© Wuhan University 2022

0 Introduction

Research on labeling of graceful graph is on the rise following the process of computing science^[1-7]. Researches on the gracefulness of fan graph, especially double fan graph, are very important for the study of graph^[8]. The gracefulness of union graphs of double fan graph, linear graph and star graph has been proved^[9], and the gracefulness of union graphs of double fan graph and all graceful graphs has also been proved^[10]. The gracefulness of union graphs of fan graph and non-graceful graphs has rarely been studied, thus it is rather significant to have further research on the gracefulness of fan graph and non-graceful graphs.

The graphs in the following are all simple undirected graphs. Let $G(V,E)$ be a graph, $V=V(G)$ be the set of vertices of graph G, $E=E(G)$ be the set of edges of graph G, $\left|A\right|$ be the order of the set A, $P_n$ be the path with n vertices, $P_{\mathrm{1}}$ be the trivial graph with one vertex, $G_{\mathrm{1}}\vee G_{\mathrm{2}}$ be the joint graph of $G_{\mathrm{1}}$ and $G_{\mathrm{2}}$, $\overline{G}$ be the complementary graph of G.

Definition 1   Let $G=(V,E)$ be a graph, and let k be a positive integer. If there is a injection $f$:$V\to${$\mathrm{0,1},\cdots ,\left|E\right|+$ $k-\mathrm{1}$}, such that for every edge $uv\in E,$ $f^{\text{'}}(uv)=\left|f(u)-f(v)\right|$ induces a bijection $f^{\text{'}}:E\to${$k,k+\mathrm{1},\cdots ,\left|E\right|$ $+k-\mathrm{1}$}, then we call $G$ a k-graceful graph, call f a k-graceful label of graph $G$. 1-graceful graph is also called graceful graph, and 1-graceful label is called graceful label.

1 The Gracefulness of Graph ${\mathit{P}}_{\mathrm{1}}^{(\mathrm{1})}\vee ({\mathit{P}}_{\mathrm{2}\mathit{n}}^{(\mathrm{1})}\bigcup {\mathit{P}}_{\mathrm{2}\mathit{n}}^{(\mathrm{2})})\bigcup {\mathit{P}}_{\mathrm{2}\mathit{n}+\mathrm{1}}\bigcup ({\mathit{P}}_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{{\mathit{K}}_{\mathrm{2}\mathit{n}}})$

Theorem 1   For natural number $n\in \mathbf{N}$, $n\ge \mathrm{1}$, $P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})})\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$ is a graceful graph.

Proof   For $V(P_{\mathrm{1}}^{(\mathrm{1})})=${$x_{\mathrm{0}}$},$V(P_{\mathrm{2}n}^{(\mathrm{1})})=\left\{x_{\mathrm{1}},x_{\mathrm{2}},x_{\mathrm{3}},\cdots ,x_{\mathrm{2}n}\right\}$,$V(P_{\mathrm{2}n}^{(\mathrm{2})})=\left\{x_{\mathrm{2}n+\mathrm{1}},x_{\mathrm{2}n+\mathrm{2}},x_{\mathrm{2}n+\mathrm{3}},\cdots ,x_{\mathrm{4}n}\right\}$, $V(P_{\mathrm{2}n+\mathrm{1}})=\left\{y_{\mathrm{1}},y_{\mathrm{2}},y_{\mathrm{3}},\cdots ,y_{\mathrm{2}n+\mathrm{1}}\right\}$,$V(P_{\mathrm{1}}^{(\mathrm{2})})=\left\{z_{\mathrm{0}}\right\}$,$V(\overline{K_{\mathrm{2}n}})=\left\{z_{\mathrm{1}},z_{\mathrm{2}},z_{\mathrm{3}},\cdots ,z_{\mathrm{2}n}\right\}$, $E=E(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$, then $\left|E\right|=\mathrm{12}n-\mathrm{2}$, $\left|V\right|=\mathrm{8}n+\mathrm{3}$.

Define the vertex label f of graph $P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})})\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$ as follows:

$f(x_{\mathrm{0}})=\mathrm{0},f(x_{\mathrm{2}n-\mathrm{1}})=\mathrm{12}n-\mathrm{2},f(x_{\mathrm{2}n})=\mathrm{2},$

$f(x_{\mathrm{2}n-i})=\{\begin{array}{l}i-\mathrm{1},i=\mathrm{2,4},\mathrm{6},\cdots ,\mathrm{2}n-\mathrm{2},\\ \mathrm{12}n-\mathrm{2}-i,i=\mathrm{3,5},\mathrm{7},\cdots ,\mathrm{2}n-\mathrm{1}.\end{array}$

$f(x_{\mathrm{2}n+i})=\{\begin{array}{l}\mathrm{2}n-\mathrm{3}+(i+\mathrm{1}),i=\mathrm{1,3},\mathrm{5},\cdots ,\mathrm{2}n-\mathrm{1},\\ \mathrm{10}n-\mathrm{1}-i,i=\mathrm{2,4},\mathrm{6},\cdots ,\mathrm{2}n.\end{array}$

$f(y_{\mathrm{2}n+\mathrm{1}})=\mathrm{12}n-\mathrm{6},f(y_{\mathrm{2}n})=\mathrm{4}n-\mathrm{6},f(y_{\mathrm{2}n-\mathrm{1}})=\mathrm{4}n-\mathrm{2},$

$f(y_{\mathrm{2}n-i})=\{\begin{array}{l}\mathrm{4}n+\mathrm{6}-i,i=\mathrm{2,4},\mathrm{6},\cdots ,\mathrm{2}n-\mathrm{2},\\ \mathrm{4}n+\mathrm{9}+i,i=\mathrm{3,5},\mathrm{7},\cdots ,\mathrm{2}n-\mathrm{1}.\end{array}$

$f(z_{\mathrm{0}})=\mathrm{12}n-\mathrm{4},f(z_i)=\mathrm{4}n-\mathrm{3}+\mathrm{2}i,\hspace{1em}i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n$

We shall prove that label f is a graceful label of graph $P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})})\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$.

(ⅰ) By the definition of f :

$\begin{array}{l}f(x_{\mathrm{0}})<f(x_{\mathrm{2}n-\mathrm{2}})<f(x_{\mathrm{2}n})<f(x_{\mathrm{2}n-\mathrm{4}})<\cdots <f(x_{\mathrm{2}})<f(x_{\mathrm{2}n+\mathrm{1}})<f(x_{\mathrm{2}n+\mathrm{3}})<\cdots <f(x_{\mathrm{2}n+\mathrm{9}})\\ <f(y_{\mathrm{2}})<f(x_{\mathrm{2}n+\mathrm{11}})<f(y_{\mathrm{4}})<\cdots <f(x_{\mathrm{4}n-\mathrm{5}})<f(y_{\mathrm{2}n})<f(x_{\mathrm{4}n-\mathrm{3}})<f(x_{\mathrm{4}n-\mathrm{1}})<f(y_{\mathrm{2}n-\mathrm{1}})\\ <f(z_{\mathrm{1}})<f(y_{\mathrm{2}n-\mathrm{6}})<f(z_{\mathrm{2}})<f(y_{\mathrm{2}n-\mathrm{4}})<f(z_{\mathrm{3}})<f(y_{\mathrm{2}n-\mathrm{2}})<\cdots <f(z_{\mathrm{11}})<f(y_{\mathrm{2}n-\mathrm{3}})<f(z_{\mathrm{13}})<f(y_{\mathrm{2}n-\mathrm{5}})\\ <f(y_{\mathrm{1}})<f(z_{\mathrm{2}n})<f(x_{\mathrm{4}n})<f(x_{\mathrm{4}n-\mathrm{2}})<f(x_{\mathrm{4}n-\mathrm{4}})<\cdots <f(x_{\mathrm{2}n+\mathrm{2}})<f(x_{\mathrm{1}})<f(x_{\mathrm{3}})\\ <f(x_{\mathrm{5}})<\cdots <f(x_{\mathrm{2}n-\mathrm{5}})<f(y_{\mathrm{2}n+\mathrm{1}})<f(x_{\mathrm{2}n-\mathrm{3}})<f(z_{\mathrm{0}})<f(x_{\mathrm{2}n-\mathrm{1}})\end{array}$

Hence mapping $f$ : $V((P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}}))\to${$\mathrm{0,1},\cdots ,\mathrm{12}n-\mathrm{2}$}is an injection.

(ⅱ) For every edge $uv\in E(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$, let $f^{\text{'}}(uv)=\left|f(u)-f(v)\right|$.

Then

$f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-i})=\{\begin{array}{l}i-\mathrm{1},i=\mathrm{2,4},\cdots ,\mathrm{2}n-\mathrm{2},\\ \mathrm{12}n-\mathrm{2}-i,i=\mathrm{3,5},\cdots ,\mathrm{2}n-\mathrm{1}.\end{array}$

$f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n+i})=\{\begin{array}{l}\mathrm{2}n-\mathrm{1}+(i-\mathrm{1}),i=\mathrm{1,3},\cdots ,\mathrm{2}n-\mathrm{1},\\ \mathrm{10}n-\mathrm{1}-i,i=\mathrm{2,4},\cdots ,\mathrm{2}n.\end{array}$

$f^{\text{'}}(x_i{x}_{i+\mathrm{1}})=\mathrm{8}n+\mathrm{2}i,i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n-\mathrm{3},$

$f^{\text{'}}(x_{\mathrm{2}n+i}{x}_{\mathrm{2}n+i+\mathrm{1}})=\mathrm{8}n-\mathrm{2}i,i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n-\mathrm{1},$

$f^{\text{'}}(y_{\mathrm{2}n-i}{y}_{\mathrm{2}n-i+\mathrm{1}})=\mathrm{2}(i+\mathrm{1}),i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n-\mathrm{1},$

$f^{\text{'}}(z_{\mathrm{0}}{z}_i)=\mathrm{8}n-\mathrm{1}-\mathrm{2}i,i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n$

$\begin{array}{l}{f}^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n})=\mathrm{2},\hspace{1em}{f}^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{1}})=\mathrm{12}n-\mathrm{2},\hspace{1em}{f}^{\text{'}}(x_{\mathrm{2}n-\mathrm{1}}{x}_{\mathrm{2}n})=\mathrm{12}n-\mathrm{4},\hspace{1em}\\ f^{\text{'}}(x_{\mathrm{2}n-\mathrm{2}}{x}_{\mathrm{2}n-\mathrm{1}})=\mathrm{12}n-\mathrm{3},\hspace{1em}{f}^{\text{'}}(y_{\mathrm{2}n}{y}_{\mathrm{2}n+\mathrm{1}})=\mathrm{8}n.\end{array}$

Hence

$\begin{array}{l}{f}^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n}{}_{-\mathrm{4}}{}^{})<f^{\text{'}}(y_{\mathrm{2}n-\mathrm{1}}{y}_{\mathrm{2}n})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n}{}_{-\mathrm{6}}{}^{})<f^{\text{'}}(y_{\mathrm{2}n-\mathrm{2}}{y}_{\mathrm{2}n}{}_{-\mathrm{1}}{}^{})\\ <\cdots <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}})<f^{\text{'}}(y_{n+\mathrm{2}}{y}_{n+\mathrm{3}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n+\mathrm{1}})<f^{\text{'}}(y_{n+\mathrm{1}}{y}_{n+\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n+\mathrm{3}})<f^{\text{'}}(y_n{y}_{n+\mathrm{1}})\\ <\cdots <f\mathrm{\text{'}}(x\mathrm{0}x\mathrm{4}n-\mathrm{1})<f\mathrm{\text{'}}(y\mathrm{2}y\mathrm{3})<f\mathrm{\text{'}}(z\mathrm{0}z\mathrm{2}n)<f\mathrm{\text{'}}(y\mathrm{1}y\mathrm{2})<f\mathrm{\text{'}}(z\mathrm{0}z\mathrm{2}n-\mathrm{1})<f\mathrm{\text{'}}(x\mathrm{4}n-\mathrm{1}x\mathrm{4}n)\\ <f^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{2}n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{4}n-\mathrm{2}}{x}_{\mathrm{4}n-\mathrm{1}})<\cdots <f^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{1}})<f^{\text{'}}(x_{\mathrm{2}n+\mathrm{1}}{x}_{\mathrm{2}n+\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{4}n})\\ <f^{\text{'}}(y_{\mathrm{2}n}{y}_{\mathrm{2}n+\mathrm{1}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{4}n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{1}}{x}_{\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{4}n-\mathrm{4}})<f^{\text{'}}(x_{\mathrm{2}}{x}_{\mathrm{3}})<\cdots <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{1}})<f^{\text{'}}(x_n{x}_{n+\mathrm{1}})\\ <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{3}})<f^{\text{'}}(x_{n+\mathrm{1}}{x}_{n+\mathrm{2}})<\cdots <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{3}})<f^{\text{'}}(x_{\mathrm{2}n-\mathrm{1}}{x}_{\mathrm{2}n})<f^{\text{'}}(x_{\mathrm{2}n-\mathrm{2}}{x}_{\mathrm{2}n}{}_{-\mathrm{1}}{}^{})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n}{}_{-\mathrm{1}}{}^{})\end{array}$

Hence

$f^{\text{'}}$ : $E(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})\to$ {$\mathrm{1,2},\cdots ,\mathrm{12}n-\mathrm{2}$} is a bijection, graph $(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup$ $(P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$ is a graceful graph.

2 The Gracefulness of Graph $({\mathit{P}}_{\mathrm{1}}^{(\mathrm{1})}\vee ({\mathit{P}}_{\mathrm{2}\mathit{n}}^{(\mathrm{1})}\bigcup {\mathit{P}}_{\mathrm{2}\mathit{n}}^{(\mathrm{2})}))\bigcup ({\mathit{P}}_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{{\mathit{K}}_{\mathrm{2}\mathit{n}}^{(\mathrm{1})}})\bigcup ({\mathit{P}}_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{{\mathit{K}}_{\mathrm{2}\mathit{n}}^{(\mathrm{2})}})$

Theorem 2   For natural number $n\in \mathbf{N}$, $n\ge \mathrm{1}$, $(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}})$ is a graceful graph.

Proof   For $V(P_{\mathrm{1}}^{(\mathrm{1})})=${$x_{\mathrm{0}}$},$V(P_{\mathrm{2}n}^{(\mathrm{1})})=\left\{x_{\mathrm{1}},x_{\mathrm{2}},x_{\mathrm{3}},\cdots ,x_{\mathrm{2}n}\right\}$,

$V(P_{\mathrm{2}n}^{(\mathrm{2})})=\left\{x_{\mathrm{2}n+\mathrm{1}},x_{\mathrm{2}n+\mathrm{2}},x_{\mathrm{2}n+\mathrm{3}},\cdots ,x_{\mathrm{4}n}\right\}$

$V(P_{\mathrm{1}}^{(\mathrm{2})})=\left\{y_{\mathrm{0}}\right\},V(\overline{K_{\mathrm{2}n}^{(\mathrm{1})}})=\left\{y_{\mathrm{1}},y_{\mathrm{2}},y_{\mathrm{3}},\cdots ,y_{\mathrm{2}n}\right\},$

$V(P_{\mathrm{1}}^{(\mathrm{3})})=\left\{z_{\mathrm{0}}\right\},V(\overline{K_{\mathrm{2}n}^{(\mathrm{2})}})=\left\{z_{\mathrm{1}},z_{\mathrm{2}},z_{\mathrm{3}},\cdots ,z_{\mathrm{2}n}\right\},$

$E=E((P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}}))$, then $\left|E\right|=\mathrm{12}n-\mathrm{2}$.

Define the vertex label f of graph $(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}})$ as follows:

$f(x_{\mathrm{0}})=\mathrm{0},f(x_{\mathrm{2}n-\mathrm{1}})=\mathrm{12}n-\mathrm{2},f(x_{\mathrm{2}n})=\mathrm{2},$

$f(x_{\mathrm{2}n-i})=\{\begin{array}{l}i-\mathrm{1},\begin{array}{c}\end{array}i=\mathrm{2,4},\mathrm{6},\cdots ,\mathrm{2}n-\mathrm{2},\\ \mathrm{12}n-\mathrm{2}-i,i=\mathrm{3,5},\mathrm{7},\cdots ,\mathrm{2}n-\mathrm{1}.\end{array}$

$f(x_{\mathrm{2}n+i})=\{\begin{array}{l}\mathrm{2}n-\mathrm{1}+(i-\mathrm{1}),\begin{array}{c}\end{array}i=\mathrm{1,3},\mathrm{5},\cdots ,\mathrm{2}n-\mathrm{1},\\ \mathrm{10}n-\mathrm{1}-i,i=\mathrm{2,4},\mathrm{6},\cdots ,\mathrm{2}n.\end{array}$

$f(y_{\mathrm{0}})=\mathrm{12}n-\mathrm{4},f(z_{\mathrm{0}})=\mathrm{12}n-\mathrm{6},f(z_{\mathrm{2}n})=\mathrm{4}n-\mathrm{6},$

$f(y_i)=\mathrm{8}n-\mathrm{1}-\mathrm{2}i,\hspace{1em}i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n,$

$f(z_i)=\mathrm{12}n-\mathrm{6}-\mathrm{2}(i+\mathrm{1}),\hspace{1em}i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n-\mathrm{1}.$

We shall prove that label f is a graceful label of graph $(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}})$.

(ⅰ) By the definition of f :

$\begin{array}{l}f(x_{\mathrm{0}})<f(x_{\mathrm{2}n-\mathrm{2}})<f(x_{\mathrm{2}n})<f(x_{\mathrm{2}n-\mathrm{4}})<\cdots <f(x_{\mathrm{2}})<f(x_{\mathrm{2}n+\mathrm{1}})<f(x_{\mathrm{2}n+\mathrm{3}})<f(z_{\mathrm{2}n})\\ <f(x_{\mathrm{2}n+\mathrm{5}})<\cdots <f(x_{\mathrm{4}n-\mathrm{5}})<f(z_{\mathrm{2}n})<f(x_{\mathrm{4}n-\mathrm{3}})<f(y_{\mathrm{2}n})<f(y_{\mathrm{2}n-\mathrm{1}})<\cdots <f(y_{\mathrm{3}})\\ <f(z_{\mathrm{2}n-\mathrm{1}})<f(y_{\mathrm{2}})<f(z_{\mathrm{2}n-\mathrm{2}})<f(y_{\mathrm{1}})<f(z_{\mathrm{2}n-\mathrm{3}})<f(x_{\mathrm{4}n})<f(z_{\mathrm{2}n-\mathrm{4}})<f(x_{\mathrm{4}n-\mathrm{2}})\\ <f(z_{\mathrm{2}n-\mathrm{6}})<\cdots <f(z_n)<f(x_{\mathrm{2}n+\mathrm{6}})<f(z_{n-\mathrm{1}})<f(x_{\mathrm{2}n+\mathrm{4}})<f(x_{\mathrm{1}})<f(z_{n-\mathrm{4}})<f(x_{\mathrm{3}})\\ <f(z_{n-\mathrm{5}})<\cdots <f(z_{\mathrm{2}})<f(z_{\mathrm{1}})<f(x_{\mathrm{2}n-\mathrm{7}})<f(x_{\mathrm{2}n-\mathrm{5}})<f(z_{\mathrm{0}})<f(x_{\mathrm{2}n-\mathrm{3}})<f(y_{\mathrm{0}})<f(x_{\mathrm{2}n-\mathrm{1}})\end{array}$

Hence mapping $f$ :$V((P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}}))\to${$\mathrm{0,1},\cdots ,\mathrm{12}n-\mathrm{2}$}}is an injection.

(ⅱ) Let

$f^{\text{'}}(uv)=\left|f(u)-f(v)\right|$

$f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-i})=\{\begin{array}{l}i-\mathrm{1},i=\mathrm{2,4},\mathrm{6},\cdots ,\mathrm{2}n-\mathrm{2},\\ \mathrm{12}n-\mathrm{2}-i,i=\mathrm{3,5},\mathrm{7},\cdots ,\mathrm{2}n-\mathrm{1}.\end{array}$

$f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n+i})=\{\begin{array}{l}\mathrm{2}n-\mathrm{1}+(i-\mathrm{1}),i=\mathrm{1,3},\cdots ,\mathrm{2}n-\mathrm{1},\\ \mathrm{10}n-\mathrm{1}-i,i=\mathrm{2,4},\cdots ,\mathrm{2}n.\end{array}$

$f^{\text{'}}(x_i{x}_{i+\mathrm{1}})=\mathrm{8}n+\mathrm{2}i,i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n-\mathrm{3},$

$f^{\text{'}}(x_{\mathrm{2}n+i}{x}_{\mathrm{2}n+i+\mathrm{1}})=\mathrm{8}n-\mathrm{2}i,i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n-\mathrm{1},$

$f^{\text{'}}(y_{\mathrm{0}}{y}_i)=\mathrm{4}n-\mathrm{3}+\mathrm{2}i,i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n,$

$f^{\text{'}}(z_{\mathrm{0}}{z}_i)=\mathrm{2}(i+\mathrm{1}),i=\mathrm{1,2},\mathrm{3},\cdots ,\mathrm{2}n-\mathrm{1},$

$\begin{array}{l}{f}^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n})=\mathrm{2},\hspace{1em}{f}^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{1}})=\mathrm{12}n-\mathrm{2},\hspace{1em}{f}^{\text{'}}(x_{\mathrm{2}n}{x}_{\mathrm{2}n-\mathrm{1}})=\mathrm{12}n-\mathrm{4},\hspace{1em}\\ f^{\text{'}}(x_{\mathrm{2}n-\mathrm{1}}{x}_{\mathrm{2}n-\mathrm{2}})=\mathrm{12}n-\mathrm{3},\hspace{1em}\hspace{1em}{f}^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{2}n})=\mathrm{8}n.\end{array}$

Hence

$\begin{array}{l}{f}^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n}{}_{-\mathrm{4}}{}^{})<f^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{1}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n}{}_{-\mathrm{6}}{}^{})<f^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{2}})<\cdots \\ <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}})<f^{\text{'}}(z_{\mathrm{0}}{z}_{n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n}{}_{+\mathrm{1}}{}^{})<f^{\text{'}}(z_{\mathrm{0}}{z}_{n-\mathrm{1}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n}{}_{+\mathrm{3}}{}^{})<f^{\text{'}}(z_{\mathrm{0}}{z}_n)<\cdots \\ <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{4}n-\mathrm{1}})<f^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{2}n-\mathrm{2}})<f^{\text{'}}(y_{\mathrm{0}}{y}_{\mathrm{1}})<f^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{2}n-\mathrm{1}})<f^{\text{'}}(y_{\mathrm{0}}{y}_{\mathrm{2}})<f^{\text{'}}(x_{\mathrm{4}n-\mathrm{1}}{x}_{\mathrm{4}n})\\ <f^{\text{'}}(x_{\mathrm{4}n-\mathrm{1}}{x}_{\mathrm{4}n})<f^{\text{'}}(y_{\mathrm{0}}{y}_{\mathrm{3}})<f^{\text{'}}(x_{\mathrm{4}n-\mathrm{2}}{x}_{\mathrm{4}n-\mathrm{1}})<f^{\text{'}}(y_{\mathrm{0}}{y}_{\mathrm{4}})<\cdots <f^{\text{'}}(y_{\mathrm{0}}{y}_{\mathrm{2}n})<f^{\text{'}}(x_{\mathrm{2}n+\mathrm{1}}{x}_{\mathrm{2}n+\mathrm{2}})\\ <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{4}n})<f^{\text{'}}(z_{\mathrm{0}}{z}_{\mathrm{2}n})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{4}n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{1}}{x}_{\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{4}n-\mathrm{4}})<f^{\text{'}}(x_{\mathrm{2}}{x}_{\mathrm{3}})<\cdots \\ <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n+\mathrm{2}})<f^{\text{'}}(x_{n-\mathrm{1}}{x}_n)<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{1}})<f^{\text{'}}(x_n{x}_{n+\mathrm{1}})<\cdots <f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{5}})\\ <f^{\text{'}}(x_{\mathrm{2}n-\mathrm{3}}{x}_{\mathrm{2}n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{3}})<f^{\text{'}}(x_{\mathrm{2}n}{x}_{\mathrm{2}n-\mathrm{1}})<f^{\text{'}}(x_{\mathrm{2}n-\mathrm{1}}{x}_{\mathrm{2}n-\mathrm{2}})<f^{\text{'}}(x_{\mathrm{0}}{x}_{\mathrm{2}n-\mathrm{1}})\end{array}$

Hence

$f^{\text{'}}:E((P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}}))\to \{\mathrm{1,2},\cdots ,\mathrm{12}n-\mathrm{2}\}$ is a bijection, and graph $(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))$ $\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}})$ is a graceful graph.

3 Conclusion

The gracefulness on two kinds of unconnected double fan graphs with even vertices,$(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))$ $\bigcup P_{\mathrm{2}n+\mathrm{1}}\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}})$ and $(P_{\mathrm{1}}^{(\mathrm{1})}\vee (P_{\mathrm{2}n}^{(\mathrm{1})}\bigcup P_{\mathrm{2}n}^{(\mathrm{2})}))\bigcup (P_{\mathrm{1}}^{(\mathrm{2})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{1})}})\bigcup (P_{\mathrm{1}}^{(\mathrm{3})}\vee \overline{K_{\mathrm{2}n}^{(\mathrm{2})}})$, were proved. The vertex label f of graph was defined by the definition of graceful graph, and f was proved as injection/bijection. The graceful theorem of the union of unconnected triple graphs was proved. The research will be beneficial to the study of gracefulness on union graph of multiple graphs.

References