Edge Odd Graceful Labeling in Some Wheel-Related Graphs

Abstract

A graph’s edge labeling involves the allocation of symbols (colors or numbers) to the edges of a graph governed by specific criteria. Such labeling of a graph G with order n and size m is named edge odd graceful if there is a bijective map $\phi$ from the set of edges $E\left(G\right)=\{e_1,\dots ,e_m\}$ to the set $\{1,3,\dots ,2m-1\}$ in a way that the derived transformation ${\phi }^{\ast }$ from the vertex-set $V\left(G\right)=\{v_1,\dots ,v_n\}$ to the set $\{0,1,2,\dots ,2m-1\}$ given by ${\phi }^{\ast }\left(u\right)={\sum }_{uv\in E\left(G\right)}\phi \left(uv\right)mod\left(2m\right)$ is injective. Any graph is named edge odd graceful if it permits an edge odd graceful allocation (Solairaju and Chithra). The primary aim of this study is to define and explore the edge odd graceful labeling of five new families of wheel-related graphs. Consequently, necessary and sufficient conditions for these families to be edge odd graceful are provided.

1. Introduction

Let $G=\left(V\right(G),E(G\left)\right)$ be a simple, connected, finite, and undirected graph, where n is the number of its vertices and m is the number of its edges. An edge labeling of G is an allocation of symbols (integers) to the edges of G, governed by specific criteria. Graph labeling, whether it is vertex or edge labeling, plays a significant role in understanding and resolving issues associated with graphs and networks across a wide range of fields. It facilitates effective representation, identification of patterns, optimization, and communication in diverse applications. As a fundamental tool in graph theory, graph labeling holds important practical implications (see [1,2,3]). Although graph labeling is crucial for applications, it remains an active area of research within graph theory. Two fundamental questions for edge labeling are: What is the family of graphs that admit an edge labeling? What are the necessary and sufficient conditions for these graphs to have such labeling?

The paper is organized into two main sections and multiple subsections to ensure a clear structure of the content. The introductory part offers contextual information concerning graph labeling, elucidating the importance of the present study and introducing the research problem, which focuses on investigating the edge odd graceful labeling in diverse graph families. The results section is further divided into subsections numbered Section 2.1, Section 2.2, Section 2.3, Section 2.4 and Section 2.5. Within each of these subsections, we define specific categories of graphs, namely, closed flower graphs, cog wheel graphs, triangulated wheel graphs, double crown-wheel graphs, and crown-triangulated wheel graphs, respectively. For every graph family, we provide comprehensive conditions that are both necessary and sufficient to establish their edge odd graceful nature. It is worth noting that all the graphs examined in this paper have no loops, no multi-edges, and no weighted edges.

Most of the references in the literature attribute the beginning of graph labeling to the work of Rosa [4] in 1967. In Rosa’s paper, a labeling of G called $\beta$-valuation was introduced as an injection $\phi$ from the set of vertices $V\left(G\right)$ to the set $\{0,1,2,\dots ,m\}$ such that when each edge $e=uv$ is designated as $\left|\phi \right(u)-\phi (v\left)\right|$, the derived edges are assigned distinct symbols. The same labeling is named “graceful labeling” by Solomon W. Golomb [5].

Another kind of labeling was defined in 1991 by Gnanajothi [6]. It is called an odd graceful labeling, which is an injection $\phi$ from the set of vertices $V\left(G\right)$ to the set $\{0,1,2,\dots ,2m-1\}$ such that when each edge $e=uv$ is designated as $\left|\phi \right(u)-\phi (v\left)\right|$, the derived edges are assigned $\{1,3,\dots ,2m-1\}$.

In 1985, Lo [7] considered a modified version of the graceful labeling of a graph G and called it edge graceful labeling. It is defined as a bijection $\phi$ from the set of edges $E\left(G\right)$ to the set $\{1,2,\dots ,m\}$ in a way that the derived transformation ${\phi }^{\ast }$ from the set of vertices $V\left(G\right)$ to $\{0,1,2,\dots ,n-1\}$ given by ${\phi }^{\ast }\left(u\right)={\sum }_{uv\in E\left(G\right)}\phi \left(uv\right)modm$ is a bijection.

In 2009, Solairaju and Chithra [8] defined a new labeling of G by combining the ideas of Gnanajothi and Lo and they call it edge odd graceful labeling. This labeling is a bijection $\phi$ from the set of edges $E\left(G\right)=\{e_1,\dots ,e_m\}$ to the set $\{1,3,\dots ,2m-1\}$ in a way that the derived transformation ${\phi }^{\ast }$ from the vertex-set $V\left(G\right)=\{v_1,\dots ,v_n\}$ to the set $\{0,1,2,\dots ,2m-1\}$ given by ${\phi }^{\ast }\left(u\right)={\sum }_{uv\in E\left(G\right)}\phi \left(uv\right)mod\left(2m\right)$ is injective. A graph is named an edge odd graceful if it admits an edge odd graceful labeling. A graph is called graceful (resp. odd graceful, edge graceful) if it admits a labeling that is graceful (resp. odd graceful, edge graceful). For more results on edge odd graceful graphs, see [9,10,11,12,13].

It is not difficult to see that not all graphs are edge odd graceful. For instance, not all stars are edge odd graceful, as we can observe in the following:

Observation 1.

For $n\ge 3,$ the star graph $S_n$ is edge odd graceful if and only if n is an even integer.

Proof. 

Let $S_n$ be a star graph with the vertex set $\{u_0,u_1,\dots ,u_n\}$, where $n\ge 2$. Then $|E\left(S_n\right)|=|E\left(K_{1,n}\right)|=n$. Each edge in $S_n$ is incident with $u_0$ and exactly one leaf vertex $u_i$ for $1\le i\le n$. All the ways to label the edges of $S_n$ are equivalent.

Now, each leaf vertex has the same labeling as the edge incident with it. Therefore, the outer vertices $\{u_1,\dots ,u_n\}$ receive distinct labels from $\{1,3,\dots ,2n-1\}$, while the central vertex $u_0$ is labeled as $(1+3+\cdots +2n-1)mod2n=n^{2}mod2n$. Therefore, $u_0$ is labeled as 0 if n is even, which means that $S_n$ is edge odd graceful in this case. However, if n is odd, $(1+3+\cdots +2n-1)mod2n=k$, where k is an odd positive integer less than $2m$, this means that $S_n$ is not edge odd graceful in this case. □

Note that identifying the outer vertices $\{u_1,u_2,\dots ,u_n\}$ of a star $S_n$ with the vertices of an n-cycle, $C_n$, with vertex set $\{v_1,v_2,\dots ,v_n\}$, results in the wheel $W_n$. For the graph $W_n$, the vertices $v_i$ and $u_i$ are combined into a single vertex $w_i$. It was shown in [14] that the wheel graph $W_n$ is edge odd graceful. This manuscript investigates some graphs that are defined in a similar way.

2. Results

2.1. Closed Flower Graphs

The first graph is the closed flower graph, error. It is obtained by connecting a cycle $C_n$ with vertex set $V\left(C_n\right)=\{v_1,v_2,v_3,\dots ,v_n\}$ and a star graph $S_n$ with vertex set $V\left(S_n\right)=\{u_0,u_1,u_2,u_3,\dots ,u_n\}$, such that each vertex $u_i$ is connected to vertices $v_i$ and $v_{i+1}$ for $1\le i\le n-1$. In addition the vertex $u_n$ is adjacent to $v_1$ and $v_n$.

Theorem 1.

The closed flower graph, $F{C}_n$, is edge odd graceful for n greater than or equal to 3.

Proof. 

Let $C{F}_n$ be a closed flower graph, where n is an integer greater than or equal to 3. The size of $C{F}_n$ is $\left|E\left(F{C}_n\right)\right|=4n=m$. Assume that the vertex set of $C{F}_n$ is $\{u_0,u_1,u_2,u_3,\dots ,u_n,v_1,v_2,v_3,\dots ,v_n\}$. We show that this graph is edge odd graceful by considering three different cases:

• Case (1): Let $F{C}_n$ be as in Figure 1 and $n\equiv k(mod8),k\ne 5$.
  

  Figure 1. $F{C}_n,$$n\equiv k(mod8),k\ne 5$.
  Define the map

  $$\phi :E\left(F{C}_n\right)\to \{1,3,\dots ,8n-1\}\mathrm{by}$$

  $$\begin{array}{cccc}& \phi \left(u_i{v}_i\right)=2i-1,1\le i\le n;\hfill & & \phi \left(u_n{v}_1\right)=2n+1;\hfill \\ & \phi \left(u_{n-i}{v}_{n-i+1}\right)=2n+2i+1,1\le i\le n-1;\hfill & & \phi \left(u_0{u}_i\right)=4n+2i-1,1\le i\le n;\hfill \\ & \phi \left(v_i{v}_{i+1}\right)=6n+2i-1,1\le i\le n-1;\mathrm{and}\hfill & & \phi \left(v_n{v}_1\right)=8n-1.\hfill \end{array}$$

  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_i\right)=(2i-1)mod\left(8n\right),1\le i\le n;\hfill \\ & {\phi }^{\ast }\left(v_i\right)=(4i-2)mod\left(8n\right),1\le i\le n;\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv [8n+1+4n+3+\dots +6n-1]\equiv 5n^{2}mod\left(8n\right)\hfill \end{array}$$

  Furthermore,

  $$\begin{array}{c}\hfill \begin{array}{cc}& n\equiv 0mod8,{\phi }^{\ast }\left(u_0\right)=0;n\equiv 1mod8,{\phi }^{\ast }\left(u_0\right)=5n;n\equiv 2mod8,{\phi }^{\ast }\left(u_0\right)=2n;\hfill \\ & n\equiv 3mod8,{\phi }^{\ast }\left(u_0\right)=7n;n\equiv 4mod8,{\phi }^{\ast }\left(u_0\right)=4n;n\equiv 6mod8,{\phi }^{\ast }\left(u_0\right)=6n\hfill \\ & n\equiv 7mod8,{\phi }^{\ast }\left(u_0\right)=3n.\hfill \end{array}\end{array}$$

  In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_2,\dots ,v_n\}$ to $V_{11}=\{2,6,10,\dots ,4n-6,4n-2\}$, respectively, is one to one. Additionally, it is a one to one map from $\{u_1,u_2,\dots ,u_n\}$ to $U_{11}=\{1,3,5,\dots ,2n-3,2n-1\}$, respectively. It is evident that $U_{11}$ and $V_{11}$ do not share any common elements. Moreover, ${\phi }^{\ast }\left(u_0\right)$ belongs to $U_{01}=\{0,2n,3n,4n,5n,6n,7n\}$, and $U_{01}$ has no elements in common with $U_{11}$ or $V_{11}$. Consequently, ${\phi }^{\ast }$ is a bijective function, and the closed flower graph is edge odd graceful when $n\equiv k(mod8),k\ne 5$.
• Case (2): Let $F{C}_n$ be as in Figure 2 and $n\equiv 5(mod8)$.
  

  Figure 2. $F{C}_n,$$n\equiv 5(mod8)$.
  Define the map

  $$\phi :E\left(F{C}_n\right)\to \{1,3,\dots ,8n-1\}\mathrm{by}$$

  $$\begin{array}{cccc}& \phi \left(u_0{u}_i\right)=2i-1,1\le i\le n;\hfill & & \phi \left(u_n{v}_1\right)=2n+1;\hfill \\ & \phi \left(u_{n-i}{v}_{n-i+1}\right)=2n+2i+1,1\le i\le n-1;\hfill & & \phi \left(u_i{v}_i\right)=4n+2i-1,1\le i\le n;\hfill \\ & \phi \left(v_i{v}_{i+1}\right)=6n+2i-1,1\le i\le n-1;\mathrm{and}\hfill & & \phi \left(v_n{v}_1\right)=8n-1.\hfill \end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_i\right)=(2i-1)mod\left(8n\right),1\le i\le n;\hfill \\ & {\phi }^{\ast }\left(v_i\right)=(4n+4i-2)mod\left(8n\right),1\le i\le n;\mathrm{and}\hfill \\ & {\varphi }^{\ast }\left(u_0\right)=[1+3+\cdots 2n-1]\equiv n^{2}mod8n=5n.\hfill \end{array}$$

In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_2,\dots ,v_n\}$ to $V_{12}=\{4n+2,4n+6,4n+10,\dots ,8n-6,8n-2\}$, respectively, is one to one. Additionally, it is a one to one map from $\{u_0,u_1,u_2,\dots ,\}$$\left\{u_n\right\}$ to $U_{12}=\{5n,1,3,5,\dots ,2n-3,2n-1\}$, respectively. It is evident that $U_{12}$ and $V_{12}$ do not share any common elements. Consequently, ${\phi }^{\ast }$ is a bijective function, and the closed flower graph exhibits an edge odd graceful property when $n\equiv 5(mod8)$. After considering case (1) and case (2), it can be concluded that every closed flower graph possesses the property of being edge odd graceful for values of n greater than or equal to $3.$ □

Examples: The closed flower graphs $C{F}_{14}$, $C{F}_{15}$, …, $C{F}_{21}$ and their explicit labeling are depicted in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10, respectively.

Figure 3. $C{F}_{14}$.

Figure 4. $C{F}_{15}$.

Figure 5. $C{F}_{16}$.

Figure 6. $C{F}_{17}$.

Figure 7. $C{F}_{18}$.

Figure 8. $C{F}_{19}$.

Figure 9. $C{F}_{20}$.

Figure 10. $C{F}_{21}$.

2.2. Cog Wheel Graphs

The second graph is the cog wheel graph, $C{W}_n$. It is obtained by combining a wheel graph $W_n=\{u_0,v_1,v_2,v_3,\dots ,v_n\}$ with a set of vertices $\{u_1,u_2,u_3,\dots ,u_n\}$ such that a vertex $u_i$ is adjacent to vertices $v_i$ and $v_{i+1}$, for $1\le i\le n-1.$ Furthermore, the vertex $u_n$ is adjacent to vertices $v_1$ and $v_n$.

Theorem 2.

The Cog wheel graph, $C{W}_n$, is edge odd graceful for n greater than or equal to 3.

Proof. 

Let $C{W}_n$ be a cog wheel graph where n is an integer greater than or equal to 3. It has a size of $\left|E\left(C{W}_n\right)\right|=4n=m$. Assuming that the vertex set of $C{W}_n$ is $\{u_0,u_1,u_2,u_3,\dots ,u_n,\}$$\{v_1,v_2,v_3,\dots ,v_n\}$, we divide the proof into two cases and provide explicit edge labeling in each case.

• Case (1): Let $C{W}_n$ be as in Figure 11 and $n\equiv k(mod8),k\ne 5$.
  

  Figure 11. $C{W}_n,n\equiv k(mod8),k\ne 5$.
  Define the map

  $$\phi :E\left(C{W}_n\right)\to \{1,3,\dots ,8n-1\}\mathrm{by}$$

  $$\begin{array}{cccc}& \phi \left(v_i{v}_{i+1}\right)=2i-1,1\le i\le n-1;\hfill & & \phi \left(v_n{v}_1\right)=2n-1;\hfill \\ & \phi \left(u_n{v}_1\right)=2n+1;\hfill & & \phi \left(u_{n-i}{v}_{n-i+1}\right)=2n+2i+1,1\le i\le n-1;\hfill \\ & \phi \left(u_0{v}_i\right)=4n+2i-1,1\le i\le n;\mathrm{and}\hfill \\ & \phi \left(u_i{v}_i\right)=8n-2i+1,1\le i\le n.\hfill \end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_i\right)=(4n-4i+2)mod\left(8n\right),1\le i\le n;\hfill \\ & {\phi }^{\ast }\left(v_i\right)=(2i-1)mod\left(8n\right),1\le i\le n;\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv [4n+1+4n+3+\dots +6n-1]\equiv 5n^{2}mod\left(8n\right).\hfill \end{array}$$
  Furthermore,

  $$\begin{array}{cc}& n\equiv 0mod8,{\phi }^{\ast }\left(u_0\right)=0,\hfill \\ & n\equiv 1mod8,{\phi }^{\ast }\left(u_0\right)=5n,\hfill \\ & n\equiv 2mod8,{\phi }^{\ast }\left(u_0\right)=2n,\hfill \\ & n\equiv 3mod8,{\phi }^{\ast }\left(u_0\right)=7n,\hfill \\ & n\equiv 4mod8,{\phi }^{\ast }\left(u_0\right)=4n,\hfill \\ & n\equiv 6mod8,{\phi }^{\ast }\left(u_0\right)=6n,\hfill \\ & n\equiv 7mod8,{\phi }^{\ast }\left(u_0\right)=3n.\hfill \end{array}$$

  In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_2,\dots ,v_n\}$ to $V_{21}=\{1,3,5,\dots ,2n-1\}$, respectively, is one to one. Additionally, it is a one to one map from $\{u_1,u_2,\dots ,u_n\}$ to $U_{21}=\{4n-2,4n-6,\dots ,6,2\}$, respectively. It is evident that $U_{21}$ and $V_{21}$ do not share any common elements. Moreover, ${\phi }^{\ast }\left(u_0\right)$ belongs to $U_{02}=\{0,2n,3n,4n,5n,6n,7n\}$, and $U_{01}$ has no elements in common with $U_{21}$ or $V_{21}$. Consequently, ${\phi }^{\ast }$ is a bijective function, and the cog wheel graph is edge odd graceful when $n\equiv k(mod8),k\ne 5$.
• Case (2): Let $C{W}_n$ be as in Figure 12 and $n\equiv 5(mod8)$.
  

  Figure 12. $C{W}_n,n\equiv 5(mod8)$.
  Define the map

  $$\phi :E\left(C{W}_n\right)\to \{1,3,\dots ,8n-1\}\mathrm{by}$$

  $$\begin{array}{cc}& \phi \left(v_i{v}_{i+1}\right)=2i-1,1\le i\le n-1;\hfill \\ & \phi \left(v_n{v}_1\right)=2n-1;\hfill \\ & \phi \left(u_0{v}_i\right)=2n+2i-1,1\le i\le n;\hfill \\ & \phi \left(u_n{v}_1\right)=4n+1\hfill \\ & \phi \left(u_{n-i}{v}_{n-i+1}\right)=4n+4i+1,1\le i\le n-1;\mathrm{and}\hfill \\ & \phi \left(u_{n-i}{v}_{n-i}\right)=4n+4i+3,0\le i\le n-1.\hfill \end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_i\right)=(8n-8i+4)mod\left(8n\right),1\le i\le n;\hfill \\ & {\phi }^{\ast }\left(v_1\right)=1;{\phi }^{\ast }\left(v_i\right)=((2n-2i+3)mod\left(8n\right),2\le i\le n;\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv [2n+1+2n+3+\dots +4n-1]=3n^2\equiv 7nmod\left(8n\right).\hfill \end{array}$$

In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_n,\dots ,v_2\}$ to $V_{22}=\{1,3,\dots ,2n-1\}$, respectively, is one to one. Additionally, it is a one to one map from $\{u_0,u_1,u_2,\dots ,u_{n-1},u_n\}$ to $U_{22}=\{7n,8n-4,8n-12,\dots ,12,4\}$, respectively. It is evident that $U_{22}$ and $V_{22}$ do not share any common elements. Consequently, ${\phi }^{\ast }$ is a bijective function, and the cog wheel graph exhibits an edge odd graceful property when $n\equiv 5(mod8)$. After considering case (1) and case (2), it can be concluded that every cog wheel graph possesses the property of being edge odd graceful for values of n greater than or equal to $3.$ □

Examples: The cog wheel graphs $C{W}_{16},\cdots ,C{W}_{23}$ and their explicit labeling are depicted in Figure 9, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20, respectively.

Figure 13. $C{W}_{16}$.

Figure 14. $C{W}_{17}$.

Figure 15. $C{W}_{18}$.

Figure 16. $C{W}_{19}$.

Figure 17. $C{W}_{20}$.

Figure 18. $C{W}_{21}$.

Figure 19. $C{W}_{22}$.

Figure 20. $C{W}_{23}$.

2.3. Triangulated Wheel Graphs

The third graph is the triangulated wheel graph,$T{W}_n$. It is obtained by combining the wheel graph $W_n=\{u_0,v_1,v_2,v_3,\dots ,v_n\}$ with a set of vertices $\{u_1,u_2,u_3,\dots ,u_n\}$ such that vertex $u_i$ is adjacent to vertices $v_i$ and $v_{i+1}$, and vertex $u_n$ is adjacent to vertices $v_1$ and $v_n$. Additionally, each vertex $u_i$ is adjacent to the vertex $u_0$.

Theorem 3.

The triangulated wheel graph, $T{W}_n,$ is edge odd graceful for n greater than or equal to 3.

Proof. 

Let $T{W}_n$ be a triangulated wheel graph where n is an integer greater than or equal to 3. It has a size of $\left|E\left(T{W}_n\right)\right|=5n=m$. Assuming that the vertex set of $T{W}_n$ is $\{u_0,u_1,u_2,u_3,\dots ,u_n,v_1,v_2,v_3,\dots ,v_n\}$, we provide an explicit edge odd labeling in three different cases based on the number of vertices.

• Case (1): Let $T{W}_n$ be as in Figure 21 and $n\equiv 0mod3$.
  

  Figure 21. $T{W}_n,n\equiv 0(mod3)$.
  Define the map

  $$\phi :E\left(T{W}_n\right)\to \{1,3,\dots ,10n-1\}\mathrm{by}$$

  $$\begin{array}{cc}& \phi \left(u_0{u}_i\right)=2i-1,1\le i\le n;\hfill \\ & \phi \left(u_0{v}_1\right)=4n+3;\hfill \\ & \phi \left(u_0{v}_2\right)=4n+1;\hfill \\ & \phi \left(u_0{v}_{n-i}\right)=4n+2i+5,0\le i\le n-3;\hfill \\ & \phi \left(u_{n-i}{v}_{n-i}\right)=2n+2i+3,0\le i\le n-2;\hfill \\ & \phi \left(u_i{v}_{i+1}\right)=6n+2i-3,2\le i\le n-1;\hfill \\ & \phi \left(u_n{v}_1\right)=8n-3\mathrm{and}\hfill \\ & \phi \left(u_1{v}_1\right)=2n+1;\hfill \\ & \phi \left(u_1{v}_2\right)=8n-1;\hfill \\ & \phi \left(v_i{v}_{i+1}\right)=8n+2i-1,1\le i\le n-1;\hfill \\ & \phi \left(v_n{v}_1\right)=10n-1.\hfill \end{array}$$

  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_i\right)=(2i-1)mod\left(10n\right),1\le i\le n;\hfill \\ & {\phi }^{\ast }\left(v_i\right)=(2n+2i-1)mod\left(10n\right),1\le i\le n;\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv [1+3+\dots +2n-1]+[4n+1+4n+3+\cdots +6n-1]\equiv 6n^{2}mod\left(10n\right).\hfill \end{array}$$

  Furthermore

  $$\begin{array}{cc}& n\equiv 0mod15,{\phi }^{\ast }\left(u_0\right)=0,\hfill \\ & n\equiv 3mod15,{\phi }^{\ast }\left(u_0\right)=8n,\hfill \\ & n\equiv 6mod15,{\phi }^{\ast }\left(u_0\right)=6n,\hfill \\ & n\equiv 9mod15,{\phi }^{\ast }\left(u_0\right)=4n,\hfill \\ & n\equiv 12mod15,{\phi }^{\ast }\left(u_0\right)=2n.\hfill \end{array}$$

  In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_2,\dots ,v_{n-1},v_n\}$ to $V_{31}=\{2n+1,2n+3,\dots ,4n-3,4n-1\}$, respectively, is one to one. Additionally, it is a one to one map from $\{u_1,u_2,\dots ,u_n\}$ to $U_{31}=\{1,3,\dots ,2n-1\}$, respectively. It is evident that $U_{31}$ and $V_{31}$ do not share any common elements. Moreover, ${\phi }^{\ast }\left(u_0\right)$ belongs to $U_{031}=\{0,2n,4n,6n,8n\}$, and $U_{031}$ has no elements in common with $U_{31}$ or $V_{31}$. Consequently, ${\phi }^{\ast }$ is a bijective function, and the triangulated wheel graph is edge odd graceful when $n\equiv 0mod3$.
• Case (2): Let $T{W}_n$ be as in Figure 22 and $n\equiv 1(mod3)$.
  

  Figure 22. $T{W}_n$, $n\equiv 1(mod3)$.
  Define the map

  $$\phi :E\left(T{W}_n\right)\to \{1,3,\dots ,10n-1\}\mathrm{by}$$

  $$\begin{array}{c}\hfill \begin{array}{ccc}& \phi \left(u_0{u}_i\right)=2i-1,1\le i\le n;\hfill & \hfill \phi \left(u_{n-i}{v}_{n-i}\right)=2n+2i+1,0\le i\le n-1;\\ & \phi \left(u_0{v}_i\right)=4n+2i-1,1\le i\le n;\hfill & \hfill \phi \left(u_i{v}_{i+1}\right)=6n+2i-1,1\le i\le n-1;\\ & \phi \left(u_n{v}_1\right)=8n-1\hfill & \hfill \phi \left(v_1{v}_n\right)=8n+1;\mathrm{and}\\ & \phi \left(v_{n-i}{v}_{n-i-1}\right)=8n+2i+3,0\le i\le n-2.\hfill \end{array}\end{array}$$
  Therefore, the derived transformation ${\varphi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_i\right)=(2i-1)mod\left(10n\right),1\le i\le n;\hfill \\ & {\phi }^{\ast }\left(v_{n-i}\right)=(2n+2i+1)mod\left(10n\right),0\le i\le n-1;\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv [1+3+\dots +2n-1]+[4n+1+4n+3+\cdots +6n-1]\equiv 6n^{2}mod\left(10n\right).\hfill \end{array}$$
  Furthermore,

  $$\begin{array}{cc}& n\equiv 1mod15,{\phi }^{\ast }\left(u_0\right)=6n,\hfill \\ & n\equiv 4mod15,{\phi }^{\ast }\left(u_0\right)=4n,\hfill \\ & n\equiv 7mod15,{\phi }^{\ast }\left(u_0\right)=2n,\hfill \\ & n\equiv 10mod15,{\phi }^{\ast }\left(u_0\right)=0,\hfill \\ & n\equiv 13mod15,{\phi }^{\ast }\left(u_0\right)=8n.\hfill \end{array}$$

  In this case, the map ${\phi }^{\ast }$ from $\{v_n,v_{n-1},\dots ,v_2,v_1\}$ to $V_{32}=\{2n+1,2n+3,\dots ,4n-3,4n-1\}$, respectively, is one to one. Additionally, it is a one to one map from $\{u_1,u_2,\dots ,u_n\}$ to $U_{32}=\{1,3,\dots ,2n-1\}$, respectively. It is evident that $U_{32}$ and $V_{32}$ do not share any common elements. Moreover, ${\phi }^{\ast }\left(u_0\right)$ belongs to $U_{032}=\{0,2n,4n,6n,8n\}$, and $U_{032}$ has no elements in common with $U_{32}$ or $V_{32}$. Consequently, ${\phi }^{\ast }$ is a bijective function, and the triangulated wheel graph is edge odd graceful when $n\equiv 1mod3$.
• Case (3): Let $T{W}_n$ be as in Figure 23 and $n\equiv 2(mod3)$.
  

  Figure 23. $T{W}_n,n\equiv 2(mod3)$.
  Define the map

  $$\phi :E\left(T{W}_n\right)\to \{1,3,\dots ,10n-1\}\mathrm{by}$$

  $$\begin{array}{c}\hfill \begin{array}{cccc}& \phi \left(u_0{u}_1\right)=1;\hfill & & \phi \left(u_0{u}_{n-i}\right)=2i+3,0\le i\le n-2;\hfill \\ & \phi \left(u_0{v}_i\right)=4n+2i-1,1\le i\le n;\hfill & & \phi \left(u_i{v}_{i+1}\right)=2n+2i-1,1\le i\le n-1;\hfill \\ & \phi \left(u_n{v}_1\right)=4n-1;\hfill & & \phi \left(u_{n-i}{v}_{n-i}\right)=6n+2i+1,0\le i\le n-1;\hfill \\ & \phi \left(v_2{v}_1\right)=8n+1;\hfill & & \phi \left(v_1{v}_n\right)=8n+3;\mathrm{and}\hfill \\ & \phi \left(v_{n-i}{v}_{n-i-1}\right)=8n+2i+5,0\le i\le n-3.\hfill \end{array}\end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_1\right)=1;{\phi }^{\ast }\left(u_i\right)=(2n-2i+3)mod\left(10n\right),2\le i\le n;\hfill \\ & {\phi }^{\ast }\left(v_1\right)=2n+3,{\phi }^{\ast }\left(v_2\right)=2n+1;\hfill \\ & {\phi }^{\ast }\left(v_i\right)=(4n-2i+5)mod\left(10n\right),3\le i\le n;\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv [1+3+\dots +2n-1]+[4n+1+4n+3+\cdots +6n-1]\equiv 6n^{2}mod\left(10n\right).\hfill \end{array}$$
  Furthermore,

  $$\begin{array}{cc}& n\equiv 2mod15,{\phi }^{\ast }\left(u_0\right)=2n,\hfill \\ & n\equiv 5mod15,{\phi }^{\ast }\left(u_0\right)=0,\hfill \\ & n\equiv 8mod15,{\phi }^{\ast }\left(u_0\right)=8n,\hfill \\ & n\equiv 11mod15,{\phi }^{\ast }\left(u_0\right)=6n,\hfill \\ & n\equiv 14mod15,{\phi }^{\ast }\left(u_0\right)=4n.\hfill \end{array}$$

In this case, the map ${\phi }^{\ast }$ from $\{v_3,v_4\dots ,v_{n-1},v_n,v_1,v_2\}$ to $V_{33}=\{4n-1,4n-3,\dots ,2n+7,2n+5,2n+3,2n+1\}$, respectively, is one to one. Additionally, it is a one to one map from $\{u_2,u_3,\dots ,u_n,u_1\}$ to $U_{33}=\{2n-1,2n-3,\dots ,3,1\}$, respectively. It is evident that $U_{33}$ and $V_{33}$ do not share any common elements. Moreover, ${\phi }^{\ast }\left(u_0\right)$ belongs to $U_{033}=\{0,2n,4n,6n,8n\}$, and $U_{033}$ has no elements in common with $U_{33}$ or $V_{33}$. Consequently, ${\phi }^{\ast }$ is a bijective function, and the triangulated wheel graph is edge odd graceful when $n\equiv 2mod3$. After considering case (1), case (2), and case (3), it can be concluded that every triangulated wheel graph possesses the property of being edge odd graceful for values of n greater than or equal to $3.$ □

Examples: The triangulated wheel graphs $T{W}_{15},T{W}_{16}$ and $T{W}_{17}$, and their explicit labeling are depicted in Figure 24, Figure 25 and Figure 26, respectively.

Figure 24. $T{W}_{15}$.

Figure 25. $T{W}_{16}$.

Figure 26. $T{W}_{17}$.

2.4. Double Crown-Wheel Graphs

The fourth graph is the double crown-wheel graph, $DC{W}_n$. It is obtained by combining a wheel graph $W_n=\{u_0,v_1,v_2,v_3,\dots ,v_n\}$ with two sets of vertices. The first set is $\{w_1,w_2,w_3,\dots ,w_n\}$, where each vertex $w_i$ is adjacent to vertices $v_i$ and $v_{i+1}$, for $1\le i\le n-1$, and the vertex $w_n$ is adjacent to vertices $v_1$ and $v_n$. The second set is $\{u_1,u_2,u_3,\dots ,u_n\}$, where each vertex $u_i$ is adjacent to vertices $v_i$ and $v_{i+1}$, for $1\le i\le n-1$, and the vertex $u_n$ is adjacent to vertices $v_1$ and $v_n$.

Theorem 4.

The double crown-wheel graph, $DC{W}_n$ is edge odd graceful for n greater than or equal to 3.

Proof. 

Clearly, $\left|E\right(DC{W}_n\left)\right|=6n=m$; assuming that the set of vertices of $DC{W}_n$ is $\{u_0,u_1,u_2,u_3,\dots ,u_n,v_1,v_2,v_3,\dots ,v_n,w_1,w_2,w_3,\dots ,w_n\}$, we provide an explicit edge odd labeling in two different cases based on the number of vertices.

• Case (1): Let $DC{W}_n$ be as in Figure 27 and $n\equiv 11(mod12)$.
  

  Figure 27. $DC{W}_n$, $n\equiv 11(mod12)$.
  Define the map

  $$\phi :E\left(DC{W}_n\right)\to \{1,3,\dots ,12n-1\}\mathrm{by}$$

  $$\begin{array}{c}\hfill \begin{array}{cccc}& \phi \left(v_1{v}_n\right)=1;\hfill & & \phi \left(v_{n-i}{v}_{n-(i+1)}\right)=2i+3,0\le i\le n-2;\hfill \\ & \phi \left(u_0{v}_1\right)=2n+1;\hfill & & \phi \left(u_0{v}_{n-i}\right)=2n+2i+3,0\le i\le n-2;\hfill \\ & \phi \left(u_i{v}_i\right)=4n+2i-1,1\le i\le n;\hfill & & \phi \left(u_i{v}_{i+1}\right)=6n+2i-1,1\le i\le n-1;\hfill \\ & \phi \left(u_n{v}_1\right)=8n-1;\hfill & & \phi \left(w_i{v}_i\right)=8n+2i-1,1\le i\le n;\hfill \\ & \phi \left(w_i{v}_{i+1}\right)=10n+2i-1,1\le i\le n-1;\mathrm{and}\hfill & & \phi \left(w_n{v}_1\right)=12n-1.\hfill \end{array}\end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{c}\hfill \begin{array}{cccc}& {\phi }^{\ast }\left(v_i\right)=(2i-1)mod12n;\hfill & & {\phi }^{\ast }\left(u_i\right)=(10n+4i-2)mod12n\hfill \\ & {\phi }^{\ast }\left(w_i\right)=(6n+4i-2)mod12n,1\le i\le n;\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv \frac{n}{2}(2n+1+4n-1)\equiv 3n^{2}mod12n\equiv 9n.\hfill \end{array}\end{array}$$

  In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_2,\dots ,v_{n-1},v_n,u_0,u_1,\dots ,u_{n-2},u_{n-1},u_n\}$ to $V{U}_{41}=\{1,3,\dots ,2n-3,2n-1,9n,(10n+2)mod12n,(10n+6)mod12n,\dots ,(14n-6)mod12n,(14n-2)mod12n\}$, respectively is one to one. Additionally, it is one to one map from $\{w_1,w_2,\dots ,w_{n-1},w_n\}$ to $W_{41}=\{6n+2,6n+6,\dots ,10n-6,10n-2\}$, respectively. It is evident that $V{U}_{41}$ and $W_{41}$ do not share any common elements. Consequently, ${\phi }^{\ast }$ is a bijective function, and the double crown-wheel graph is edge odd graceful when $n\equiv 11(mod12)$.
• Case (2): Let $DC{W}_n$ be as in Figure 28 and $n\equiv k(mod12),k\ne 11$.
  

  Figure 28. $DC{W}_n$, $n\equiv k(mod12),k\ne 11$.
  Define the map

  $$\phi :E\left(DC{W}_n\right)\to \{1,3,\dots ,12n-1\}\mathrm{by},$$

  $$\begin{array}{cccc}& \phi \left(v_1{v}_n\right)=1;\hfill & & \phi \left(v_{n-i}{v}_{n-(i+1)}\right)=2i+3,0\le i\le n-2;\hfill \\ & \phi \left(w_i{v}_i\right)=2n+2i-1,1\le i\le n;\hfill & & \phi \left(w_i{v}_{i+1}\right)=4n+2i-1,1\le i\le n-1;\hfill \\ & \phi \left(w_n{v}_1\right)=6n-1;\hfill & & \phi \left(u_i{v}_i\right)=6n+2i-1,1\le i\le n;\hfill \\ & \phi \left(u_i{v}_{i+1}\right)=8n+2i-1,1\le i\le n-1;\hfill & & \phi \left(u_n{v}_1\right)=10n-1;\hfill \\ & \phi \left(u_0{v}_1\right)=10n+1;\hfill & & \phi \left(u_0{v}_{n-i}\right)=10n+2i+3,0\le i\le n-2.\hfill \end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{c}\hfill \begin{array}{ccc}& {\phi }^{\ast }\left(v_i\right)=(2i-1)mod12n;\hfill & \hfill {\phi }^{\ast }\left(u_i\right)=(2n+4i-2)mod12n\\ & {\phi }^{\ast }\left(w_i\right)=(6n+4i-2)mod12n,1\le i\le n.\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(u_0\right)\equiv \frac{n}{2}(10n+1+12n-1)\equiv 11n^{2}mod12n\mathrm{Furthermore},.\hfill \end{array}\end{array}$$

  $$\begin{array}{cccc}& n\equiv 0mod12,{\phi }^{\ast }\left(u_0\right)=0;\hfill & & n\equiv 1mod12,{\phi }^{\ast }\left(u_0\right)=11n,\hfill \\ & n\equiv 2mod12,{\phi }^{\ast }\left(u_0\right)=10n;\hfill & & n\equiv 3mod12,{\phi }^{\ast }\left(u_0\right)=9n,\hfill \\ & n\equiv 4mod12,{\phi }^{\ast }\left(u_0\right)=8n;\hfill & & n\equiv 5mod12,{\phi }^{\ast }\left(u_0\right)=7n,\hfill \\ & n\equiv 6mod12,{\phi }^{\ast }\left(u_0\right)=6n;\hfill & & n\equiv 7mod12,{\phi }^{\ast }\left(u_0\right)=5n,\hfill \end{array}$$

  $$\begin{array}{cccc}& n\equiv 8mod12,{\phi }^{\ast }\left(u_0\right)=4n;\hfill & & n\equiv 9mod12,{\phi }^{\ast }\left(u_0\right)=3n,\hfill \end{array}$$

  $$\begin{array}{cc}& n\equiv 10mod12,{\phi }^{\ast }\left(u_0\right)=2n.\hfill \end{array}$$

In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_2,\dots ,v_{n-1}{v}_n,u_1,u_2,\dots ,u_{n-1},u_n\}$ to $V{U}_{42}=\{1,3,\dots ,2n-3,2n-1,2n+2,2n+6,\dots ,6n-6,6n-2\}$, respectively, is one to one. Additionally, it is a one to one map from $\{w_1,w_2,\dots w_{n-1},w_n\}$ to $W_{42}=\{6n+2,6n+6,\dots ,10n-6,10n-2\}$, respectively. It is evident that $V{U}_{42}$ and $W_{42}$ do not share any common elements. Moreover, ${\phi }^{\ast }\left(u_0\right)$ belongs to $U_{04}=\{0,2n,3n,4n,5n,6n,7n,8n,9n,10n,11n\}$, and $U_{04}$ has no elements in common with $V{U}_{42}$ or $W_{42}$. Consequently, ${\phi }^{\ast }$ is a bijective function, and the double crown-wheel graph is edge odd graceful when $n\equiv k(mod12),k\ne 11$. After considering case (1) and case (2), it can be concluded that every double crown-wheel graph possesses the property of being edge odd graceful for values of n greater than or equal to $3.$ □

Examples: The double crown-wheel graphs $DC{W}_{16}\cdots DC{W}_{19}$ and $DC{W}_{23}$ and their explicit labeling are depicted in Figure 29, Figure 30, Figure 31, Figure 32 and Figure 33, respectively.

Figure 29. $DC{W}_{16}$.

Figure 30. $DC{W}_{17}$.

Figure 31. $DC{W}_{18}$.

Figure 32. $DC{W}_{19}$.

Figure 33. $DC{W}_{23}$.

2.5. Crown-Triangulated Wheel Graphs

The fifth graph is the crown-triangulated wheel graph, $CT{W}_n$. It is obtained by combining the triangulated wheel graphs $T{W}_n=\{u_0,v_1,v_2,v_3,\dots ,v_n,u_1,u_2,u_3,\dots ,u_n\}$ with the set $\{w_1,w_2,w_3,\dots ,w_n\}$, where each vertex $w_i$ is adjacent to vertices $v_i$ and $v_{i+1}$, for $1\le i\le n-1$, and the vertex $w_n$ is adjacent to vertices $v_1$ and $v_n$.

Theorem 5.

The crown-triangulated wheel graph, $CT{W}_n$ is edge odd graceful for n greater than or equal to 3.

Proof. 

Clearly, $\left|E\right(CT{W}_n\left)\right|=7n=m$, the set of vertices of $CT{W}_n$ is $\{u_0,u_1,u_2,u_3,\dots ,u_n,v_1,v_2,v_3,\dots ,v_n,w_1,w_2,w_3,\dots ,w_n\}.$ There are two cases:

• Case (1): Let $CT{W}_n$ be as in Figure 34 and $n\equiv 5(mod14)$.
  

  Figure 34. $CT{W}_n$, $n\equiv 5(mod14)$.
  Define the map

  $$\phi :E\left(CT{W}_n\right)\to \{1,3,\dots ,14n-1\}\mathrm{by}$$

  $$\begin{array}{cccc}& \phi \left(u_0{u}_{n-i}\right)=2i-1,1\le i\le n-1;\hfill & & \phi \left(u_0{u}_n\right)=2n-1;\hfill \\ & \phi \left(u_0{v}_{n-i}\right)=2n+2i-1,1\le i\le n-1;\hfill & & \phi \left(u_0{v}_n\right)=4n-1;\hfill \\ & \phi \left(w_i{v}_i\right)=4n+2i+1,1\le i\le n-1;\hfill & & \phi \left(w_n{v}_n\right)=4n+1;\hfill \end{array}$$

  $$\begin{array}{cccc}& \phi \left(w_i{v}_{i+1}\right)=6n+2i+1,1\le i\le n-1;\hfill & & \phi \left(w_n{v}_1\right)=6n+1;\hfill \\ & \phi \left(u_i{v}_i\right)=8n+2i+1,1\le i\le n-1;\hfill & & \phi \left(u_n{v}_n\right)=8n+1;\hfill \\ & \phi \left(u_i{v}_{i+1}\right)=10n+2i+1,1\le i\le n-1;\hfill & & \phi \left(u_n{v}_1\right)=10n+1;\hfill \\ & \phi \left(v_{n-i}{v}_{n-(i+1)}\right)=12n+2i+1,0\le i\le n-2;\mathrm{and}\hfill & & \phi \left(v_1{v}_n\right)=14n-1.\hfill \end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(v_i\right)=(4n+2i-1)mod14n,1\le i\le n;\hfill \\ & {\phi }^{\ast }\left(u_n\right)=(6n+1);{\phi }^{\ast }\left(u_i\right)=(6n+2i+1)mod14n,1\le i\le n-1;\hfill \\ & {\phi }^{\ast }\left(w_n\right)=(10n+2);{\phi }^{\ast }\left(w_i\right)=(10n+4i+2)mod14n,1\le i\le n-1.\hfill \end{array}$$
  Furthermore,

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_0\right)\equiv \frac{2n}{2}(1+4n-1)\equiv 4n^{2}mod14n\equiv 6n.\hfill \end{array}$$

  In this case, the map ${\phi }^{\ast }$ from $\{v_1,v_2,\dots ,v_{n-1},v_n,u_0,u_n,u_1,\dots ,u_{n-2},u_{n-1}\}$ to $V{U}_{51}=\{4n+1,4n+3,\dots ,6n-5,6n-3,6n-1,6n,6n+1,6n+3,\dots ,8n-3,8n-1\}$, respectively, is one to one. Additionally, it is one to one map from $\{w_n,w_1,w_2,\dots ,w_{n-1}\}$ to $W_{51}=\{10n+2,10n+6,10n+10,\dots ,14n-2\}$, respectively. It is evident that $V{U}_{51}$ and $W_{51}$ do not share any common elements. Consequently, ${\phi }^{\ast }$ is a bijective function, and the crown-triangulated wheel graph is edge odd graceful when $n\equiv 5(mod14)$.
• Case (2): Let $CT{W}_n$ be as in Figure 35 and $n\equiv k(mod14),k\ne 5$.
  

  Figure 35. $CT{W}_n$, $n\equiv k(mod14),k\ne 5$.
  Define the map

  $$\phi :E\left(CT{W}_n\right)\to \{1,3,\dots ,14n-1\}\mathrm{by}$$

  $$\begin{array}{cc}& \phi \left(u_0{v}_{n-i}\right)=2i+3,0\le i\le n-2;\hfill \\ & \phi \left(u_0{v}_1\right)=1;\hfill \\ & \phi \left(u_0{u}_{n-i}\right)=2n+2i+1,0\le i\le n-1;\hfill \\ & \phi \left(u_i{v}_i\right)=4n+2i-1,1\le i\le n;\hfill \\ & \phi \left(u_i{v}_{i+1}\right)=6n+2i-1,1\le i\le n-1;\hfill \\ & \phi \left(u_n{v}_1\right)=8n-1;\hfill \\ & \phi \left(w_i{v}_i\right)=8n+2i-1,1\le i\le n;\hfill \\ & \phi \left(w_n{v}_1\right)=12n-1;\hfill \\ & \phi \left(w_i{v}_{i+1}\right)=10n+2i-1,1\le i\le n-1;\hfill \\ & \phi \left(v_1{v}_n\right)=12n+1;\hfill \\ & \phi \left(v_{n-i}{v}_{n-(i+1)}\right)=12n+2i+3,0\le i\le n-2.\hfill \end{array}$$
  Therefore, the derived transformation ${\phi }^{\ast }$ on the set of vertices is given in the following manner:

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_i\right)=(2i-1)mod14n;\hfill \\ & {\phi }^{\ast }\left(w_i\right)=(4n+4i-2)mod14n,1\le i\le n\mathrm{and}\hfill \\ & {\phi }^{\ast }\left(v_i\right)=(2n+2i-1)mod14n,1\le i\le n.\hfill \end{array}$$
  Furthermore,

  $$\begin{array}{cc}& {\phi }^{\ast }\left(u_0\right)\equiv \frac{2n}{2}(1+4n-1)\equiv 4n^{2}mod14n.\hfill \end{array}$$
  Furthermore,

  $$\begin{array}{cc}& n\equiv 0mod14,{\phi }^{\ast }\left(u_0\right)=0,\hfill \\ & n\equiv 1mod14,{\phi }^{\ast }\left(u_0\right)=4n,\hfill \\ & n\equiv 2mod14,{\phi }^{\ast }\left(u_0\right)=8n,\hfill \\ & n\equiv 3mod14,{\phi }^{\ast }\left(u_0\right)=12n,\hfill \\ & n\equiv 4mod14,{\phi }^{\ast }\left(u_0\right)=2n,\hfill \\ & n\equiv 6mod14,{\phi }^{\ast }\left(u_0\right)=10n,\hfill \\ & n\equiv 7mod14,{\phi }^{\ast }\left(u_0\right)=0,\hfill \\ & n\equiv 8mod14,{\phi }^{\ast }\left(u_0\right)=4n,\hfill \\ & n\equiv 9mod14,{\phi }^{\ast }\left(u_0\right)=8n,\hfill \\ & n\equiv 10mod14,{\phi }^{\ast }\left(u_0\right)=12n,\hfill \\ & n\equiv 11mod14,{\phi }^{\ast }\left(u_0\right)=2n,\hfill \\ & n\equiv 12mod14,{\phi }^{\ast }\left(u_0\right)=6n,\hfill \\ & n\equiv 13mod14,{\phi }^{\ast }\left(u_0\right)=10n.\hfill \end{array}$$

In this case, the map ${\phi }^{\ast }$ from $\{u_1,u_2,\dots ,u_n,v_1,v_2,\dots ,v_n\}$ to $V{U}_{52}=\{1,3,\dots ,2n-1,2n+1,2n+3,\dots ,4n-1\}$, respectively, is one to one. Additionally, it is a one to one map from $\{w_1,w_2,\dots ,w_n\}$ to $W_{52}=\{4n+2,4n+6,4n+10,\dots ,8n-2\}$, respectively. It is evident that $V{U}_{52}$ and $W_{52}$ do not share any common elements. Moreover, ${\phi }^{\ast }\left(u_0\right)$ belongs to $U_{05}=\{0,2n,4n,6n,8n,10n,12n\}$, and $U_{05}$ has no elements in common with $V{U}_{52}$ or $W_{52}$. Consequently, ${\phi }^{\ast }$ is a bijective function, and the crown-triangulated wheel graph is edge odd graceful when $n\equiv k(mod14),k\ne 5$. After considering case (1) and case (2), it can be concluded that every crown-triangulated wheel graph possesses the property of being edge odd graceful for values of n greater than or equal to $3.$ □

Examples: The crown-triangulated wheel graphs $DC{W}_{16},\cdots ,DC{W}_{19}$ and their explicit labeling are depicted in Figure 36, Figure 37, Figure 38 and Figure 39, respectively.

Figure 36. $CT{W}_{16}$.

Figure 37. $CT{W}_{17}$.

Figure 38. $CT{W}_{18}$.

Figure 39. $CT{W}_{19}$.

3. Conclusions

The concepts of edge graceful labeling and edge odd graceful labeling have been recognized in the field of graph theory since 2009. Despite numerous studies on these topics, the edge-odd-gracefulness of many graphs remains unknown. One particular group of graphs, stars $S_n$, is proven to not be edge odd graceful if n is odd, as illustrated in Observation 1. In this research, we introduce five novel graph families that bear some resemblance to wheels. These families encompass closed flower graphs, cogwheel graphs, triangulated wheel graphs, double crown-wheel graphs, and crown-triangulated wheel graphs. By providing explicit labeling for each class, we establish the edge-odd-gracefulness of these graphs. Theorems 1–5 serve as the basis for demonstrating the edge-odd-gracefulness of their respective graph classes.