---
abstract: 'Let $G$ be a simple graph. A total dominator coloring of $G$, is a proper coloring of the vertices of $G$ in which each vertex of the graph is adjacent to every vertex of some color class. The total dominator chromatic (TDC) number $\chi_d^t(G)$ of $G$, is the minimum number of colors among all total dominator coloring of $G$. 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 study the total dominator chromatic number of the neighbourhood of two graphs and investigate the total dominator chromatic number of $r$-gluing of two graphs. Stability (bondage number) of total dominator chromatic number of $G$ is the minimum number of vertices (edges) of $G$ whose removal changes the TDC-number of $G$. We study the stability and bondage number of certatin graphs.'
author:
- 'Nima Ghanbari and Saeid Alikhani$^{}$[^1]'
title:  More on the total dominator chromatic number of a graph
---

Department of Mathematics, Yazd University, 89195-741, Yazd, Iran\
[n.ghanbari.math@gmail.com, alikhani@yazd.ac.ir]{}\

[**Keywords:**]{} total dominator chromatic number; neighbourhood corona; stability.

[**AMS Subj. Class.:**]{} 05C15, 05C69

Introduction
============

In this paper, we are concerned with simple finite graphs. Let $G=(V,E)$ be such a graph and $\lambda \in \mathbb{N}$. A mapping $f : V (G)\longrightarrow \{1, 2,...,\lambda\}$ is called a $\lambda$-proper coloring of $G$, if $f(u) \neq f(v)$ whenever the vertices $u$ and $v$ are adjacent in $G$. A color class of this coloring, is a set consisting of all those vertices assigned the same color. If $f$ is a proper coloring of $G$ with the coloring classes $V_1, V_2,..., V_{\lambda}$ such that every vertex in $V_i$ has color $i$, then sometimes write simply $f = (V_1,V_2,...,V_{\lambda})$. The chromatic number $\chi(G)$ of $G$ is the minimum number of colors needed in a proper coloring of a graph. The concept of a graph coloring and chromatic number is very well-studied in graph theory.

Vijayalekshmi and Kazemi [@Adel; @Adel2; @Vij1; @Vij2] studied the total dominator coloring, abbreviated TD-coloring. Let $G$ be a graph with no isolated vertex, the total dominator coloring is a proper coloring of $G$ in which each vertex of the graph is adjacent to every vertex of some (other) color class. The total dominator chromatic number, abbreviated TDC-number, $\chi_d^t(G)$ of $G$ is the minimum number of color classes in a TD-coloring of $G$. Computation of the TDC-number is NP-complete ([@Adel]). The TD-chromatic number of some graphs, such as paths, cycles, wheels and the complement of paths and cycles has computed in [@Adel]. Also Henning in [@GCOM] established the lower and the upper bounds on the TDC-number of a graph in terms of its total domination number $\gamma_t(G)$. He has shown that, every graph $G$ with no isolated vertex satisfies $\gamma_t(G) \leq \chi_d^t (G)\leq \gamma_t(G) + \chi(G)$. The properties of TD-colorings in trees has studied in [@GCOM; @Adel]. Trees $T$ with $\gamma_t(T) =\chi_d^t(T)$ has characterized in [@GCOM]. We have examined the effects on $\chi_d^t(G)$ when $G$ is modified by operations on vertex and edge of $G$ in [@nima2].

The corona of two graphs $G$ and $H$ which denoted by $G\circ H$ is defined in [@Harary] and there have been some results on the corona of two graphs [@Frucht]. In [@nima1] we have studied the total dominator chromatic number of corona of two graphs. In this paper we consider another variation of corona of two graphs and study its total dominator chromatic number. Given simple graphs $G_1$ and $G_2$, the neighbourhood corona of $G_1$ and $G_2$, 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 $G_2$ ([@Gopalapillai]). Figure \[fig1\] shows $P_4\star P_3$, where $P_n$ is the path of order $n$. Liu and Zhu in [@linear] determined the adjacency spectrum of $G_1\star G_2$ for arbitrary $G_1$ and $G_2$ and the Laplacian spectrum and signless Laplacian spectrum of $G_1\star G_2$ for regular $G_1$ and arbitrary $G_2$, in terms of the corresponding spectrum of $G_1$ and $G_2$. Also Gopalapillai in [@Gopalapillai] has studied the eigenvalues and spectrum of $G_1\star G_2$, when $G_2$ is regular.

A domination-critical (domination-super critical, respectively) vertex in a graph $G$, is a vertex whose removal decreases (increases, respectively) the domination number. Bauer et al. [@1] introduced the concept of domination stability in graphs. The domination stability, or just $\gamma$-stability, of a graph $G$ is the minimum number of vertices whose removal changes the domination number. Motivated by domination stability, we introduce the total dominator chromatic (TDC)-stability (TDC-bondage number) of a graph.

The total dominator chromatic (TDC)-stability (TDC-bondage number) of graph $G$, is the minimum number of vertices (edges) of $G$, whose removal changes the TDC-number of $G$.

![\[fig1\] The neighbourhood corona of $P_4 \star P_3$.](fig100){width="37.00000%"}

In the next section, we study the total dominator chromatic number of neighbourhood corona of two graphs. We investigate the total dominator chromatic number of $r$-gluing of two graphs in Section 3. We study the TDC-stability and TDC-bondage number of certain graphs in Section 4.

TDC-number of neighbourhood corona of two graphs
================================================

The TDC-number of binary graph operations, aside from Cartesian and corona product ([@nima1; @Adel2]), have not been widely studied. In this section, we shall study the total dominator chromatic (TDC) number of neighbourhood corona of two graphs. We begin with the following theorem which gives an upper bound for $\chi_d^t(G_1\star G_2)$.

\[1\] For any two connected graphs $G_1$ and $G_2$, $$\chi_d^t(G_1\star G_2)\leq\vert V(G_1)\vert+\vert V(G_2)\vert.$$

[[**Proof. **]{}]{} First we give the colors $a_1,a_2,\ldots,a_{|V(G_1)|}$ to the vertices of $G_1$. Now for every copy of $G_2$ we give the colors $b_1,b_2,\ldots,b_{|V(G_2)|}$ to its vertices. Since $G_1$ is connected, each vertex of $G_1$ uses one of its adjacent colors for TD-coloring. Also each vertex of each $G_2$ is adjacent to at least one vertex of $G_1$ and can use this color for TD-coloring. Therefore this is a TD-coloring for the $G_1\star G_2$ and we have the result. [$\square$]{}

The following theorem gives another upper bound for the total dominator chromatic number of the neighbourhood of two graphs.

\[2\] For any two connected graphs $G_1$ and $G_2$, $$\chi_d^t(G_1\star G_2)\leq\chi_d^t(G_1)+\vert V(G_2)\vert.$$

[[**Proof. **]{}]{}First we give the colors $a_1,a_2,\ldots,a_{\chi_d^t(G_1)}$ to the vertices of $G_1$ as we colored $G_1$ for a TD-coloring. Now for every copy of $G_2$ we give the colors $b_1,b_2,\ldots,b_{|V(G_2)|}$ to its vertices. Now we consider one copy of $G_2$ and we call the corresponding vertex for this $G_2$, say $w$. Each vertex of $G_2$ is now adjacent to all of the adjacent vertices of $w$. So each color of $w$ for the TD-coloring, can use for each vertices of $G_2$, too. So this is a TD-coloring of $G_1\star G_2$ and we have $$\chi_d^t(G_1\star G_2)\leq\chi_d^t(G_1)+\vert V(G_2)\vert.$$ [$\square$]{}

\[3\] For any two connected graphs $G_1$ and $G_2$, $$\chi_d^t(G_1\star G_2)\leq\chi_d^t(G_1)+\chi_d^t(G_2).$$

[[**Proof. **]{}]{}We give the colors $a_1,a_2,\ldots,a_{\chi_d^t(G_1)}$ to the vertices of $G_1$ as we colored $G_1$ for a TD-coloring. So each vertex of $G_1$ uses the old color class. Now for every copy of $G_2$ we give the colors $b_1,b_2,\ldots,b_{\chi_d^t(G_2)}$ to its vertices as we color $G_2$ for a TD-coloring. Now by the same argument as the proof of the Theorem \[2\] we have the result. [$\square$]{}

The following theorem present an upper bound for $\chi_d^t(G_1\star G_2)$ based on total dominator chromatic number of $G_1$ and chromatic number of $G_2$.

\[4\] For any two connected graphs $G_1$ and $G_2$, $$\chi_d^t(G_1\star G_2)=\chi_d^t(G_1)+\chi(G_2).$$

[[**Proof. **]{}]{}We give the colors $a_1,a_2,\ldots,a_{\chi_d^t(G_1)}$ to the vertices of $G_1$ as we colored $G_1$ for a TD-coloring. So each vertex of $G_1$ uses the old color class. Now for every copy of $G_2$ we give the colors $b_1,b_2,\ldots,b_{\chi(G_2)}$ to its vertices as we color $G_2$ for a coloring. Now by the same argument as the Proof of the Theorem \[2\] we have: $$\chi_d^t(G_1\star G_2)\leq\chi_d^t(G_1)+\chi(G_2).$$ Now we consider one color class such as $j\in\{a_1,a_2,\ldots,a_{\chi_d^t(G_1)},b_1,b_2,\ldots,b_{\chi(G_2)}\}$. we cannot omit the color class $j$, because the coloring is proper and all vertices of $G_2$ are adjacent to at least one vertex of $G_1$ and since $G_1$ is connected, every vertex of $G_1$ is adjacent to at least all of the vertices of a $G_2$. So we have the result. [$\square$]{}

[**Remark 1.**]{} The bounds in the Theorems \[1\],\[2\], and \[3\] are sharp. It suffices to consider the complete graph $K_4$ as $G_1$ and $K_3$ as $G_2$.

The friendship graph $F_n$ $(n \geq  2)$ can be constructed by joining $n$ copies of the cycle graph $C_3$ with a common vertex. It is easy to see that $\chi_d^t(F_n)=3$ ([@nima1]). We end this section with the following corollary which follows from Theorem \[4\]:

1.  For every natural number $n$, $\chi_d^t(F_n\star K_n)=n+3$.

2.  For every natural number $n$, $\chi_d^t(F_n\star C_{2n})=5.$

TDC-number of $r$-gluing of two graphs
======================================

Let $G_1$ and $G_2$ be two graphs and $r\in \mathbb{N}\cup \{0\}$ with $r\leq min \{\omega(G_1),\omega(G_2)\}$, where $\omega(G)$ shows the clique number of $G$. Choose a $K_r$ from each $G_i$, $i=1,2$, and form a new graph $G$ from the union of $G_1$ and $G_2$ by identifying the two chosen $K_r$’s in an arbitrary manners. The graph $G$ is called $r$-gluing of $G_1$ and $G_2$ and denoted by $G_1\cup_{K_r}G_2$. If $r=0$ then $G_1\cup_{K_0}G_2$ is just disjoint union. The $G_1\cup_{K_i}G_2$ for $i=1,2$, is called vertex and edge gluing, respectively. In this section, we study the total dominator chromatic number of $r$-gluing of two graphs.

\[glue\] For any two connected graphs $G_1$ and $G_2$, $$max \{\chi_d^t(G_1),\chi_d^t(G_2)\}\leq\chi_d^t(G_1\cup_{K_r} G_2)\leq\chi_d^t(G_1)+\chi_d^t(G_2)-r.$$

[[**Proof. **]{}]{}Since we need at least $\chi_d^t(G_1)$ colors to color $G_1$ and $\chi_d^t(G_2)$ colors to color $G_2$, so we need at least max $\{\chi_d^t(G_1),\chi_d^t(G_2)\}$ colors to color $G_1\cup_{K_r} G_2$. So we have $max \{\chi_d^t(G_1),\chi_d^t(G_2)\}\leq\chi_d^t(G_1\cup_{K_r} G_2)$.

On the other hand, first we can give $a_1,a_2,\ldots,a_r$ to the vertices of $K_r$. Then we give $a_{r+1},\ldots,a_{\chi_d^t(G_1)}$ to the other vertices of $G_1$ to have a TD-coloring for $G_1$. Also we give $b_1,b_2,\ldots,b_{\chi_d^t(G_1)-r}$ to the other vertices of $G_2$ to have a TD-coloring for $G_2$. So every vertex of $G_1\cup_{K_r} G_2$ uses the color class which used before and this is a TD-coloring for $G_1\cup_{K_r} G_2$. So $\chi_d^t(G_1\cup_{K_r} G_2)\leq\chi_d^t(G_1)+\chi_d^t(G_2)-r.$ [$\square$]{}

[**Remark 2.**]{} The bounds in the Theorem \[glue\] are sharp. For the lower bound it suffices to consider the complete graph $K_4$ as $G_1$ and $K_5$ as $G_2$ and $r=4$. For the upper bound it suffices to consider the cycle graph $C_4$ as $G_1$ and $K_3$ as $G_2$ and $r=1$.

TDC-stability and TDC-bondage number of certain graphs
======================================================

In this section, we study the stability and bondage number of total dominator chromatic number of certain graphs. First we consider stability of certain graphs.

TDC-stability of certain graphs
-------------------------------

Stability of total dominator chromatic number of a graph $G$, $St_d^t(G)$, is the minimum number of vertices of $G$و whose removal changes the TDC-number of $G$. To obtain the stability of specific graphs, we need the following results:

[([@Adel])]{}\[CnPn\]

Let $P_n$ be a path of order $n\geq 2$. Then $$\chi_d^t(P_n)=\left\{
    \begin{array}{lr}
    {\displaystyle
        2\lceil\frac{n}{3}\rceil-1},&
    \quad\mbox{if $n\equiv 1$ $(mod\,3)$,}\\[15pt]
    {\displaystyle
        2\lceil\frac{n}{3}\rceil},&
    \quad\mbox{otherwise.}
    \end{array}
    \right.$$

Let $C_n$ be a cycle of order $n\geq 4$. Then $$\chi_d^t(C_n)=\left\{
    \begin{array}{lr}
    {\displaystyle
        2,} &
    \quad\mbox{if $n=4$}\\[15pt]
    {\displaystyle
        4\lfloor\frac{n}{6}\rfloor+r,}&
    \quad\mbox{if $n\neq 4$, $n\equiv r$ $(mod\,6)$, $r=0,1,2,4$,}\\[15pt]
    {\displaystyle
        4\lfloor\frac{n}{6}\rfloor+r-1,}&
    \quad\mbox{if $n\equiv r$ $(mod\,6)$, $r=3,5$.}
    \end{array}
    \right.$$

\[Pn\] For any $n\geq 4$, $St_d^t(P_n)=1$.

[[**Proof. **]{}]{}We prove the theorem for three following cases:

-   If $n\equiv 0$ $(mod\,3)$, then we have shown a TDC coloring of $P_{3s}$ in Figure \[P3s\]. By removing the vertex $v_{3s-1}$, we have a TDC coloring by $3s-1$ colors. So we have $St_d^t(P_{3s})=1$.

-   If $n\equiv 1$ $(mod\,3)$, then in this case we remove an end vertex $v$. By Theorem \[CnPn\], we know that $\chi_d^t(P_{3s+1})=2s+1$ and $\chi_d^t(P_{3s})=2s$. So we have $St_d^t(P_{3s+1})=1$.

-   If $n\equiv 2$ $(mod\,3)$, the proof is similar to the proof of Part (ii).

[$\square$]{}

![Coloring of $P_{3s}$ and removing the vertex $v_{3s-1}$. []{data-label="P3s"}](P3s){width="5in"}

\[cycle\] For every $n\geq 3$, $   {St}_d^t(C_n)=\left\{
    \begin{array}{lr}
    {\displaystyle
        1,} &
    \quad\mbox{if  $n\equiv r$ $(mod\,6)$, $r=1,2,4,5$,}\\[15pt]
    {\displaystyle
        1,}&
    \quad\mbox{if $n=3$}\\[15pt]
    {\displaystyle
        2,} &
    \quad\mbox{if $n\neq3$  $n\equiv r$ $(mod\,6)$, $r=0,3$.}
    \end{array}
\right. $

[[**Proof. **]{}]{}There are six cases, $n\equiv r$ $(mod\,6)$ and $r=0,1,2,3,4,5$. We only prove one case and the proof of another cases are similar. Suppose that $n=6k+3$. By Theorem \[CnPn\], we have $\chi_d^t(C_{6k+3})=4k+2$. By removing each vertex of this graph, we have a path graph of order $6k+2$ and By Theorem \[CnPn\], we have $\chi_d^t(P_{6k+2})=4k+2$. Now by Theorem \[Pn\], we have $St_d^t(P_{6k+2})=1$. Therefore we have $St_d^t(C_{6k+3})=2$. [$\square$]{}

The friendship graph $F_n$ $(n\geqslant 2)$ can be constructed by joining $n$ copies of the cycle graph $C_3$ with a common vertex. The $n$-book graph $(n\geqslant 2)$ is defined as the Cartesian product $K_{1,n}\square P_2$. We call every $C_4$ in the book graph $B_n$, a page of $B_n$. All pages in $B_n$ have a common side $v_1v_2$. The following easy result gives the TDC-stability of $F_n$ and $B_n$.

1.  For every $n\geq 2$, $St_d^t(F_n)=1$.

2.  For every $n\geq 3$, $St_d^t(B_n)=1$.

[[**Proof. **]{}]{}

1.  As we see in the Figure \[friend\], by removing the vertex $v$ we have a TDC coloring for that graph using $2n$ colors. So we have $St_d^t(F_n)=1$.

2.  By removing the vertex $v$ in Figure \[book\], we have a TDC coloring for that graph using $n$ colors. So we have $St_d^t(B_n)=1$. [$\square$]{}

![Coloring of $F_n$ and removing the vertex $v$. []{data-label="friend"}](fr){width="3.5in"}

![Coloring of $B_n$ and removing vertex $v$. []{data-label="book"}](br){width="3.5in"}

For every $n\in \mathbb{N}$, there exists a graph $G$ such that $ St_d^t(G)=n$.

[[**Proof. **]{}]{}Consider the graph $G$ of order $2n$ in Figure \[nregular\]. As observe that, each vertex with color $1$ is adjacent to every vertex with color $2$ and each vertex with color $2$ is adjacent to every vertex with color $1$, and so $\chi_d^t(G)=2$. By removing just one vertex of $G$, the coloring does not change. Suppose that $A$ is the set of vertices whose have color $1$. The TDC-number of the induced graph $G-A$ is $1$. The set $A$ has the minimum number of vertices which changes the TDC number of these kind of graphs (since $K_2$ is always a subgraph of these graphs and we do not need to change the color of the graph by removing each vertex). Therefore $ St_d^t(G)=n$. [$\square$]{}

![$n$-regular graph of order $2n$. []{data-label="nregular"}](nregular){width="1.6in"}

In 1956, Nordhaus and Gaddum obtained the lower and upper bounds for the sum of the chromatic numbers of a graph and its complement (actually, the upper bound was first proved by Zykov [@Zykov] in 1949). Since then, Nordhaus-Gaddum type bounds were obtained for many graph invariants. An exhaustive survey is given in [@Aouchiche]. So investigation of Nordhaus-Gaddum type inequalities for the TDC-stability number is an interesting problem and here we just present a lower bound and finding sharp upper bounds remain as an open problem. Since we have to remove at least one vertex of both graph $G$ and $G^c$ to change the TDC-number of $G$ and $G^c$, so we have the following easy result.

\[GGc\] For every graph $G$, $St_d^t(G)+St_d^t(G^c)\geq 2$.

[**Remark 4.**]{} There are graphs $G$ whose satisfy the equality in Theorem \[GGc\]. Two graphs $P_3$ and $C_5$ are two examples. As another example we can consider the graph $K_n-e$, where $e\in E(K_n)$.

We end this subsection by proposing the following conjecture:

Let $G$ be a graph with a vertex of degree one or two. Then $St_d^t(G)=1$ or $St_d^t(G)=2$.

TDC-bondage number number of certain graphs
-------------------------------------------

Bondage number of the total dominator coloring of a graph $G$, $B_d^t(G)$, is the minimum number of edges of $G$, whose removal changes the TDC-number of $G$. In this subsection, we study the TDC-bondage number of specific graphs.

\[BPn\] For the path graphs $P_n$ with $n\geq 3$, $B_d^t(P_n)=1$.

[[**Proof. **]{}]{}We have three cases:

-   If $n\equiv 2$ $(mod\,3)$. In this case, as we see in the Figure \[P3s2\], by removing the edge between two vertices $v_{3s+1}$ and $v_{3s+2}$, we have a TDC coloring for new graph by $3s+1$ colors. So $B_d^t(P_{3s+2})=1$.

-   If $n\equiv 1$ $(mod\,3)$. In this case, by removing the edge $v_2v_3$ we have two paths $P_2$ and $P_{3s-1}$ and by Theorem \[CnPn\], $\chi_d^t(P_{3s+1})=2s+1$, $\chi_d^t(P_{3s-1})=2s$ and $\chi_d^t(P_2)=2$. So we have $B_d^t(P_{3s+1})=1$.

-   If $n\equiv 0$ $(mod\,3)$ and so $n=3s$ for some $s\in \mathbb{N}$. By removing a suitable edge which makes two paths $P_{3t+2}$ and $P_{3k+1}$ ($t+k+1=s$), by Theorem \[CnPn\], we have $\chi_d^t(P_{3t+2})=2t+2$, and $\chi_d^t(P_{3k+1})=2k+1$. So $\chi_d^t(P_{3t+2})+\chi_d^t(P_{3k+1})=2t+2k+3$. On the other hand $\chi_d^t(P_{3s})=2s=2s+2k+2$. Therefore $B_d^t(P_{3s+1})=1$. [$\square$]{}

![Coloring of $P_{3s+2}$ and removing an edge. []{data-label="P3s2"}](P3s2){width="5in"}

For the cycle graphs $C_n$ with $n\geq 5$ we have: $$B_d^t(C_n)=\left\{
    \begin{array}{lr}   
   {\displaystyle
        1,} &
    \quad\mbox{if  $n\equiv 4$ $(mod\,6)$,}\\[15pt]
    {\displaystyle
        2,} &
    \quad\mbox{Otherwise.}
        \end{array}
    \right.$$

[[**Proof. **]{}]{}There exist six cases, $n\equiv r$ $(mod\,6)$ and $r=0,1,2,3,4,5$. We only prove one case and the proof of another cases are similar. Suppose that $n=6k+3$. By Theorem \[CnPn\], we have $\chi_d^t(C_{6k+3})=4k+2$. By removing each edge of this graph, we have a path graph of order $6k+3$ and by Theorem \[CnPn\], we have $\chi_d^t(P_{6k+3})=4k+2$. Now by Theorem \[BPn\], we have $B_d^t(P_{n})=1$. Therefore we need to remove two edges and so $B_d^t(C_{6k+3})=2$. [$\square$]{}

The bondage number of the friendship graph $F_n$ ($n\geq 2$) is $B_d^t(F_n)=1$.

[[**Proof. **]{}]{}As we see in the Figure \[friend2\], by removing the edge $e$ we have a TDC coloring for that graph by $4$ colors. So we have $B_d^t(F_n)=1$. [$\square$]{}

![Coloring of $F_n$ and removing edge $e$. []{data-label="friend2"}](fer){width="3.5in"}

We end the paper by the following theorem and its remark:

For every graph $G$, $B_d^t(G)+B_d^t(G^c)\geq 2$.

[[**Proof. **]{}]{}Since for changing the TDC-number of $G$ and $G^c$, we have to remove at least one edge of these two graphs, so we have the result. [$\square$]{}

[**Remark 5.**]{} The lower bound in the Theorem \[GGc\] is sharp. It is sufficient to consider Path graph $P_3$ as $G$. Also we can consider the graph $K_5-e$ as shown in Figure \[K5\].

![Coloring of $K_5-e$ and removing edge $e^\prime$. []{data-label="K5"}](BDcomp){width="2.8in"}

[99]{}

S. Alikhani, N. Ghanbari, [*Total dominator chromatic number of graphs with specific constructions*]{}. Submitted.

M. Aouchiche and P. Hansen, [*A survey of Nordhaus-Gaddum type relations*]{}, Discrete Appl. Math. 161 (2013), 466-546.

D. Bauer, F. Harary, J. Nieminen, C. Suffel, [*Domination alternation sets in graphs*]{}, Discrete Math. 47 (1983), 153-161.

R. Frucht, F.Harary, [*On the corona two graphs*]{}, Aequationes Math. 4, (1970) 322-325.

N. Ghanbari, S. Alikhani, [*Total dominator chromatic number of some operations on a graph*]{}. Submitted. Available at `https://arxiv.org/abs/1606.00666`

I. Gopalapillai, [*The spectrum of neighborhood corona of graphs*]{}, Kragujevac Journal of Mathematics. 35, (2011) 493-500.

F. Harary, [*Graph Theory*]{}, Addition-Wesley Publishing Co., Reading, MA/Menlo Park, CA/London (1969).

M.A. Henning, [*Total dominator colorings and total domination in graphs*]{}, Graphs Combin. 31 (2015) 953-–974.

A.P. Kazemi, [*Total dominator chromatic number of a graph*]{}, Trans. Combin. Vol. 4 No. 2 (2015), 57–68.

A.P. Kazemi, [*Total dominator coloring in product graphs*]{}, Utilatas Math., 94, (2014) 329–345.

X. Liu, S. Zhou, [*Spectra of the neighbourhood corona of two graphs*]{}, Linear Multilinear Alg. 62, 9, (2014) 1205–1219.

A. Vijayalekshmi, [*Total dominator colorings in Graphs*]{}, Int. J. Adv. Research Tech. 4, (2012) 1–6.

A. Vijayalekshmi, [*Total dominator colorings in Paths*]{}, Int. J. Math. Combin., 2, (2012) 89–95.

A.A. Zykov, [*On Some Properties of Linear Complexes*]{}, Math. Sbornik. NS, 24 (1949), 163-188.

[^1]: Corresponding author
