The Local Antimagic Total Chromatic Number of Some Wheel-Related Graphs

Abstract

Let $G=(V,E)$ be a connected graph with $\left|V\right|=n$ and $\left|E\right|=m$. A bijection $f:V\left(G\right)\cup E\left(G\right)\to \{1,2,\cdots ,n+m\}$ is called local antimagic total labeling if, for any two adjacent vertices u and v, ${\omega }_t\left(u\right)\ne {\omega }_t\left(v\right)$, where ${\omega }_t\left(u\right)=f\left(u\right)+{\sum }_{e\in E\left(u\right)}f\left(e\right)$, and $E\left(u\right)$ is the set of edges incident to u. Thus, any local antimagic total labeling induces a proper coloring of G, where the vertex x in G is assigned the color ${\omega }_t\left(x\right)$. The local antimagic total chromatic number, denoted by ${\chi }_{lat}\left(G\right)$, is the minimum number of colors taken over all colorings induced by local antimagic total labelings of G. In this paper, we present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph $F_n$, the bowknot graph $B_{n,n}$, the Dutch windmill graph $D_4^n$, the analogous Dutch graph $A{D}_4^n$ and the flower graph ${\mathcal{F}}_n$.

1. Introduction

Let $G=(V,E)$ be a connected simple graph with n vertices and m edges. In 2017, Arumugam et al. [1] and Bensmail et al. [2] independently introduced the notation of local antimagic labeling of graphs. A bijection $f:E\left(G\right)\to \{1,2,\cdots ,m\}$ is called a local antimagic labeling of G if any two adjacent vertices u and v in G satisfy $\omega \left(u\right)\ne \omega \left(v\right)$, where $\omega \left(u\right)={\sum }_{e\in E\left(u\right)}f\left(e\right)$, and $E\left(u\right)$ is the set of edges incident to u. It is clear that assigning $\omega \left(x\right)$ to x for each $x\in V\left(G\right)$ naturally induces a proper vertex coloring of G, which is called a local antimagic coloring of G. A graph G is called local antimagic if G has a local antimagic labeling. The local antimagic chromatic number [1] of G, denoted by ${\chi }_{la}\left(G\right)$, is the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G.

Arumugam et al. [1] presented the local antimagic chromatic numbers of some families of graphs, such as star $S_n$, path $P_n$, cycle $C_n$, wheel $W_n$, friendship graph ${\mathcal{F}}_n$, complete graph $K_n$, complete bipartite graph $K_{2,n}$ and the join graph $G\vee \overline{K_2}$, where G is a graph of order $n\ge 4$ and $\overline{K_2}$ is the complement graph of the complete graph $K_2$. Meanwhile, many researchers have studied the local antimagic chromatic numbers of classes of many graphs. In [3], Lau et al. gave counterexamples to the lower bound of ${\chi }_{la}(G\vee \overline{K_2})$ that was obtained in [1]. Another counterexample was independently found by Shaebani [4]. Lau et al. gave affirmative solutions on Problem 3.3 of [1] and settled Theorem 2.15 of [1]. Moreover, they also completely determined the local antimagic chromatic number of a complete bipartite graph. In [5], Lau et al. presented some sufficient conditions for ${\chi }_{la}\left(H\right)\le {\chi }_{la}\left(G\right)$, where H is obtained from G by deleting or adding a certain edge. They then determined the exact values of the local antimagic chromatic numbers of many cycle-related join graphs. Nazula et al. [6] determined the local antimagic chromatic number of unicyclic graphs. Premalatha et al. [7] determined the local antimagic chromatic number of the corona product of two graphs, such as paths with null graphs. In [8], Bača et al. estimated that for the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph, there are trees and graphs with vertices of even degrees and with chromatic index 3. From the results proved by Haslegrave [9], Bača et al. obtained that the local antimagic chromatic numbers of disjoint union of arbitrary graphs are finite if and only if none of these graphs contain an isolated edge as a subgraph.

Recently, Putri et al. [10] extended this notion by labeling the vertices and edges of a graph G to establish a vertex coloring. The local antimagic total labeling on a graph G is defined to be an assignment $f:V\left(G\right)\cup E\left(G\right)\to \{1,2,\cdots ,|V\left(G\right)|+|E\left(G\right)\left|\right\}$ so that the weights of any two adjacent vertices u and v are distinct, that is, ${\omega }_t\left(u\right)\ne {\omega }_t\left(v\right)$, where ${\omega }_t\left(u\right)=f\left(u\right)+{\sum }_{e\in E\left(u\right)}f\left(e\right)$. Analogous to the local antimagic labeling, any local antimagic total labeling induces a proper vertex coloring of G, where the vertex x in G is assigned the color ${\omega }_t\left(x\right)$. The local antimagic total chromatic number of G, denoted by ${\chi }_{lat}\left(G\right)$, is the minimum number of colors taken over all colorings induced by local antimagic total labelings of G. Clearly, ${\chi }_{la}\left(G\right)\ge {\chi }_{lat}\left(G\right)\ge \chi \left(G\right)$ for any graph G. Putri et al. [10] presented the local antimagic total chromatic numbers of some families tree, such as star, double star, banana tree graph, centipede graph, and the amalgamation of the star graph. In [11], Kurniawati et al. determined the exact values of local antimagic total chromatic numbers of graphs $G\odot K_2$, when G is the star, path, cycle and friendship graphs. Moreover, Kurniawati et al. [12] determined the exact value of graphs $G\odot m{K}_1$, when G is the star, path, cycle and friendship graphs.

In this paper, we present the local antimagic total chromatic numbers of some wheel-related graphs, such as the fan graph $F_n$, the bowknot graph $B_{n,n}$, the Dutch windmill graph $D_4^n$, the analogous Dutch windmill graph $A{D}_4^n$ and the flower graph ${\mathcal{F}}_n$.

2. Main Results

In this section, we compute the local antimagic total chromatic numbers of some wheel-related graphs. In [1], Arumugam et al. presented the exact value of ${\chi }_{la}\left(W_n\right)$ with three cases: (i) ${\chi }_{la}\left(W_n\right)=4$ for $n\equiv 1(mod4)$, (ii) ${\chi }_{la}\left(W_n\right)=4$ for $n\equiv 3(mod4)$ and (iii) ${\chi }_{la}\left(W_n\right)=3$ for $n\equiv 2(mod4)$. However, they only presented the range of ${\chi }_{la}\left(W_n\right)$ for $n\equiv 0(mod4)$. Then Lau et al. [3] gave the exact value of ${\chi }_{la}\left(W_n\right)$ that ${\chi }_{la}\left(W_n\right)=3$ for $n\equiv 0(mod4)$. In [5], Lau et al. corrected three errors of the local antimagic labeling for $n\equiv 1(mod4)$ and one error of the local antimagic labeling for $n\equiv 3(mod4)$. Arumugam et al. and Lau et al. completely determined the exact value of the local antimagic chromatic number of the wheel in the following lemma.

Lemma 1

([1,3]). For the wheel $W_n$ of order $n+1$, we have

$${\chi }_{la}\left(W_n\right)=\left\{\begin{array}{cc}3,\hfill & \mathrm{for}\mathrm{even}n,\hfill \\ 4,\hfill & \mathrm{for}\mathrm{odd}n.\hfill \end{array}\right.$$

Lemma 2

([10]). For any graph G, we have ${\chi }_{lat}\left(G\right)\ge \chi \left(G\right)$.

The fan graph $F_n$ of order $n+1$ is obtained by deleting a rim edge of the wheel $W_n$, where the central vertex of $W_n$ is also the central vertex of $F_n$. The fan graph $F_5$ is shown in Figure 1. In fact, Slamin et al. [13] in 2018 and Amalia et al. [14] in 2021 presented the exact value of the local antimagic total chromatic number of the fan graph. However, the following local antimagic total labeling of the fan graph in Theorem 1 is different from that of these authors.

Theorem 1.

For the fan graph $F_n$ and $n\ge 4$, then ${\chi }_{lat}\left(F_n\right)=3$.

Proof. 

Let $V\left(F_n\right)=\left\{c\right\}\cup \left\{u_i\right|1\le i\le n\}$ and $E\left(F_n\right)=\left\{c{u}_i\right|1\le i\le n\}\cup \left\{u_i{u}_{i+1}\right|1\le i\le n-1\}$ be vertex set and edge set of the fan graph, respectively. Then we obtain $\left|V\right(F_n\left)\right|=n+1$ and $\left|E\right(F_n\left)\right|=2n-1$. It is clear that ${\chi }_{lat}\left(F_n\right)\ge \chi \left(F_n\right)=3$. In order to prove that ${\chi }_{lat}\left(F_n\right)=3$, it suffices to prove that ${\chi }_{lat}\left(F_n\right)\le 3$, which means that we should obtain a local antimagic total labeling using three distinct colors. Define $f:V\left(F_n\right)\cup E\left(F_n\right)\to \{1,2,\cdots ,3n\}$. Let $f\left(c\right)=3n$, and label the edges $u_i{u}_{i+1}$ for i such that $1\le i\le n-1$ as follows:

$$f\left(u_i{u}_{i+1}\right)=\left\{\begin{array}{cc}n-\frac{i+1}{2},\hfill & \mathrm{for}\mathrm{odd}i,\hfill \\ \frac{i}{2},\hfill & \mathrm{for}\mathrm{even}i.\hfill \end{array}\right.$$

Then label the remaining edges and vertices of graph $F_n$. Let us discuss two cases for n.

Case 1. For odd n.

$$\begin{array}{c}f\left(c{u}_i\right)=\left\{\begin{array}{cc}2n+\frac{i-1}{2},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i<n,\hfill \\ \frac{5n-3}{2}+\frac{i}{2},\hfill & \mathrm{for}\mathrm{even}i,\mathrm{and}1\le i\le n,\hfill \\ 3n-1\hfill & \mathrm{for}i=n.\hfill \end{array}\right.\hfill \\ f\left(u_i\right)=\left\{\begin{array}{cc}\frac{3n}{2}-\frac{i}{2},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ 2n-\frac{i}{2},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

From the above labelings, we have

$$\begin{array}{c}{\omega }_t\left(u_i\right)=\left\{\begin{array}{cc}\frac{9n-3}{2},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ \frac{11n-3}{2},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \\ {\omega }_t\left(c\right)=\frac{5n^2+5n}{2}.\hfill \end{array}$$

Thus f is a local antimagic total labeling using three colors, and we obtain ${\chi }_{lat}\left(F_n\right)\le 3$.

Case 2. For even n.

$$\begin{array}{c}f\left(c{u}_i\right)=\left\{\begin{array}{cc}\frac{5n-1}{2}-\frac{i}{2},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ 3n-1-\frac{i}{2},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i<n,\hfill \\ 3n-1,\hfill & \mathrm{for}i=n.\hfill \end{array}\right.\hfill \\ f\left(u_i\right)=\left\{\begin{array}{cc}n+\frac{i-1}{2},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ \frac{3n-2}{2}+\frac{i}{2},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

From the above labelings, we have

$$\begin{array}{c}{\omega }_t\left(u_i\right)=\left\{\begin{array}{cc}\frac{9n-4}{2},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ \frac{11n-4}{2},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \\ {\omega }_t\left(c\right)=\frac{5n^2+5n}{2}.\hfill \end{array}$$

Thus f is a local antimagic total labeling using three colors, and we obtain ${\chi }_{lat}\left(F_n\right)\le 3$. The proof is complete. □

All figures in this paper, the red front represents the local antimagic total labeling of edges and vertices of graph; other colors represent the sum of weights of vertex and the edges incident with the vertex in the local antimagic total labeling of graphs. Different colors are selected to clearly see the number of different colors of vertices in the figure.

Example 1.

The local antimagic total labelings of the fan graph $F_9$ and $F_8$ are shown in Figure 2 and Figure 3.

The bowknot graph, denoted by $B_{n,n}$, is the graph by gluing two central vertices of double fan graphs $F_n$. Obviously, the bowknot graph $B_{n,n}$ is obtained from the wheel $W_{2n}$ by deleting two edges every $n-1$ edges on the rim of the wheel, and shown in Figure 4. It has $2n+1$ vertices and $4n-2$ edges.

Theorem 2.

For the bowknot graph $B_{n,n}(n\ge 5)$, we have ${\chi }_{lat}\left(B_{n,n}\right)=3$.

Proof. 

Let $V\left(B_{n,n}\right)=\left\{c\right\}\cup \{u_i,v_i|1\le i\le n\}$ be the vertex set of graph $B_{n,n}$, and let $E\left(B_{n,n}\right)=\{c{u}_i,c{v}_i|1\le i\le n\}\cup \{u_i{u}_{i+1},v_i{v}_{i+1}|1\le i\le n-1\}$ be the edge set of $B_{n,n}$. Since $K_3$ is an induced subgraph of $B_{n,n}$, we have ${\chi }_{lat}\left(B_{n,n}\right)\ge \chi \left(B_{n,n}\right)=3$. Define $f:V\left(B_{n,n}\right)\cup E\left(B_{n,n}\right)\to \{1,2,\cdots ,6n-1\}$ and consider the following two cases.

Case 1. If n is odd.

Firstly, label the edges of $B_{n,n}$ as follows:

$$\begin{array}{c}f\left(u_i{u}_{i+1}\right)=\left\{\begin{array}{cc}i,\hfill & \mathrm{if}i\mathrm{is}\mathrm{odd}\mathrm{and}1\le i\le n-1,\hfill \\ 2n-i,\hfill & \mathrm{if}i\mathrm{is}\mathrm{even}\mathrm{and}1\le i\le n-1.\hfill \end{array}\right.\hfill \\ f\left(v_i{v}_{i+1}\right)=\left\{\begin{array}{cc}i+1,\hfill & \mathrm{if}i\mathrm{is}\mathrm{odd}\mathrm{and}1\le i\le n-1,\hfill \\ 2n-1-i,\hfill & \mathrm{if}i\mathrm{is}\mathrm{even}\mathrm{and}1\le i\le n-1.\hfill \end{array}\right.\hfill \\ f\left(c{u}_i\right)=\left\{\begin{array}{cc}4n-3,\hfill & \mathrm{if}i=1,\hfill \\ 2n-4+i\hfill & \mathrm{if}3\le i\le n-1\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 3n-5+i,\hfill & \mathrm{if}1\le i\le n-1\mathrm{and}i\mathrm{is}\mathrm{even}.\hfill \\ 4n-4,\hfill & \mathrm{if}i=n.\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{c}f\left(c{v}_i\right)=\left\{\begin{array}{cc}3n-4,\hfill & \mathrm{if}i=1,\hfill \\ 2n-3+i\hfill & \mathrm{if}3\le i\le n-1\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 3n-4+i,\hfill & \mathrm{if}1\le i\le n-1\mathrm{and}i\mathrm{is}\mathrm{even}.\hfill \\ 4n-2,\hfill & \mathrm{if}i=n.\hfill \end{array}\right.\hfill \end{array}$$

Secondly, label the vertices of $B_{n,n}$ by the following way:

$$\begin{array}{c}f\left(u_i\right)=\left\{\begin{array}{cc}5n-i\hfill & \mathrm{if}1\le i\le n\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 6n-i,\hfill & \mathrm{if}1\le i\le n\mathrm{and}i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.\hfill \\ f\left(v_i\right)=\left\{\begin{array}{cc}6n-1,\hfill & \mathrm{if}i=1,\hfill \\ 5n-1-i\hfill & \mathrm{if}3\le i\le n\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 6n-1-i,\hfill & \mathrm{if}1\le i\le n\mathrm{and}i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.\hfill \\ f\left(c\right)=5n-2.\hfill \end{array}$$

Accordingly, we have

$$\begin{array}{cc}{\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=9n-3,\hfill & \mathrm{if}i\mathrm{is}\mathrm{odd}\mathrm{and}1\le i\le n,\hfill \\ {\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=11n-6,\hfill & \mathrm{if}i\mathrm{is}\mathrm{even}\mathrm{and}1\le i\le n,\hfill \\ {\omega }_t\left(c\right)=6n^2+2n-2.\hfill \end{array}$$

It is clear that f is a local antimagic total labeling of $B_{n,n}$ using three distinct colors and ${\chi }_{lat}\left(B_{n,n}\right)\le 3$ for odd n.

Case 2. If n is even.

Label the edges of $B_{n,n}$ as follows:

$$\begin{array}{c}f\left(u_i{u}_{i+1}\right)=\left\{\begin{array}{cc}i,\hfill & \mathrm{if}i\mathrm{is}\mathrm{odd}\mathrm{and}1\le i\le n-1,\hfill \\ 2n-i,\hfill & \mathrm{if}i\mathrm{is}\mathrm{even}\mathrm{and}1\le i\le n-1.\hfill \end{array}\right.\hfill \\ f\left(v_i{v}_{i+1}\right)=\left\{\begin{array}{cc}i+1,\hfill & \mathrm{if}i\mathrm{is}\mathrm{odd}\mathrm{and}1\le i\le n-1,\hfill \\ 2n-1-i,\hfill & \mathrm{if}i\mathrm{is}\mathrm{even}\mathrm{and}1\le i\le n-1.\hfill \end{array}\right.\hfill \\ f\left(c{u}_i\right)=\left\{\begin{array}{cc}4n-3,\hfill & \mathrm{if}i=1,\hfill \\ 3n-5+i,\hfill & \mathrm{if}3\le i\le 5\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 3n-4+i,\hfill & \mathrm{if}7\le i\le n\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 2n-3+i,\hfill & \mathrm{if}1\le i\le n\mathrm{and}i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.\hfill \\ f\left(c{v}_i\right)=\left\{\begin{array}{cc}4n-2,\hfill & \mathrm{if}i=1,\hfill \\ 3n-1,\hfill & \mathrm{if}i=3,\hfill \\ 3n-3+i,\hfill & \mathrm{if}5\le i\le n\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 2n-2+i,\hfill & \mathrm{if}1\le i<n\mathrm{and}i\mathrm{is}\mathrm{even}.\hfill \\ 4n-1,\hfill & \mathrm{if}i=n.\hfill \end{array}\right.\hfill \end{array}$$

Then, give the exact values of the vertices of $B_{n,n}$:

$$\begin{array}{c}f\left(u_i\right)=\left\{\begin{array}{cc}6n-1,\hfill & \mathrm{if}i=1,\hfill \\ 5n+1-i\hfill & \mathrm{if}3\le i\le 5\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 5n-i\hfill & \mathrm{if}7\le i\le n\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 6n-2-i,\hfill & \mathrm{if}1\le i<n\mathrm{and}i\mathrm{is}\mathrm{even},\hfill \\ 6n-2,\hfill & \mathrm{if}i=n.\hfill \end{array}\right.\hfill \\ f\left(v_i\right)=\left\{\begin{array}{cc}6n-3,\hfill & \mathrm{if}i=1,\hfill \\ 5n-3,\hfill & \mathrm{if}i=3,\hfill \\ 5n-1-i\hfill & \mathrm{if}5\le i\le n\mathrm{and}i\mathrm{is}\mathrm{odd},\hfill \\ 6n-3-i,\hfill & \mathrm{if}1\le i<n\mathrm{and}i\mathrm{is}\mathrm{even},\hfill \\ 5n-5,\hfill & \mathrm{if}i=n.\hfill \end{array}\right.\hfill \\ f\left(c\right)=3n+1.\hfill \end{array}$$

From above labelings under f, we obtain

$$\begin{array}{cc}{\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=10n-3,\hfill & \mathrm{if}i\mathrm{is}\mathrm{odd}\mathrm{and}1\le i\le n,\hfill \\ {\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=10n-6,\hfill & \mathrm{if}i\mathrm{is}\mathrm{even}\mathrm{and}1\le i\le n,\hfill \\ {\omega }_t\left(c\right)=6n^2+n-1.\hfill \end{array}$$

Clearly, f is a local antimagic total labeling of $B_{n,n}$ using three distinct colors and ${\chi }_{lat}\left(B_{n,n}\right)\le 3$. Hence, we obtain ${\chi }_{lat}\left(B_{n,n}\right)=3$ for $n\ge 2$, and the proof is completed. □

Example 2.

Let $n=5$ and $n=6$. We have the local antimagic total labelings of the bowknot graph $B_{5,5}$ and $B_{6,6}$ in Figure 5 and Figure 6.

The Dutch windmill graph, denoted by $D_4^n(n\ge 2)$, is a graph of order $3n+1$ and size $4n$ by gluing a common vertex of n cycles $C_4$. A example of the Dutch windmill graph is shown in Figure 7. Note that the Dutch windmill graph $D_4^n$ is obtained from the wheel $W_{3n}$ by deleting one edge continuously every 2 edges on the rim of the wheel $W_{3n}$, and then delete the middle spoke edge of each vane in the resulting graph. Let $V\left(D_4^n\right)=\left\{c\right\}\cup \{u_i,v_i,w_i|1\le i\le n\}$ and $E\left(D_4^n\right)=\{c{u}_i,c{v}_i,w_i{u}_i,w_i{v}_i|1\le i\le n\}$ be the vertex and edge sets of graph $D_4^n$, respectively.

Theorem 3.

For the Dutch windmill graph $D_4^n$, we have

$${\chi }_{lat}\left(D_4^n\right)=\left\{\begin{array}{cc}2,\hfill & \mathrm{if}n\le 7,\hfill \\ 3,\hfill & \mathrm{otherwises}.\hfill \end{array}\right.$$

Proof. 

Obviously, the lower bound of the local antiamgic total chromatic number for graph $D_4^n$ is two since ${\chi }_{lat}\left(D_4^n\right)\ge \chi \left(D_4^n\right)=2$. Then we consider the upper bound of ${\chi }_{lat}\left(D_4^n\right)$.

Define $f:V\left(D_4^n\right)\cup E\left(D_4^n\right)\to \{1,2,\cdots ,7n+1\}$. Let $\sigma$ be the sum of the weights of the vertex c and the edges incident with the vertex c and let $\xi$ be the sum of the weights of the vertices $w_i$ for $1\le i\le n$. If we use the minimum weights, label the vertex c and the edges incident with the vertex c, then $\sigma \ge 1+2+\cdots +2n+1$, and if we use the maximum weights, label the edges $w_i{u}_i,w_i{v}_i$ and the vertices $w_i$ for $1\le i\le n$, then $\xi \le (4n+2)+(4n+3)+\cdots +7n+1$. Suppose that there is a local antiamgic total labeling using two distinct colors labeling the graph $D_4^n$; thus, the color of the vertex c is same as the vertices $w_i$ for $1\le i\le n$. It means $\sigma \le \frac{\xi }{n}$ and so $n\le 7$. When $\sigma \le \frac{\xi }{n}$, there possibly is obtained a local antimagic total labeling f using two distinct colors such that the vertex c and the vertices $w_i$ have the same color for $n\le 7$. However, when $\sigma >\frac{\xi }{n}$, there must exist three different colors. Consider two cases as follows:

Case 1. For $n>7$.

According to the parity of n, there are two subcases to confirm the exact values of the local antimagic total labeling.

Subcase 1. If n is odd.

Label the edges and vertices of the graph $D_4^n$ by the following way:

$$\begin{array}{c}f\left(c{u}_i\right)=i,\mathrm{for}1\le i\le n,\hfill \\ f\left(c{v}_i\right)=\left\{\begin{array}{cc}\frac{3n-1}{2}-2i,\hfill & \mathrm{for}1\le i\le \lfloor \frac{n-3}{4}\rfloor ,\hfill \\ \frac{5n-1}{2}-2i,\hfill & \mathrm{for}\lfloor \frac{n+1}{4}\rfloor \le i\le 2\lceil \frac{n-1}{4}\rceil -\lfloor \frac{n-3}{4}\rfloor ,\hfill \\ \frac{7n-1}{2}-2i.\hfill & \mathrm{for}2\lceil \frac{n-1}{4}\rceil -\lfloor \frac{n-3}{4}\rfloor <i\le n.\hfill \end{array}\right.\hfill \\ f\left(w_i{u}_i\right)=\left\{\begin{array}{cc}\frac{11n+3}{2}+i,\hfill & \mathrm{for}1\le i\le \frac{n-1}{2},\hfill \\ \frac{9n+3}{2}+i,\hfill & \mathrm{for}\frac{n+1}{2}\le i\le n.\hfill \end{array}\right.\hfill \\ f\left(w_i{v}_i\right)=\left\{\begin{array}{cc}7n+1-2i,\hfill & \mathrm{for}1\le i\le \frac{n-1}{2},\hfill \\ 8n+1-2i,\hfill & \mathrm{for}\frac{n+1}{2}\le i<n,\hfill \\ 7n+1,\hfill & \mathrm{for}i=n.\hfill \end{array}\right.\hfill \end{array}$$

Then

$$\begin{array}{c}f\left(c\right)=5n+1,\hfill \\ f\left(u_i\right)=\left\{\begin{array}{cc}5n+1-2i,\hfill & \mathrm{for}1\le i\le \frac{n-1}{2},\hfill \\ 6n+1-2i,\hfill & \mathrm{for}\frac{n+1}{2}\le i\le n.\hfill \end{array}\right.\hfill \\ f\left(v_i\right)=\left\{\begin{array}{cc}2n+2+4i,\hfill & \mathrm{for}1\le i\le \frac{n-1}{2}-\lceil \frac{n+1}{4}\rceil ,\hfill \\ n+2+4i,\hfill & \mathrm{for}\frac{n+1}{2}-\lceil \frac{n+1}{4}\rceil \le i\le \frac{n-1}{2},\hfill \\ 2+4i,\hfill & \mathrm{for}\frac{n+1}{2}\le i\le \frac{n-3}{2}+\lceil \frac{n+1}{4}\rceil ,\hfill \\ 2-n+4i,\hfill & \mathrm{for}\frac{n-1}{2}+\lceil \frac{n+1}{4}\rceil \le i<n,\hfill \\ 2n+2,\hfill & \mathrm{for}i=n.\hfill \end{array}\right.\hfill \\ f\left(w_i\right)=\left\{\begin{array}{cc}3n+1+i,\hfill & \mathrm{for}1\le i\le n-1,\hfill \\ 3n+1,\hfill & \mathrm{for}i=n,\hfill \end{array}\right.\hfill \end{array}$$

From the above vertex weights, we have

$$\begin{array}{cc}{\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=\frac{21n+5}{2},\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(w_i\right)=\frac{31n+7}{2},\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(c\right)=2n^2+6n+1.\hfill \end{array}$$

Therefore, f is a local antimagic total labeling of the graph $D_4^n$ using three distinct colors and so ${\chi }_{lat}\left(D_4^n\right)=3$ for odd n.

Subcase 2. If n is even.

Label the edges and vertices of the graph $D_4^n$ by the following way:

$$\begin{array}{c}f\left(c{u}_i\right)=i,\mathrm{for}1\le i\le n,\hfill \\ f\left(c{v}_i\right)=n+i,\mathrm{for}1\le i\le n,\hfill \\ f\left(w_i{v}_i\right)=4n+1-2i,\mathrm{for}1\le i\le n,\hfill \\ f\left(w_i{u}_i\right)=\left\{\begin{array}{cc}5n+2,\hfill & \mathrm{for}i=1,\hfill \\ 6n+1+i,\hfill & \mathrm{for}2\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

Then

$$\begin{array}{c}f\left(c\right)=5n+1,\hfill \\ f\left(v_i\right)=5n+2+i,\mathrm{for}1\le i\le n,\hfill \\ f\left(u_i\right)=\left\{\begin{array}{cc}5n,\hfill & \mathrm{for}i=1,\hfill \\ 4n+2-2i,\hfill & \mathrm{for}2\le i\le n,\hfill \end{array}\right.\hfill \\ f\left(w_i\right)=\left\{\begin{array}{cc}5n-1,\hfill & \mathrm{for}i=1,\hfill \\ 4n-2+i,\hfill & \mathrm{for}2\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

For the vertex weights under the labeling f, we have

$$\begin{array}{cc}{\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=10n+3,\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(w_i\right)=14n,\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(c\right)=2n^2+6n+1.\hfill \end{array}$$

The above arguments indicate that f is a local antimagic total labeling of $D_4^n$ with three colors, and so ${\chi }_{lat}\left(D_4^n\right)\le 3$ for even n.

Case 2. For $n\le 7$.

Present the detailed local antimagic total labeling for each n when $n\le 7$. The following figure is shown the local antimagic total labeling for $n=7$ in Figure 8.

The exact value of each edge and vertex of the local antimagic total labelings for $n\le 6$ are given in

Table 1. $D_4^2$.

The Weights of Local Antimagic Total Labeling of ${\mathit{D}}_4^2$
i	1	2
$f\left(c{u}_i\right)$	1	3
$f\left(c{v}_i\right)$	2	4
$f\left(w_i{u}_i\right)$	10	12
$f\left(w_i{v}_i\right)$	6	5
$f\left(u_i\right)$	11	7
$f\left(v_i\right)$	14	13
$f\left(w_i\right)$	9	8
$f\left(c\right)$	15

,

Table 2. $D_4^3$.

The Weights of Local Antimagic Total Labeling of ${\mathit{D}}_4^3$
i	1	2	3
$f\left(c{u}_i\right)$	1	2	3
$f\left(c{v}_i\right)$	6	4	5
$f\left(w_i{u}_i\right)$	19	17	18
$f\left(w_i{v}_i\right)$	7	8	9
$f\left(u_i\right)$	14	15	13
$f\left(v_i\right)$	21	22	20
$f\left(w_i\right)$	11	12	10
$f\left(c\right)$	16

,

Table 3. $D_4^4$.

The Weights of Local Antimagic Total Labeling of ${\mathit{D}}_4^4$
i	1	2	3	4
$f\left(c{u}_i\right)$	2	4	14	6
$f\left(c{v}_i\right)$	5	1	8	3
$f\left(w_i{u}_i\right)$	12	11	7	10
$f\left(w_i{v}_i\right)$	15	17	22	20
$f\left(u_i\right)$	25	24	18	23
$f\left(v_i\right)$	19	21	9	16
$f\left(w_i\right)$	29	28	27	26
$f\left(c\right)$	13

,

Table 4. $D_4^5$.

The Weights of Local Antimagic Total Labeling of ${\mathit{D}}_4^5$
i	1	2	3	4	5
$f\left(c{u}_i\right)$	1	2	3	4	5
$f\left(c{v}_i\right)$	10	8	6	9	7
$f\left(w_i{u}_i\right)$	30	31	27	28	29
$f\left(w_i{v}_i\right)$	34	32	35	33	36
$f\left(u_i\right)$	24	22	25	23	21
$f\left(v_i\right)$	11	15	14	13	12
$f\left(w_i\right)$	17	18	19	20	16
$f\left(c\right)$	26

and

Table 5. $D_4^6$.

The Weights of Local Antimagic Total Labeling of ${\mathit{D}}_4^6$
i	1	2	3	4	5	6
$f\left(c{u}_i\right)$	3	6	2	5	4	1
$f\left(c{v}_i\right)$	12	8	11	9	10	7
$f\left(w_i{u}_i\right)$	26	27	28	29	31	36
$f\left(w_i{v}_i\right)$	41	42	40	43	39	35
$f\left(u_i\right)$	38	34	37	33	32	30
$f\left(v_i\right)$	14	17	16	15	18	25
$f\left(w_i\right)$	24	22	23	19	21	20
$f\left(c\right)$	13

.

When $n=2$, we obtain that ${\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=22$ and ${\omega }_t\left(w_i\right)={\omega }_t\left(c\right)=25$ for $i=1,2$. Similarly, when $n=3$, we obtain that ${\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=34$ and ${\omega }_t\left(w_i\right)={\omega }_t\left(c\right)=37$ for $i=1,2,3$. When $n=4$, we obtain that ${\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=39$ and ${\omega }_t\left(w_i\right)={\omega }_t\left(c\right)=56$ for $1\le i\le 4$. Therefore ${\chi }_{lat}\left(D_4^2\right)={\chi }_{lat}\left(D_4^3\right)={\chi }_{lat}\left(D_4^4\right)=2$.

When $n=5$, ${\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=55$ and ${\omega }_t\left(w_i\right)={\omega }_t\left(c\right)=81$ for $1\le i\le 5$. Similarly, when $n=6$, ${\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=67$ and ${\omega }_t\left(w_i\right)={\omega }_t\left(c\right)=91$ for $1\le i\le 6$. Therefore ${\chi }_{lat}\left(D_4^5\right)={\chi }_{lat}\left(D_4^6\right)=2$.

In conclusion, the local antimagic total chromatic number of the graph $D_4^n$ is ${\chi }_{lat}\left(D_4^n\right)=2$ for $n\le 7$, and ${\chi }_{lat}\left(D_4^n\right)=3$ for $n>7$, respectively. The proof is done. □

Example 3.

The local antimagic total labelings of the graph $D_4^9$ and $D_4^8$ are shown in Figure 9 and Figure 10.

The analogous Dutch windmill graph, denoted by $A{D}_4^n$, is obtained from the Dutch windmill graph $D_4^n$ by adding edges $c{w}_i$ for each $i\in 1,2,\dots ,n$. The graph $A{D}_4^n$ has $3n+1$ vertices and $5n$ edges. It can be seen that the analogous Dutch windmill graph $A{D}_4^n$ can be viewed as from the wheel $W_{3n}$ by deleting one edge continuously every 2 edges on the rim of the wheel. The analogous Dutch windmill graph $A{D}_4^4$ is shown in Figure 11.

Theorem 4.

For the analogous Dutch windmill graph, $A{D}_4^n$, we have ${\chi }_{lat}\left(A{D}_4^n\right)=3$.

Proof. 

The lower bound of the local antimagic total chromatic number of graph $A{D}_4^n$ is 3 since $K_3$ is an induced subgraph of graph $A{D}_4^n$ and ${\chi }_{lat}\left(A{D}_4^n\right)\ge \chi \left(A{D}_4^n\right)$. Define $f:V\left(A{D}_4^n\right)\cup E\left(A{D}_4^n\right)\to \{1,2,\cdots ,8n+1\}$. We discuss two cases for the exact value of each vertex and edge as follows.

Case 1. If n is odd.

For $n=1$, the local antimagic total labeling of the graph $A{D}_4^1$ is shown in Figure 12.

Label the edges of $A{D}_4^n$ by the following way:

$$\begin{array}{c}f\left(c{u}_i\right)=\left\{\begin{array}{cc}\frac{n-1}{2}+i,\hfill & \mathrm{for}1\le i\le \frac{n+1}{2},\hfill \\ i-\frac{n+1}{2},\hfill & \mathrm{for}\frac{n+3}{2}\le i\le n.\hfill \end{array}\right.\hfill \\ f\left(c{v}_i\right)=\left\{\begin{array}{cc}\frac{3n+1}{2}-i,\hfill & \mathrm{for}1\le i\le \frac{n-1}{2},\hfill \\ \frac{5n+1}{2}-i,\hfill & \mathrm{for}\frac{n+1}{2}\le i\le n.\hfill \end{array}\right.\hfill \\ f\left(c{w}_i\right)=8n+1-i,\mathrm{for}1\le i\le n.\hfill \\ f\left(w_i{u}_i\right)=5n+i,\mathrm{for}1\le i\le n.\hfill \\ f\left(w_i{v}_i\right)=7n+1-i,\mathrm{for}1\le i\le n.\hfill \end{array}$$

Then label the vertices of $A{D}_4^n$ as follows:

$$\begin{array}{c}f\left(c\right)=8n+1,\hfill \\ f\left(w_i\right)=3n+i,\mathrm{for}1\le i\le n,\hfill \\ f\left(u_i\right)=\left\{\begin{array}{cc}5n+2-2i,\hfill & \mathrm{for}1\le i\le \frac{n+1}{2},\hfill \\ 6n+2-2i,\hfill & \mathrm{for}\frac{n+3}{2}\le i\le n.\hfill \end{array}\right.\hfill \\ f\left(v_i\right)=\left\{\begin{array}{cc}2n+2i,\hfill & \mathrm{for}1\le i\le \frac{n-1}{2},\hfill \\ n+2i,\hfill & \mathrm{for}\frac{n+1}{2}\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

For the vertex weights under the labeling f, we have

$$\begin{array}{cc}{\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=\frac{21n+3}{2},\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(w_i\right)=23n+2,\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(c\right)=\frac{19n^2+19n+2}{2}.\hfill \end{array}$$

The above arguments indicate that f is a local antimagic labeling of $A{D}_4^n$ with three colors, and so ${\chi }_{lat}\left(A{D}_4^n\right)\le 3$ for odd n.

Case 2. If n is even.

We give the following exact values of edges of $A{D}_4^n$:

$$\begin{array}{c}f\left(c{u}_i\right)=\left\{\begin{array}{cc}i,\hfill & \mathrm{for}1\le i<n,\hfill \\ 2n,\hfill & \mathrm{for}i=n.\hfill \end{array}\right.\hfill \\ f\left(c{v}_i\right)=2n-i,\mathrm{for}1\le i\le n,\hfill \\ f\left(c{w}_i\right)=8n+1-i,\mathrm{for}1\le i\le n,\hfill \\ f\left(w_i{u}_i\right)=6n+i,\mathrm{for}1\le i\le n,\hfill \\ f\left(w_i{v}_i\right)=6n+1-i,\mathrm{for}1\le i\le n,\hfill \end{array}$$

Then label the vertices as follows:

$$\begin{array}{c}f\left(c\right)=8n+1,\hfill \\ f\left(w_i\right)=2n+1+i,\mathrm{for}1\le i\le n,\hfill \\ f\left(u_i\right)=\left\{\begin{array}{cc}5n+1-2i,\hfill & \mathrm{for}1\le i<n,\hfill \\ 2n+1,\hfill & \mathrm{for}i=n.\hfill \end{array}\right.\hfill \\ f\left(v_i\right)=3n+2i,\mathrm{for}1\le i\le n.\hfill \end{array}$$

We obtain that,

$$\begin{array}{cc}{\omega }_t\left(u_i\right)={\omega }_t\left(v_i\right)=11n+1,\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(w_i\right)=22n+3,\hfill & \mathrm{for}1\le i\le n,\hfill \\ {\omega }_t\left(c\right)=\frac{19n^2+19n+2}{2}.\hfill \end{array}$$

It is clear that f is a local antimagic total labeling of $A{D}_4^n$ using three colors and so ${\chi }_{lat}\left(A{D}_4^n\right)\le 3$ for even n. The proof is complete. □

Example 4.

The local antimagic total labelings of the graph $A{D}_4^5$ and $A{D}_4^6$ are shown in Figure 13 and Figure 14.

The flower graph ${\mathcal{F}}_n$ of order $2n+1$ is a graph obtained by adjoining firstly a pendant edge to each vertex on the rim of the wheel graph $W_n$, then joining every pendant vertex to the central vertex of $W_n$ by an edge. The flower graph ${\mathcal{F}}_5$ is shown in Figure 15. Then we respectively obtain the lower and upper bounds of the local antimagic vertex total chromatic number of the flower graph ${\mathcal{F}}_n$.

Theorem 5.

For the flower graph ${\mathcal{F}}_n$, we have

$$\left\{\begin{array}{cc}4\le {\chi }_{lat}\left({\mathcal{F}}_n\right)\le 5,\hfill & \mathrm{for}\mathrm{odd}n,\hfill \\ 3\le {\chi }_{lat}\left({\mathcal{F}}_n\right)\le 4,\hfill & \mathrm{for}\mathrm{even}n.\hfill \end{array}\right.$$

Proof. 

${\chi }_{lavt}\left({\mathcal{F}}_n\right)\ge \chi \left({\mathcal{F}}_n\right)=4$ for odd n and ${\chi }_{lavt}\left({\mathcal{F}}_n\right)\ge \chi \left({\mathcal{F}}_n\right)=3$ for even n since $W_n$ is the induced subgraph of graph ${\mathcal{F}}_n$. Then give the upper bound of the local antimigic total chromatic number of the flower graph.

The flower graph ${\mathcal{F}}_n$ has $2n+1$ vertices and $4n$ edges. The vertex set of ${\mathcal{F}}_n$ is $V\left({\mathcal{F}}_n\right)=\left\{u_i\right|1\le i\le n\}\cup \left\{v_i\right|1\le i\le n\}\cup \left\{c\right\}$ and the edge set of ${\mathcal{F}}_n$ is $E\left({\mathcal{F}}_n\right)=\left\{u_i{u}_{i+1}\right|1\le i\le n\}\cup \left\{c{u}_i\right|1\le i\le n\}\cup \left\{u_i{v}_i\right|1\le i\le n\}\cup \left\{c{v}_i\right|1\le i\le n\}$, where the edge $u_n{u}_{n+1}$ is the edge $u_n{u}_1$. Let f be the local antimagic labeling of the graph $W_n$ defined in Lemma 1 such that the vertices are assigned four distinct colors for odd n and three colors for even n.

Define a bijection $g:V\left({\mathcal{F}}_n\right)\cup E\left({\mathcal{F}}_n\right)\to \{1,2,\cdots ,6n+1\}$. Firstly, label the edges of the subgraph $W_n$ of the ${\mathcal{F}}_n$ such that $g\left(e\right)=f\left(e\right)$, where $e\in W_n$. Secondly, label the remaining edges and vertices of ${\mathcal{F}}_n$ using $\{2n+1,\cdots ,6n+1\}$ since the wheel has $2n$ edges. Let us discuss two cases for n.

Case 1. If n is odd.

By Lemma 1, for $n\equiv 1(mod4)$, the vertex weights of graph $W_n$ are, respectively, $\omega \left(u_1\right)=\frac{5n+11}{4}$, $\omega \left(u_i\right)=\frac{11n+13}{4}$ for odd i and $i\ne 1$, $\omega \left(u_i\right)=\frac{9n+11}{4}$ for even i and $\omega \left(c\right)=\frac{6n^2+n+1}{4}$. For $n\equiv 3(mod4)$, the vertex weights of graph $W_n$ are, respectively, $\omega \left(u_1\right)=2n+2$, $\omega \left(u_i\right)=\frac{9n+9}{4}$ for odd i and $i\ne 1$, $\omega \left(u_i\right)=\frac{11n+7}{4}$ for even i and $\omega \left(c\right)=\frac{(3n+1)n}{2}$. Then

$$\begin{array}{c}g\left(c\right)=6n+1,\hfill \\ g\left(c{v}_i\right)=4n+1-2i,\mathrm{for}1\le i\le n,\hfill \\ g\left(u_i{v}_i\right)=\left\{\begin{array}{cc}5n-1+i,\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ 4n-1+i,\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

$$\begin{array}{c}g\left(v_i\right)=\left\{\begin{array}{cc}2n+1+i,\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ 3n+1+i,\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \\ g\left(u_i\right)=\left\{\begin{array}{cc}5n+2-i,\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ 6n+2-i,\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

Conclude the vertex weights under labeling g for each vertex of the graph ${\mathcal{F}}_n$ as follows:

For $n\equiv 1(mod4)$,

$$\begin{array}{c}{\omega }_t\left(c\right)=\frac{18n^2+25n+5}{4},\hfill \\ {\omega }_t\left(v_i\right)=11n+1,\mathrm{for}1\le i\le n.\hfill \\ {\omega }_t\left(u_i\right)=\left\{\begin{array}{cc}\frac{51n+17}{4},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}2\le i\le n,\hfill \\ \frac{49n+11}{4},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n,\hfill \\ \frac{45n+15}{4},\hfill & \mathrm{for}i=1.\hfill \end{array}\right.\hfill \end{array}$$

For $n\equiv 3(mod4)$,

$$\begin{array}{c}{\omega }_t\left(c\right)=\frac{9n^2+13n+2}{2},\hfill \\ {\omega }_t\left(v_i\right)=11n+1,\mathrm{for}1\le i\le n.\hfill \\ {\omega }_t\left(u_i\right)=\left\{\begin{array}{cc}\frac{49n+13}{4},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}2\le i\le n,\hfill \\ \frac{51n+11}{4},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n,\hfill \\ 12n+3,\hfill & \mathrm{for}i=1.\hfill \end{array}\right.\hfill \end{array}$$

It is clear that g is a local antimagic total labeling of the graph ${\mathcal{F}}_n$ using five colors, and so ${\chi }_{lat}\left({\mathcal{F}}_n\right)\le 5$.

Case 2. If n is even.

By Lemma 1, for $n\equiv 2(mod4)$, the vertex weights of graph $W_n$ are, respectively, $\omega \left(u_i\right)=\frac{9n+6}{4}$ for odd i and $\omega \left(u_i\right)=\frac{11n+6}{4}$ for even i and $\omega \left(c\right)=\frac{(3n+1)n}{2}$. For $n\equiv 4(mod4)$, the vertex weights of graph $W_n$ are, respectively, $\omega \left(u_i\right)=\frac{9n+8}{4}$ for odd i and $i\ne 1$, $\omega \left(u_i\right)=\frac{11n+4}{4}$ for even i and $\omega \left(c\right)=\frac{(3n+1)n}{2}$. Then

$$\begin{array}{c}g\left(c\right)=5n+1,\hfill \\ g\left(u_i\right)=2n+2i,\mathrm{for}1\le i\le n,\hfill \\ g\left(u_i{v}_i\right)=4n+1-2i,\mathrm{for}1\le i\le n,\hfill \\ g\left(v_i\right)=\left\{\begin{array}{cc}4n+i,\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ 5n+1+i,\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \\ g\left(c{v}_i\right)=\left\{\begin{array}{cc}5n+1+i,\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ 4n+i,\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

Accordingly, we obtain the vertex weights under labeling g for each vertex of the graph ${\mathcal{F}}_n$.

For $n\equiv 2(mod4)$,

$$\begin{array}{c}{\omega }_t\left(x\right)=\frac{13n^2+13n+2}{2},\hfill \\ {\omega }_t\left(v_i\right)=13n+2,\mathrm{for}1\le i\le n.\hfill \\ {\omega }_t\left(u_i\right)=\left\{\begin{array}{cc}\frac{33n+10}{4},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ \frac{35n+10}{4},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

For $n\equiv 4(mod4)$,

$$\begin{array}{c}{\omega }_t\left(x\right)=\frac{13n^2+13n+2}{2},\hfill \\ {\omega }_t\left(v_i\right)=13n+2,\mathrm{for}1\le i\le n.\hfill \\ {\omega }_t\left(u_i\right)=\left\{\begin{array}{cc}\frac{33n+12}{4},\hfill & \mathrm{for}\mathrm{odd}i\mathrm{and}1\le i\le n,\hfill \\ \frac{35n+8}{4},\hfill & \mathrm{for}\mathrm{even}i\mathrm{and}1\le i\le n.\hfill \end{array}\right.\hfill \end{array}$$

Therefore g is a local antimagic total labeling of graph ${\mathcal{F}}_n$ with four colors and ${\chi }_{lat}\left({\mathcal{F}}_n\right)\le 4$. The proof is done. □

Example 5.

The local antimagic total labelings of the graph ${\mathcal{F}}_9$ and ${\mathcal{F}}_{11}$ are shown in Figure 16 and Figure 17.

Example 6.

The local antimagic total labelings of the graph ${\mathcal{F}}_{10}$ and ${\mathcal{F}}_{12}$ are shown in Figure 18 and Figure 19.

3. Conclusions

Note that the difference of the local antimagic chromatic coloring and the local antimagic total chromatic coloring is that the former is assigning weighted values for the edge set of graph G, which induces a proper vertex coloring of G; the latter is assigning the vertex and edge set of graph G simultaneously, which also induces a proper vertex coloring of G.

For a graph G with independent vertices, the determination of the local antimagic chromatic number is easier than that of the local antimagic total chromatic number. In fact, the weighted values of independent edges in the local antimagic coloring of G must be different, but may be the same in the local antimagic total chromatic coloring of G. In this paper, we consider the local antimagic total chromatic colorings of wheel-related graphs which have no independent edges, and present the local antimagic total chromatic numbers of the fan graph $F_n$, the bowknot graph $B_{n,n}$, the Dutch windmill graph $D_4^n$, the analogous Dutch graph $A{D}_4^n$ and the flower graph ${\mathcal{F}}_n$.