A dominator coloring of a graph \(\mathscr{G}\) is a proper coloring where each vertex of \(\mathscr{G}\) is within the closed neighborhood of at least one vertex from each color class. The minimum number of color classes required for a dominator coloring of \(\mathscr{G}\) is termed the dominator chromatic number. Additionally, a total dominator coloring of a graph \(\mathscr{G}\) is a proper coloring in which every vertex dominates at least one color class other than its own. The minimum number of color classes needed for a total dominator coloring of \(\mathscr{G}\) is known as the total dominator chromatic number. In this paper, our objective is to derive findings concerning dominator and total dominator coloring of the duplication corresponding corona of specific graphs.
A Graph
theory has indeed emerged as a foundational discipline with far-reaching
implications across various fields. Its applications span diverse
domains, including biochemistry, nanotechnology, electrical engineering,
computer science, and operations research. The ability to model
intricate systems and relationships through graphs provides invaluable
insights and solutions to complex problems. The genesis of graph theory
can be traced back to Euler’s seminal work on the Konigsberg bridge
problem in 1735. Euler’s formulation of the Eulerian graph laid the
groundwork for further exploration and development within the field
[1]. Subsequent
contributions, such as those by A.F. Mobius in 1840, introduced
fundamental concepts like complete graphs and bipartite graphs, further
enriching the theoretical framework of graph theory [1].
Kuratowski’s theorem, proved through recreational problems, demonstrated
the planarity of certain graph structures, expanding our understanding
of graph properties and paving the way for deeper investigations into
graph theory. These historical milestones underscore the evolution of
graph theory from its inception to its current significance. By
leveraging powerful combinatorial methods, graph theory continues to
shape both applied and theoretical aspects of mathematics, serving as a
cornerstone for problem solving and theorem proving in various
disciplines.
Graphs serve as a powerful tool for modeling relationships between objects in a wide array of disciplines. To expand a bit on proper coloring, it is a concept deeply rooted in graph theory and has applications in various fields, including scheduling, map coloring, and resource allocation. In proper coloring, each vertex of the graph is assigned a color such that no two adjacent vertices share the same color. This concept is often applied to solve problems where certain constraints must be met while assigning resources or scheduling tasks. The notation provided is \(f: V (\mathscr{G}) \to C\), which represents a function f that maps the vertices of a graph \(\mathscr{G}\) to a collection of color classes C. References like [2– 5] likely delve deeper into specific applications, algorithms, or extensions of proper coloring within graph theory, showcasing its relevance and versatility in solving discrete problems.
The concept of dominator coloring introduces additional constraints beyond traditional proper coloring to capture more intricate relationships within the graph. In a dominator coloring of a graph \(\mathscr{G}\), each vertex must be in the closed neighborhood of at least one vertex of every color class. This requirement adds a layer of connectivity to the coloring scheme, ensuring that each vertex has a certain influence on the entire graph. The minimum number of color classes required to achieve dominator coloring is termed the dominator chromatic number. A total dominator coloring takes this concept further by requiring that each vertex dominate at least one color class other than its own. This condition enhances the connectivity aspect of the coloring scheme, ensuring a more distributed dominance throughout the graph. The minimum number of color classes required for a total dominator coloring is termed the total dominator chromatic number. References such as [6– 13] likely delve into the theoretical properties, algorithms, and applications of dominator coloring and its variants, showcasing their significance in graph theory and related fields. The authors Michael Cary [14] et al. give the dominator coloring concepts that are studied. The authors Xiaojing Wang et al. [15] give the degree-based topological indices of the product graphs.
Our approach is also motivated by recent developments in graph resolvability and metric-based parameters. Notably, the work on mixed metric dimension and exchange properties in hexagonal nano-networks provides a foundational framework for characterizing complex structures through minimal identifiers [16]. Similarly, fault-tolerant resolvability concepts [17] and newly introduced parameters like the local edge partition dimension [18] and the mixed partition dimension [19] highlight the importance of nuanced graph invariants. Our analysis also draws from studies on graph labeling [20] and molecular structures such as the V-Phenylenic nanotube [21], bridging the gap between theoretical constructs and practical applications. Dominator and total dominator coloring can be discussed on these structures.
Let \(\mathscr{G}_ 1\) and \(\mathscr{G} _ 2\) be the two vertex-disjoint graphs with \({n_1}\) and \({n_2}\) vertices, respectively. Let \(V(\mathscr{G}_1)\) \(\cup\) U \((\mathscr{G}_1)\) where \(V(\mathscr{G} _ 1)\) = \(\{v _ 1, v _ 2, v _3,…,v _ {n_1}\}\) and U\((\mathscr{G} _ 1)\) = \(\{ x _ 1, x _ 2, x _ 3, ,…,x _ {n_1}\}\) be the vertex set of \(\mathscr{DG} _ 1\), the duplication graph of \(\mathscr{G}_1\).The vertex \(x_i\) is a duplication of \(v_i\) for each i = 1,2,…, \(n_1\). The vertex \(x_i\) is a duplication of \(v_i\) for each \(i = 1, 2, …, n_1\). Duplication corresponding corona [22– 25] of \(\mathscr{G} _ 1\) and \(\mathscr{G} _ 2,\) denoted by \(\mathscr{G} _ 1 \ \underline \divideontimes \mathscr{G} _ 2\), is the graph obtained from \(\mathscr{DG} _ 1\) and \(n_1\) copies of \(\mathscr{G} _ 2\) by making \(x_i\) and \(v_i\) adjacent to every vertex in the \(i\) – th copy of \(\mathscr{G} _ 2\) for \(i = 1,2,…,n _{1}.\)References such as [22, 24, 25] likely explore the properties, applications, and further details of the duplication corresponding corona graph, shedding light on its significance within graph theory and related areas. The author, Renny [23], gives some results on the spectra of a new duplication-based corona of the graph and gives a clear explanation of the definition of duplication for the corresponding corona graph. The authors Renny et al. [23, 24] give a new product related to the area studied. Motivated by these works, a new corona-type graph, namely, the duplication of the corresponding corona, and its adjacency, Laplacian, and signless Laplacian spectra.
In this paper, we undertake the computation of the dominator and total dominator coloring of the duplication corresponding corona of various graphs, such as path with pan, complete, and sunlet graphs. The Pan graph denotes a graph composed of vertices representing a singleton graph \(K_1\) with a vertex connected by a bridge to a cycle \(C_m\) of vertices denoted by \(p_j, 1\leq j \leq m\). The edges of the pan graph \(\rho_m\) are defined as \(E(\rho_m) = {q} \cup {q_j: 1 \leq j \leq m}\), where ‘q’ represents the edge that connects the singleton graph \(K_1\) to the cycle \(C_m\). The Sun let graph on \(2n\) vertices is formed by attaching \(n\) pendant edges to the vertices of the cycle \(C_n\). It is denoted as \(S_{n}\) [13]. A complete graph is characterized by having every pair of vertices connected by an edge. A complete graph with n vertices is denoted by \(K_n\).
This section focuses on presenting some of the preliminary results and bounds necessary for our study.
Proposition 1. [2, 7] For the path \(P_ n , n \geq 2,\) we have \(\chi_{d}(P_n) = \begin{cases} \left\lceil\frac{n}{3}\right\rceil + 1, \quad when \ \ n = 2,3,4,5,7,\\ \left\lceil\frac{n}{3}\right\rceil + 2, \quad otherwise.\\ \end{cases}\)
Proposition 2. [6] Let \(P_n\) be a path of order \(n \geq 2\). Then
\(\chi_d^t(P_n) = \begin{cases} 2\left\lceil\frac{n}{3}\right\rceil – 1, \ \quad if \ n \equiv 1(mod \ 3),\\ 2\left\lceil\frac{n}{3}\right\rceil, \quad\quad\quad otherwise.\\ \end{cases}\)
Lemma 1. [1] Let \(\mathscr{G}\) be a connected graph. Then
max \(\{\chi(G), \gamma (G)\}\) \(\leq \chi_{d}(G) \leq \chi(G) +\gamma.\)
Lemma 2. [6] For any connected graph \(G\) of order n with \(\delta(G)\geq1\). Then
max \({\{\chi(G), \gamma_t (G),2\}} \leq \chi_{d}^t(G) \leq n.\)
Theorem 1. Let \(n\geq 2, m\geq 3\), the dominator chromatic number of duplication corresponding corona of path \(P_n\) with pan \(\rho _ {m}\) is \(\chi _ {d}( P _ n \ \underline \divideontimes \ \rho _ {m}) = \begin{cases} n+4\text{ if m is odd},\\ n+3\text{ if m is even}.\\ \end{cases}\)
Proof. Let \(V(P_n) = \{p_i: 1 \leq i \leq n\}\) represent the vertices of a path graph with n vertices, and let \(DP_n\) denote the duplication of the graph \(P_n\), where the duplicated vertices are labeled as \(x_i\).
Consider \(n\) copies of the Pan graph \(\rho_m\). Let \(V(\rho_m) = \{a, q_j: 1 \leq j \leq m\}\) denote the vertices of the Pan graph, where \(a\) represents the vertex of the singleton graph \(K_1\) connected to the cycle with a bridge, and \(q_j\) represents the vertices of the cycle as defined. Let \('s'\) denote the respective copy of pan graphs taken, where \(1 \leq s \leq n\). The vertex set of the resultant graph \(P _ n \ \underline \divideontimes \ \rho _ {m}\) is defined as follows: \[V(P _ n \ \underline \divideontimes \ \rho _ {m})=\{p_i,x_i,q_{ij},a_i: 1\leq i \leq n, 1\leq j \leq m\}.\]
The cardinality of the vertex set of a resultant graph is given by \(\mid V(P _ n \ \underline \divideontimes \ \rho _ {m}) \mid = n(m+3)\). The maximum and minimum degree of a graph is given by \(\Delta\ (P _ n \ \underline \divideontimes \ \rho _ m) = m+2\) and \(\delta\ (P _ n \ \underline \divideontimes \ \rho _ m) = 3\), respectively. We associate a coloring transformation with \[\zeta:V(P _ n \ \underline \divideontimes \ \rho _ {m})\to\{\zeta_1,\zeta_2,…\chi_d(P _ n \ \underline \divideontimes \ \rho _ {m})\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq j \leq m.\)
Case 1. m is odd
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{ij}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{ij}\) and \(a_i\) in every \(s^{th}\) copy.
Assign the color \(\zeta _ 4\) to the vertices \(q_{ij}\) in every \(s^{th}\) copy when \(j=m\)
Assign the color \(\zeta_{4+s}\) to the vertices \(x_i,\) for \(1\leq s \leq n\).
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{ij}\) and \(a_i\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{ij}\) in every \(s^{th}\) copy.
Assign the color \(\zeta_{3+s}\) to the vertices \(x_i,\) for \(1\leq s \leq n\).
Allocation of distinct colors to the vertices \(x_i\) allows the other vertices \(p_i, a_i\) and \(q_{ij}\) in every \(s\)-th copy to dominate the vertices \(x_i\). These vertices \(x_i\) are self-dominating.
Using more than the colors mentioned if \(\chi_{d}(P_{n} \ \underline \divideontimes \ \rho_{m}) \geq n + 4\) when m is odd and if \(\chi_{d}(P_{n} \ \underline \divideontimes \ \rho_{m}) \geq n + 3\) when m is even we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}( P _ n \ \underline \divideontimes \ \rho _ {m}) \leq n + 4\) when m is odd and if, \(\chi _ {d}( P _ n \ \underline \divideontimes \ \rho _ {m}) \leq n + 3\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(n+4\) colors when \(m\) is odd and \(n + 3\) colors when \(m\) is even. Thus, \[\chi _ {d}( P _ n \ \underline \divideontimes \ \rho _ {m}) = \begin{cases} n+4 \ \ if \ m \ is \ odd,\\ n+3 \ \ if \ m \ is \ even.\\ \end{cases}\] ◻
Theorem 2. Let \(n\geq 2, m\geq 3\), the total dominator chromatic number of duplication corresponding corona of path \(P_n\) with pan \(\rho _ {m}\) is \[\chi _ {d}^t(P_n \ \underline \divideontimes \ \rho _ {m}) = \begin{cases} 2n+3 \ \ if \ m \ is \ odd,\\ 2n+2 \ \ if \ m \ is \ even.\\ \end{cases}\]
Proof. We take into account all the considerations and notation introduced in the preceding theorem. Similarly, we associate a coloring transformation with \[\zeta:V(P _ n \ \underline \divideontimes \ \rho _ {m})\to\{\zeta_1,\zeta_2,…\chi_d^t(P _ n \ \underline \divideontimes \ \rho _ {m})\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq j \leq m.\)
Case 1. m is odd
Assign the color \(\zeta_1\) to the vertices \(p_i\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{ij}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{ij}\) and \(a_i\) in every \(s^{th}\) copy.
Assign the color \(\zeta_{3+s}\) to the vertices \(q_{ij}\) in every \(s^{th}\) copy when j = m.
Assign the colors \(\{\zeta_{n+4},\zeta_{n+5},…\zeta_{2n+3}\}\) to the vertices \(x_i\).
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{ij}\) and \(a_i\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta_{2+s}\) to the vertices \(q_{ij}\) in every \(s^{th}\) copy.
Assign the color \(\{\zeta_{n+3},\zeta_{n+4},…\zeta_{2n+2}\}\) to the vertices \(x_i\).
Allocation of distinct colors to the vertices \(x_i\) and \(q_j\) in every copy allows the vertices to dominate at least one color class other than their own.
Using more than the colors mentioned if \(\chi _ {d}^t( P _ n \ \underline \divideontimes \ \rho _ {m}) \geq 2n + 3\) when m is odd and if, \(\chi _ {d}^t(P _ n \ \underline \divideontimes \ \rho _ {m}) \geq 2n + 2\) when m is even, we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}^t( P _ n \ \underline \divideontimes \ \rho _ {m}) \leq 2n + 3\) when m is odd and if, \(\chi _ {d}^t( P _ n \ \underline \divideontimes \ \rho _ {m}) \leq 2n + 2\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(2n + 3\) colors when \(m\) is odd and \(2n + 2\) colors when \(m\) is even. Thus, \[\chi _ {d}^t( P _ n \ \underline \divideontimes \ \rho _ {m}) = \begin{cases} 2n+3 \ \ if \ m \ is \ odd,\\ 2n+2 \ \ if \ m \ is \ even.\\ \end{cases}\] ◻
Theorem 3. Let \(n\geq 3, m\geq 2\), the dominator chromatic number of duplication corresponding corona of pan \(\rho _ n\) with path \(P_{m}\) is \[\chi _ {d}(\rho _ n \ \underline \divideontimes \ P_{m}) = n+4.\]
Proof. Let \(V(\rho_n) = \{a, p_i: 1 \leq i \leq n\}\) represent the vertices of a pan graph with \(n+1\) vertices, where \(a\) represents the vertex of the singleton graph \(K_1\) connected to the cycle with a bridge and \(p_i\) represent the vertices of the cycle as defined. Let \(D\rho_n\) denote the duplication of the graph \(\rho_n\), where the duplicated vertices are labeled as \(x_t, 1\leq t\leq n+1\).
Consider \(n+1\) copies of the Path graph \(P_m\). Let \(V(P_m) = \{ q_j : 1 \leq j \leq m\}\) denote the vertices of the Path graph. Let \('s'\) denote the respective copy of Path graphs taken, where \(1 \leq s \leq n+1\). The vertex set of the resultant graph \(\rho _ n \ \underline \divideontimes \ P _ {m}\) is defined as follows: \[V(\rho _ {n} \ \underline \divideontimes \ P _ m )=\{p_i, a, x_t, q_{tj} : 1\leq i \leq n, 1\leq t\leq n+1, 1\leq j \leq m\}.\]
The cardinality of the vertex set of a resultant graph is given by \(\mid V(\rho _ {m} \ \underline \divideontimes \ P _ n) \mid = (n+1)(m+2)\). The maximum and minimum degree of a graph is given by \(\Delta\ (\rho _ {n} \ \underline \divideontimes \ P _ m) = m+3\) and \(\delta(\rho _ {n} \ \underline \divideontimes \ P _ m) =3\) respectively. We associate a coloring transformation with \[\zeta:V(\rho _ {n} \ \underline \divideontimes \ P _ m)\to\{\zeta_1,\zeta_2,…\chi_d(\rho _ {n} \ \underline \divideontimes \ P _ m)\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq t \leq n+1, 1\leq j \leq m.\)
Assign the color \(\zeta_ 1\) to the vertices \(p_i\) and \(a\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
Assign the color \(\zeta_{3+s}\) to the vertices \(x_t\), for \(1\leq s\leq n+1.\)
Allocation of distinct colors to the vertices \(x_t\) allows the other vertices \(p_i, a\) and \(q_{tj}\) in every \(s\)-th copy to dominate the vertices \(x_t\). These vertices \(x_t\) are self-dominating.
Using more than the colors mentioned if \(\chi _ {d}(\rho _ n \ \underline \divideontimes \ P_{m}) \geq n + 4\), we will arrive at a maximum number of colors. Conversely, using fewer colors, say if \(\chi _ {d}(\rho _ n \ \underline \divideontimes \ P_{m} \leq n + 4\), would fail to satisfy the dominator coloring property. Hence, we require \(n + 4\) colors. Thus,
\[\chi _ {d}(\rho _ {n} \ \underline \divideontimes \ P _ m ) = n+4.\] ◻
Theorem 4. Let \(n\geq 3, m\geq 2\), the total dominator chromatic number of duplication corresponding corona of pan \(\rho _ n\) with path \(P_{m}\) is \[\chi _ {d}^t(\rho _ n \ \underline \divideontimes \ P_{m}) = 2n+4.\]
Proof. We take into account all the considerations and notation introduced in the preceding theorem. Similarly, we associate a coloring transformation with \[\zeta:V(\rho _ {n} \ \underline \divideontimes \ P _ m)\to\{\zeta_1,\zeta_2,…\chi_d^t(\rho _ {n} \ \underline \divideontimes \ P _ m)\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq t \leq n+1, 1\leq j \leq m.\)
Assign the color \(\zeta_ 1\) to the vertices \(p_i, a\).
When \(j\equiv 1 (mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ {2+s}\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
Assign the color \(\{\zeta_{n+4}, \zeta_{n+5},,…,\zeta_{2n+4}\}\) to the vertices \(x_t\).
Allocation of distinct colors to the vertices \(x_t\) and \(q_{tj}\) in every copy allows the vertices to dominate at least one color class other than their own.
Using more than the colors mentioned if \(\chi _ {d}^t(\rho _ n \ \underline \divideontimes \ P_{m}) \geq n + 4\), we will arrive at a maximum number of colors. Conversely, using fewer colors, say if \(\chi _ {d}^t(\rho _ n \ \underline \divideontimes \ P_{m}) \leq n + 4\), would fail to satisfy the dominator coloring property.
Hence, we require \(n + 4\) colors. Thus, \[\chi _ {d}^t(\rho _ {n} \ \underline \divideontimes \ P _ m ) = 2n+4.\] ◻
Theorem 5. Let \(n,m\geq 3\), the dominator chromatic number of duplication corresponding corona of pan \(\rho_n\) with pan \(\rho_m\) is \[\chi_{d}( \rho_n \underline \divideontimes \rho_m)= \begin{cases} n+5\text{ if m is odd},\\ n+4\text{ if m is even}.\\ \end{cases}\]
Proof. Let \(V(\rho_n) = \{a, p_i: 1 \leq i \leq n\}\) represent the vertices of a pan graph with \(n+1\) vertices, where \(a\) represents the vertex of the singleton graph \(K_1\) connected to the cycle with a bridge and \(p_i\) represent the vertices of the cycle as defined. Let \(D\rho_n\) denote the duplication of the graph \(\rho_n\), where the duplicated vertices are labeled \(x_t, 1\leq t\leq n+1\).
Consider \(n+1\) copies of the pan graph \(P_m\). Let \(V(\rho_n) = \{ b, q_j: 1 \leq j \leq m\}\) denote the vertices of the Pan graph \(\rho_m\), where \(b\) represents the vertex of the singleton graph \(K_1\) connected to the cycle with a bridge and \(q_j\) represents the vertices of the cycle as defined. Let \('s'\) denote the respective copy of Pan graphs taken, where \(1 \leq s \leq n+1\).The vertex set of the resultant graph \(\rho _ n \ \underline \divideontimes \ P _ {m}\) is defined as follows: \[V(\rho _ {n} \ \underline \divideontimes \ \rho _ {m} )=\{a, p_i,x_t,q_{tj},b_t: 1\leq i \leq n, 1\leq t\leq n+1, 1\leq j \leq m\}.\]
The cardinality of the vertex set of a resultant graph is given by \(\mid V(\rho _ {m} \ \underline \divideontimes \ \rho _ {m}) \mid = (n+1)(m+3)\). The maximum and minimum degree of a graph is given by \(\Delta\ (\rho _ {n} \ \underline \divideontimes \ \rho _ {m}) = 3(m+1)\) and \(\delta(\rho _ {n} \ \underline \divideontimes \ \rho _ {m}) =3\) respectively. We associate a coloring transformation with \[\zeta:V(\rho _ {n} \ \underline \divideontimes \ \rho _ {m})\to\{\zeta_1,\zeta_2,…\chi_d(\rho _ {n} \ \underline \divideontimes \ \rho _ {m})\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq t \leq n+1, 1\leq j \leq m.\)
Case 1. m is odd
Assign the color \(\zeta_ 1\) to the vertices \(p_i\) and \(a\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{tj}\) and \(b_t\) in every \(s^{th}\) copy.
For j=m, assign the color \(\zeta _ 4\) to the vertices \(q_{tj}\) in every \(s^{th}\) copies.
Assign the color \(\zeta_{4+s}\) to the vertices \(x_t\), for \(1\leq s\leq n+1.\)
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\) and \(a\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{tj}\) and \(b_t\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
Assign the color \(\zeta_{3+s}\) to the vertices \(x_t\), for \(1\leq s\leq n+1.\)
Allocation of distinct colors to the vertices \(x_t\) allows the other vertices in every \(s\)-th copy to dominate the vertices \(x_t\). These vertices \(x_t\) are self-dominating.
Using more than the colors mentioned if \(\chi _ {d}(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \geq n + 5\) when m is odd and if, \(\chi _ {d}(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \geq n + 4\) when m is even, we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \leq n + 5\) when m is odd and if, \(\chi _ {d}(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \leq n + 4\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(n + 5\) colors when m is odd and \(n + 4\) colors when m is even. Thus, \[\chi _ {d}( \rho _ {n}\ \underline \divideontimes \ \rho _ {m})= \begin{cases} n+5 \ \ if \ m \ is \ odd,\\ n+4 \ \ if \ m \ is \ even.\\ \end{cases}\] ◻
Theorem 6. Let \(n,m\geq 3\), the total dominator chromatic number of duplication corresponding corona of pan \(\rho _ n\) with pan \(\rho _ m\) is \[\chi _ {d}^t( \rho _ n \underline \divideontimes \rho _ m)= \begin{cases} 2n+5\text{ if m is odd},\\ 2n+4\text{ if m is even}.\\ \end{cases}\]
Proof. We take into account all the considerations and notation introduced in the preceding theorem. Similarly, we associate a coloring transformation with \[\zeta:V(\rho _ {n} \ \underline \divideontimes \ \rho _ {m})\to\{\zeta_1,\zeta_2,…\chi_d^t(\rho _ {n} \ \underline \divideontimes \ \rho _ {m}\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq t\leq n+1, 1\leq j \leq m.\)
Case 1. m is odd
Assign the color \(\zeta_ 1\) to the vertices \(p_i\) and \(a\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
For j = m, assign the color \(\{\zeta_{3+s}\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
Assign the colors \(\{\zeta_{n+5},\zeta_{n+6},…\zeta_{2n+5}\}\) to the vertices \(x_t\), for \(1\leq s\leq n+1.\)
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\) and \(a\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ {2+s}\) to the vertices \(q_{tj}\) in every \(s^{th}\) copy.
Assign the color \(\{\zeta_{n+4},\zeta_{n+5},…\zeta_{2n+4}\}\) to the vertices \(x_t\), for \(1\leq s\leq n+1.\)
Allocation of distinct colors to the vertices \(x_t\) and \(q_{tj}\) in every copy allows the vertices to dominate at least one color class other than their own.
Using more than the colors mentioned if \(\chi _ {d}^t(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \geq 2n + 5\) when m is odd and if, \(\chi _ {d}^t(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \geq 2n + 4\) when m is even, we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}^t(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \leq 2n + 5\) when m is odd and if, \(\chi _ {d}^t(\rho _ n \ \underline \divideontimes \ \rho _ {m}) \leq 2n + 4\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(n + 5\) colors when m is odd and \(n + 4\) colors when m is even. Thus, \[\chi _ {d}^t( \rho _ {n}\ \underline \divideontimes \ \rho _ {m})= \begin{cases} 2n+5 \ \ if \ m \ is \ odd,\\ 2n+4 \ \ if \ m \ is \ even.\\ \end{cases}\] ◻
Theorem 7. Let \(n\geq 2, m\geq 3\), the dominator chromatic number of duplication corresponding corona of path \(P_n\) with sun let \(S _{m}\) is \[\chi _ {d}( P _ n \underline \divideontimes S_{m})= \begin{cases} n+4\text{ if m is odd},\\ n+3\text{ if m is even}.\\ \end{cases}\]
Proof. Let \(V(P_n) = \{p_i: 1 \leq i \leq n\}\) represent the vertices of a path graph with n vertices, and let \(DP_n\) denote the duplication of the graph \(P_n\), where the duplicated vertices are labeled as \(x_i\).
Consider \(n\) copies of the sun, let the graph \(S_m\). Let \(V(S_m) = \{q_{sj},q_{sj}': 1\leq s\leq 2n, 1 \leq j \leq m\}\) be the vertices of sun let graph on 2n vertices, where \('s'\) denotes the respective copies of the sun let graphs taken. The vertex set of the resultant graph \(P _ n \ \underline \divideontimes \ S _ {m}\) is defined as follows: \[V(P _ {n} \ \underline \divideontimes \ S _ m )=\{p_i,x_i,q_{sj},q_{sj}': 1\leq i \leq n, 1\leq s \leq 2n, 1\leq j \leq m\}.\]
The cardinality of the vertex set of a resultant graph is given by \(\mid V( P_ {n} \ \underline \divideontimes \ S_m) \mid = 2n(m+1)\). The maximum and minimum degree of a graph is given by \(\Delta (P_ {n} \ \underline \divideontimes \ S _ m) = m+3\) and \(\delta (P_ {n} \ \underline \divideontimes \ S _ m) = 3\) respectively. We associate a coloring transformation with \[\zeta:V(P _ {n} \ \underline \divideontimes \ S _ m)\to\{\zeta_1,\zeta_2,…\chi_d(P _ {n} \ \underline \divideontimes \ S _ m)\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq s\leq 2n, 1\leq j \leq m.\)
Case 1. m is odd
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_3\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the color \(\zeta_4\) to the vertices \(q_{sj}\), when j=m
Assign the color \(\zeta_{4+s}\) to the vertices \(x_i\), for \(1\leq s\leq n.\)
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_3\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the color \(\zeta_{3+s}\) to the vertices \(x_i\), for \(1\leq s\leq n.\)
Allocation of distinct colors to the vertices \(x_i\) allows the other vertices in every \(s\)-th copy to dominate the vertices \(x_i\). These vertices are indeed self-dominating.
Using more than the colors mentioned if \(\chi _ {d}( P _ n \ \underline \divideontimes \ S_{m}) \geq n + 4\) when m is odd and if, \(\chi _ {d}(P _ n \ \underline \divideontimes \ S_{m}) \geq n + 3\) when m is even, we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}( P _ n \ \underline \divideontimes \ S_{m}) \leq n + 4\) when m is odd and if, \(\chi _ {d}( P _ n \ \underline \divideontimes \ S_{m}) \leq n + 3\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(n + 4\) colors when \(m\) is odd and \(n + 3\) colors when \(m\) is even. Thus, \[\chi _ {d}( P _ n \ \underline \divideontimes \ S_{m})= \begin{cases} n+4 \ \ if \ m \ is \ odd,\\ n+3 \ \ if \ m \ is \ even.\\ \end{cases}\] ◻
Theorem 8. Let \(n\geq 2, m\geq 3\), the total dominator chromatic number of duplication corresponding corona of path \(P_n\) with sunlet \(S _{m}\) is \[\chi _ {d}^t( P_n \underline \divideontimes S_{m})= \begin{cases} 2n+3\text{ if m is odd},\\ 2n+2\text{ if m is even}.\\ \end{cases}\]
Proof. We take into account all the considerations and notation introduced in the preceding theorem. Similarly, we associate a coloring transformation with \[\zeta:V( P_ {n} \ \underline \divideontimes \ S _ m)\to\{\zeta_1,\zeta_2,…\chi_d^t(P _ {n} \ \underline \divideontimes \ S _ m)\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq s\leq 2n, 1\leq j \leq m.\)
Case 1. m is odd
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_3\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the color \(\zeta_{4+s}\) to the vertices \(q_{sj}\), when j=m
Assign the colors \(\{\zeta_{n+4},\zeta_{n+5},…\zeta_{2n+3}\}\) to the vertices \(x_i\), for \(1\leq s\leq n.\)
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_{2+s}\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the colors \(\{\zeta_{n+3},\zeta_{n+4},…\zeta_{2n+2}\}\) to the vertices \(x_i\), for \(1\leq s\leq n.\)
Allocation of distinct colors to the vertices \(x_i\) and \(q_{sj}\) in every copy allows the vertices to dominate at least one color class other than their own.
Using more than the colors mentioned if \(\chi _ {d}^t(P _ n \ \underline \divideontimes \ S_{m}) \geq 2n + 3\) when m is odd and if, \(\chi _ {d}^t(P _ n \ \underline \divideontimes \ S_{m}) \geq 2n + 2\) when m is even, we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}^t( P _ n \ \underline \divideontimes \ S_{m}) \leq 2n + 3\) when m is odd and if, \(\chi _ {d}^t(P _ n \ \underline \divideontimes \ S_{m}) \leq 2n + 2\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(n + 4\) colors when \(m\) is odd and \(n + 3\) colors when \(m\) is even. Thus, \[\chi _ {d}^t( P _ n \ \underline \divideontimes \ S_{m})= \begin{cases} 2n+3 \ \ if \ m \ is \ odd,\\ 2n+2 \ \ if\ m \ is \ even.\\ \end{cases}\] ◻
Theorem 9. Let \(n\geq 3, m\geq 2\), the dominator chromatic number of duplication corresponding corona of Sun let \(S_{n}\) with path \(P_{m}\) is \[\chi _ {d} (S_{n}\ \underline \divideontimes \ P_{m}) = 2n+3.\]
Proof. Let \(V(S_n) = \{p_i, p_i': 1 \leq i \leq n\}\) represent the vertices of a sun let graph with 2n vertices, and let \(DS_n\) denote the duplication of the graph \(S_n\), where the duplicated vertices are labeled as \(x_i,x_i'\).
Consider \(2n\) copies of the Path graph \(P_m\). Let \(V(P_m) = \{ q_{sj.} 1\leq t \leq 2n, 1 \leq j \leq m\}\) be the vertices of the path graph, where \('s'\) denotes the respective copies of Path graphs taken.
The vertex set of the resultant graph \(S_{n} \ \underline \divideontimes \ P _ {m}\) is defined as follows: \[V(S_{n} \ \underline \divideontimes \ P _ {m}) =\{p_i,p_i',x_i,x_i',q_{sj.} 1\leq i \leq n, 1\leq s\leq 2n, 1\leq j \leq m\}.\]
The cardinality of the vertex set of the resultant graph is given by \(\mid V(S_{n} \ \underline \divideontimes \ P _ {m}) \mid = 2n(m+2)\). The maximum and minimum degree of a graph is given by \(\Delta\ (S_{n} \ \underline \divideontimes \ P _ {m}) = m+3\) and \(\delta(S_{n} \ \underline \divideontimes \ P _ {m})=3\) respectively.
We associate a coloring transformation with \[\zeta:V(S_{n} \ \underline \divideontimes \ P _ {m}) \to\{\zeta_1,\zeta_2,…\chi_d(S_{n} \ \underline \divideontimes \ P _ {m}) \}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq s\leq 2n, 1\leq j \leq m.\)
Assign the color \(\zeta_ 1\) to the vertices \(p_i\), \(p'_i\).
When \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{sj}\) in every \(s^{th}\) copy.
When \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ 3\) to the vertices \(q_{sj}\) in every \(s^{th}\) copy.
Assign the color \(\zeta_{3+s}\) to the vertices \(x_i,x_i'\), for \(1 \leq s \leq 2n\).
Allocation of distinct colors to the vertices \(x_i,x_i'\) allows the other vertices in every \(s\)-th copy to dominate the vertices \(x_i,x_i'\). These vertices are, in turn, self-dominating.
Using more than the colors mentioned if \(\chi_{d}(S_n \ \underline \divideontimes \ P_{m}) \geq 2n+3\), we will arrive at a maximum number of colors. Conversely, using fewer colors, say if \(\chi _ {d}(S_n \ \underline \divideontimes \ P_{m}) \leq 2n+3\), would fail to satisfy the dominator coloring property. Thus, \[\chi _ {d}(S_{n} \ \underline \divideontimes \ P _ m ) = 2n+3.\] ◻
Theorem 10. Let \(n\geq 3, m\geq 2\), the total dominator chromatic number of duplication corresponding corona of sun let \(S_{n}\) with path \(P_{m}\) is \[\chi _ {d}^t(S_{n}\ \underline \divideontimes \ P_{m}) = 4n+2.\]
Proof. We take into account all the considerations and notation introduced in the previous theorem. Similarly, we associate a coloring transformation with \[\zeta:V(S_{n} \ \underline \divideontimes \ P _ {m}) \to\{\zeta_1,\zeta_2,…\chi_d^t(S_{n} \ \underline \divideontimes \ P _ {m}) \}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq s\leq 2n, 1\leq j \leq m.\)
Assign the color \(\zeta_ 1\) to the vertices \(p_i, p'_i\).
For \(j\equiv 1(mod \ 2)\), assign the color \(\zeta _ 2\) to the vertices \(q_{sj}\) in \(s^{th}\) copy.
For \(j\equiv 0(mod \ 2)\), assign the color \(\zeta _ {2+s}\) to the vertices \(q_{sj}\) in every \(s^{th}\) copy.
Assign the colors \(\{\zeta_{n+6}\), \(\zeta_{n+7}\),,…,\(\zeta_{4n+2}\}\) to the vertices \(x_i\).
Allocation of distinct colors to the vertices \(x_t,x_t'\) and \(q_{tj}\) in every copy allows the vertices to dominate at least one color class other than their own.
Using more than the colors mentioned if \(\chi _ {d}^t(S _ n \ \underline \divideontimes \ P_{m}) \geq 4n + 2\), we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}^t(S _ n \ \underline \divideontimes \ P_{m}) \leq 4n + 2\), would fail to satisfy the dominator coloring property. Hence, we require \(4n + 2\) colors. Thus, \[\chi _ {d}^t(S_{n} \ \underline \divideontimes \ P _ m ) = 4n+2.\] ◻
Theorem 11. Let \(n,m\geq 3\), the dominator chromatic number of duplication corresponding corona of sun let \(S_{n}\) with sun let \(S_{m}\) is \[\chi _ {d}( S _ n \ \underline \divideontimes \ S_{m})= \begin{cases} 2n+4\text{ if m is odd},\\ 2n+3\text{ if m is even}.\\ \end{cases}\]
Proof. Let \(V(S_n) = \{p_i,p_i': 1 \leq i \leq n\}\) represent the vertices of a sun let graph with 2n vertices, and let \(DS_n\) denote the duplication of the graph \(S_n\), where the duplicated vertices are labeled as \(x_i,x_i'\).
Consider \(2n\) copies of the Sun, let the graph \(S_m\). Let \(V(S_m) = \{ q_{sj},q_{sj'.} 1\leq s\leq 2n, 1 \leq j \leq m\}\) be the vertices of sun let graph on 2m vertices, where \('s'\) denotes the respective copy of sun let graphs taken.
The vertex set of the resultant graph \(S_{n} \ \underline \divideontimes \ S _ {m}\) is defined as follows: \[V(S_{n} \ \underline \divideontimes \ S _ {m}) =\{p_i,p_i', x_i, x_i', q_{sj},q_{sj}': 1\leq i \leq n, 1\leq s\leq 2n, 1\leq j \leq m\}.\]
The cardinality of the vertex set of a resultant graph is given by \(\mid V(S_{n} \ \underline \divideontimes \ S _ {m}) \mid = 4n(m+1)\). The maximum and minimum degree of a graph is given by \(\Delta\ (S_{n} \ \underline \divideontimes \ S _ {m}) = 6m\) and \(\delta(S_{n} \ \underline \divideontimes \ S _ {m})=3\) respectively.
Case 1. m is odd
Assign the color \(\zeta_ 1\) to the vertices \(p_i\), \(p_i'\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_3\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the color \(\zeta_4\) to the vertices \(q_{sj}\), when j=m
Assign the color \(\zeta_{4+s}\) to the vertices \(x_i,x_i'\), for \(1\leq s\leq 2n.\)
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\),\(p_i'\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_3\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the color \(\zeta_{3+s}\) to the vertices \(x_i,x_i'\), for \(1\leq s\leq 2n.\)
Allocation of distinct colors to the vertices \(x_i,x_i'\) allows the other vertices in every \(s\)-th copy to dominate the vertices \(x_i,x_i'\). These vertices are indeed self-dominating.
Using more than the colors mentioned if \(\chi _ {d}(S _ n \ \underline \divideontimes \ S_{m}) \geq 2n + 4\) when m is odd and if, \(\chi _ {d}(S _ n \ \underline \divideontimes \ S_{m}) \geq 2n + 3\) when m is even, we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}(S _ n \ \underline \divideontimes \ S_{m}) \leq 2n + 4\) when m is odd and if, \(\chi _ {d}(S _ n \ \underline \divideontimes \ S_{m}) \leq 2n + 3\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(2n + 4\) colors when \(m\) is odd and \(2n + 3\) colors when \(m\) is even. Thus, \[\chi _ {d}( S _ n \ \underline \divideontimes \ S_{m})= \begin{cases} 2n+4 \ \ if \ m \ is \ odd,\\ 2n+3 \ \ if \ m \ is \ even.\\ \end{cases}\] ◻
Theorem 12. Let \(n,m\geq 3\), the total dominator chromatic number of duplication corresponding corona of sun let \(S_{n}\) with sun let \(S_{m}\) is \[\chi _ {d}^t( S _ n \ \underline \divideontimes \ S_{m})= \begin{cases} 4n+3\text{ if m is odd},\\ 4n+2\text{ if m is even}.\\ \end{cases}\]
Proof. We take into account all the considerations and
notations introduced in the preceding
theorem. Similarly, associate a coloring transformation with \[\zeta:V(S_{n} \ \underline \divideontimes \ S _
{m}) \to\{\zeta_1,\zeta_2,…\chi_d^t(S_{n} \ \underline \divideontimes
\ S _ {m}) \}\] This mapping is defined for the values \(1 \leq i \leq n, 1\leq s\leq 2n, 1\leq j \leq
m.\)
Case 1. m is odd
Assign the color \(\zeta_ 1\) to the vertices \(p_i\),\(p_i'\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_3\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the color \(\zeta_{3+s}\) to the vertices \(q_{sj}\), when j=m
Assign the colors \(\{\zeta_{2n+4},\zeta_{2n+5},…\zeta_{4n+3}\}\) to the vertices \(x_i,x_i'\), for \(1\leq s\leq 2n.\)
Case 2. m is even
Assign the color \(\zeta_ 1\) to the vertices \(p_i\),\(p_i'\).
Assign the color \(\zeta_2\) to the vertices \(q_{sj}\), when j is odd and \(q_{sj}'\), when j is even.
Assign the color \(\zeta_{2+s}\) to the vertices \(q_{sj}\), when j is even and \(q_{sj}'\), when j is odd.
Assign the colors \(\{\zeta_{2n+3},\zeta_{2n+4},…\zeta_{4n+2}\}\) to the vertices \(x_i,x_i'\), for \(1\leq s\leq 2n.\)
Allocation of distinct colors to the vertices \(x_i,x_i'\) and \(q_{sj}\) in every copy allows the vertices to dominate at least one color class other than their own.
Using more than the colors mentioned if \(\chi _ {d}^t(S _ n \ \underline \divideontimes \ S_{m}) \geq 4n+3\) when m is odd and if, \(\chi _ {d}^t(S _ n \ \underline \divideontimes \ S_{m}) \geq 4n+2\) when m is even, we will arrive at a maximum number of colors. Conversely, using fewer colors, say if, \(\chi _ {d}^t( S _ n \ \underline \divideontimes \ S_{m}) \leq 4n+3\) when m is odd and if, \(\chi _ {d}^t(S _ n \ \underline \divideontimes \ S_{m}) \leq 4n+2\) when m is even, would fail to satisfy the dominator coloring property.
Hence, we require \(4n+3\) colors when \(m\) is odd and \(4n+2\) colors when \(m\) is even. Thus, \[\chi _ {d}^t(S _ n \ \underline \divideontimes \ S_{m})= \begin{cases} 4n+3 \ \ if \ m\ is \ odd,\\ 4n+2 \ \ if \ m \ is \ even.\\ \end{cases}\] ◻
Theorem 13. Let \(n\geq 2, m\geq 3\), the dominator chromatic number of duplication corresponding corona of path \(P_n\) with complete graph \(\kappa _ {m}\) is \[\chi _ {d}( P _ n \ \underline \divideontimes \ \kappa _ {m}) = m+n+1.\]
Proof.
Let \(V(P_n) = \{p_i: 1 \leq i \leq n\}\) represent the vertices of a path graph with n vertices, and let \(DP_n\) denote the duplication of the graph \(P_n\), where the duplicated vertices are labeled as \(x_i\).Consider \(n\) copies of the
complete graph \(\kappa _ {m}\). Let
\(V(\kappa _ {m}) = \{ q_{sj} : 1\leq s \leq
n, 1 \leq j \leq m\}\) be the vertices of complete graph \(\kappa _ {m}\), where \('s'\) denotes the respective copy
of Path graphs taken. The vertex set of the resultant graph \(( P _ n \ \underline \divideontimes \ \kappa _
{m})\) is defined as follows: \(V( P _
n \ \underline \divideontimes \ \kappa _ {m}) =\{p_i,x_i,q_{sj.} 1\leq
i,s \leq n, 1\leq j \leq m\}\)
The cardinality of the vertex set of a resultant graph is given by \(\mid V( P _ n \ \underline \divideontimes \
\kappa _ {m}) \mid =n(m+2)\).
The maximum and minimum degree of a graph is given by \(\Delta (P _ n \ \underline \divideontimes \
\kappa _ {m}) = m+3\) and \(\delta ( P
_ n \ \underline \divideontimes \ \kappa _ {m}) = 3\),
respectively. We associate a coloring transformation with \[\zeta:V( P _ n \ \underline \divideontimes \
\kappa _ {m})\to\{\zeta_1,\zeta_2,…\chi_d(P _ n \
\underline \divideontimes \ \kappa _ {m})\}.\]
This mapping is defined for the values \(1 \leq i \leq n, 1\leq j \leq m.\)
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
Assign the color \(j+1\) to the vertices \(q_{sj}\) in every \(s^{th}\) copy.
Assign the colors \(\{\zeta_{m+2}\), \(\zeta_{m+3}\),…,\(\zeta_{m+n+1}\}\) to the vertices \(x_i\).
Allocation of distinct colors to the vertices \(x_i\) allows the other vertices in every \(s\)-th copy to dominate the vertices \(x_i\). These vertices are indeed self-dominating.
Using more than the colors mentioned if \(\chi _ {d}(P _ n \ \underline \divideontimes \ \kappa _ {m}) \geq m+n+1\), we will arrive at a maximum number of colors. Conversely, using fewer colors, say if \(\chi _ {d}(P_n \ \underline \divideontimes \ \kappa _ {m}) \leq m+n+1\), would fail to satisfy the dominator coloring property. Thus, \[\chi _ {d}( P _ n \ \underline \divideontimes \ \kappa _ {m}) = m+n+1.\] ◻
Theorem 14. Let \(n\geq 2, m\geq 3\), the total dominator chromatic number of duplication corresponding corona of \(P_n\) with \(\kappa _ {m}\) is \[\chi _ {d}^t( P _ n \ \underline \divideontimes \ \kappa _ {m}) = 2n+m.\]
Proof. . We take into account all the considerations and notation introduced in the preceding theorem. Similarly, we associate a coloring transformation with \[\zeta:V( P _ n \ \underline \divideontimes \ \kappa _ {m})\to\{\zeta_1,\zeta_2,…\chi_d^t(P _ n \ \underline \divideontimes \ \kappa _ {m})\}.\]
This mapping is defined for the values \(1 \leq i,s \leq n, 1\leq j \leq m.\)
Assign the color \(\zeta_ 1\) to the vertices \(p_i\).
For \(1\leq j\leq m-1\), assign the color \(j+1\) to the vertices \(q_{sj}\) in every s-th copy.
For \(j=m\), assign the color \(m+s\) to the vertices \(q_{sj}\) in every s-th copy.
Assign the color \(\zeta_{m+n+i}\) to the vertices \(x_i\).
Allocation of distinct colors to the vertices \(x_i\) and \(q_{sj}\) in every copy allows the vertices to dominate at least one color class other than their own.
Using more than the colors mentioned if \(\chi _ {d}^t(P _ n \ \underline \divideontimes \ \kappa _ {m}) \geq 2n+m\), we will arrive at a maximum number of colors. Conversely, using fewer colors, say if \(\chi _ {d}^t(P_n \ \underline \divideontimes \ \kappa _ {m}) \leq 2n+m\), would fail to satisfy the dominator coloring property. Thus, \[\chi _ {d}^t( P _ n \ \underline \divideontimes \ \kappa _ {m}) = 2n+m .\] ◻
Theorem 15. Let \(n\geq 3, m\geq 2\), the dominator chromatic number of duplication corresponding corona of complete graph \(\kappa _ {n}\) with path \(P_m\) is \[\chi _ {d}(\kappa _ {n} \ \underline \divideontimes \ P _ m ) = n+3.\]
Proof. The proof is similar to Theorem 3. ◻
Theorem 16. Let \(n\geq 3, m\geq 2\), the total dominator chromatic number of duplication corresponding corona of complete graph \(\kappa _ {n}\) with path \(P_m\) is \[\chi _ {d}^t( \kappa _ {n} \ \underline \divideontimes \ P _ m ) = 2n+2.\]
Proof. The proof follows from Theorem 4. ◻
Corollary 1. Let \(n,m\geq 3\), the dominator chromatic number of duplication corresponding corona of \(\kappa _ {n}\) with \(\kappa _ {m}\) is \[\chi _ {d}( \kappa _ {n} \ \underline \divideontimes \ \kappa _ {m}) = n+3.\]
Corollary 2 \(n,m\geq 3\), the total dominator chromatic number of duplication corresponding corona of \(\kappa _ {n}\) with \(\kappa _ {m}\) is \[\chi _ {d}^t( \kappa _ {n} \ \underline \divideontimes \ \kappa _ {m}) = 2n+2.\]
In this study, we compare dominator and total dominator coloring across various corona products involving path, pan, complete, and sunlet graphs. Where the dominator and total dominator colorings have greater than or equal to or less than or equal relationships, indicating different behaviors between the two in various scenarios. The results provide a detailed analysis of dominator and total dominator coloring of graphs formed by specific coronas. The work would be extended to consider duplication corresponding to the corona of some general graphs and general graphs for some other operations, like adding a vertex and adding edge corona graphs. Finally, these results have potential real-life applications, such as network design, medical resource allocation, and optimization problems.
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