, can be distinguished by at most two vertex (edge) labels. We also compute the distinguishing number and the distinguishing index of Halin and Mycielskian graphs.Comment: 10 pages, 6 figure

Distinguishing number and distinguishing index of lexicographic product of two graphs

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. The lexicographic product of two graphs $G$ and $H$, $G[H]$ can be obtained from $G$ by substituting a copy $H_u$ of $H$ for every vertex $u$ of $G$ and then joining all vertices of $H_u$ with all vertices of $H_v$ if $uv\in E(G)$. In this paper we obtain some sharp bounds for the distinguishing number and the distinguishing index of lexicographic product of two graphs. As consequences, we prove that if $G$ is a connected graph with a special condition on automorphism group of $G[G]$ and $D(G)> 1$, then for every natural $k$, $D(G)\leq D(G^k)\leq D(G)+k-1$, where $G^k=G[G[...]]$. Also we prove that all lexicographic powers of $G$, $G^k$ ($k\geq 2$) can be distinguished by at most two edge labels.Comment: 11 pages, 2 figures. arXiv admin note: text overlap with arXiv:1606.0375

Relationship between the distinguishing index, minimum degree and maximum degree of graphs

Let $\delta$ and $\Delta$ be the minimum and the maximum degree of the vertices of a simple connected graph $G$, respectively. 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. Motivated by a conjecture by Pil\'sniak (2017) that implies that for any $2$-connected graph $D'(G) \leq \lceil \sqrt{\Delta (G)}\rceil +1$, we prove that for any graph $G$ with $\delta\geq 2$, $D'(G) \leq \lceil \sqrt[\delta]{\Delta }\rceil +1$. Also, we show that the distinguishing index of $k$-regular graphs is at most $2$, for any $k\geq 5$.Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1702.03524, arXiv:1704.0415

Characterization of graphs with distinguishing number equal list distinguishing number

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. A list assignment to $G$ is an assignment $L = \{L(v)\}_{v\in V (G)}$ of lists of labels to the vertices of $G$. A distinguishing $L$-labeling of $G$ is a distinguishing labeling of $G$ where the label of each vertex $v$ comes from $L(v)$. The list distinguishing number of $G$, $D_l(G)$ is the minimum $k$ such that every list assignment to $G$ in which $|L(v)| = k$ for all $v \in V (G)$ yields a distinguishing $L$-labeling of $G$. In this paper, we determine the list-distinguishing number for two families of graphs. We also characterize graphs with the distinguishing number equal the list distinguishing number. Finally, we show that this characterization works for other list numbers of a graph.Comment: 10 pages, 2 figure

The chromatic distinguishing index of certain graphs

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 least number $d$ such that $G$ has a proper edge labeling with $d$ labels that is preserved only by the identity automorphism of $G$. In this paper we compute the distinguishing chromatic index for some specific graphs. Also we study the distinguishing chromatic index of corona product and join of two graphs.Comment: 12 pages, 2 figure

On the saturation number of graphs

Let $G=(V,E)$ be a simple connected graph. A matching $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. A matching $M$ is maximal if it cannot be extended to a larger matching in $G$. The cardinality of any smallest maximal matching in $G$ is the saturation number of $G$ and is denoted by $s(G)$. In this paper we study the saturation number of the corona product of two specific graphs. We also consider some graphs with certain constructions that are of importance in chemistry and study their saturation number.Comment: 12 pages, 7 figure

The distinguishing chromatic number of bipartite graphs of girth at least six

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. Motivated by a conjecture in \cite{colins}, we prove that if $G$ is a bipartite graph of girth at least six with the maximum degree $\Delta (G)$, then $\chi_{D}(G)\leq \Delta (G)+1$. We also obtain an upper bound for $\chi_{D}(G)$ where $G$ is a graph with at most one cycle. Finally, we state a relationship between the distinguishing chromatic number of a graph and its spanning subgraphs.Comment: 6 page

More on the sixth coefficient of the matching polynomial in regular graphs

A matching set $M$ in a graph $G$ is a collection of edges of $G$ such that no two edges from $M$ share a vertex. In this paper we consider some parameters related to the matching of regular graphs. We find the sixth coefficient of the matching polynomial of regular graphs. As a consequence, every cubic graph of order $10$ is matching unique.Comment: 11 pages, 5 figure

Distinguishing number and distinguishing index of natural and fractional powers of graphs

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. For any $n \in \mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$. The $m^{th}$ power of $G$, is a graph with same set of vertices of $G$ and an edge between two vertices if and only if there is a path of length at most $m$ between them. The fractional power of $G$, denoted by $G^{\frac{m}{n}}$ is $m^{th}$ power of the $n$-subdivision of $G$ or $n$-subdivision of $m$-th power of $G$. In this paper we study the distinguishing number and distinguishing index of natural and fractional powers of $G$. We show that the natural powers more than two of a graph distinguished by three edge labels. Also we show that for a connected graph $G$ of order $n \geqslant 3$ with maximum degree $\Delta (G)$, $D(G^{\frac{1}{k}})\leqslant min\{s: 2^k+\sum^s_{n=3}n^{k-1}\geqslant \Delta (G)\}$ and for $m\geqslant 3$, $D'(G^{\frac{m}{k}})\leqslant 3$.Comment: 13 page

Distinguishing number and distinguishing index of neighbourhood corona of two graphs

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. The neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on automorphisms, we study the distinguishing number and the distinguishing index of $G_1\star G_2$. We obtain upper bounds for $D(G_1\star G_2)$ and $D'(G_1\star G_2)$.Comment: 15 pages, 11 figure