--supermagic labeling of graphs

Abstract

A simple graph $G$ admits an $H$-covering if every edge in $E\left(G\right)$ belongs to a subgraph of $G$ isomorphic to $H$. The graph $G$ is said to be $H$-magic if there exists a total labeling $f:V\left(G\right)\cup E\left(G\right)\to \{1,2,3,\dots ,\left|V\left(G\right)\right|+\left|E\left(G\right)\right|\}$ such that for every subgraph $H^{\prime }$ of $G$ isomorphic to $H$, $\sum _{v\in V\left(H^{{}^{\prime }}\right)}f\left(v\right)+\sum _{e\in E\left(H^{{}^{\prime }}\right)}f\left(e\right)$ is constant. Additionally, the labeling $f$ is called $H$-$E$ supermagic labeling if $f\left(E\left(G\right)\right)=\{1,2,3,\dots ,\left|E\left(G\right)\right|\}$. In this paper, we study $C_m$-$E$-supermagic labeling of some families of connected graphs.

Keywords:

• $H$-magic labeling
• $H$-$E$-supermagic labeling
• $C_m$-$E$-supermagic labeling

1 Introduction

We consider finite and simple graphs. The vertex and edge sets of a graph $G$ are denoted by $V\left(G\right)$ and $E\left(G\right)$, respectively. Let $H$ be a graph. An edge- covering of $G$ is a family of subgraphs $H_1,H_2,\dots ,H_k$ such that each edge of $E\left(G\right)$ belongs to at least one of the subgraphs $H_i$, $1\le i\le k$. Then it is said that $G$ admits an $(H_1,H_2,\dots ,H_k)$-(edge) covering. If every $H_i$ is isomorphic to a given graph $H$, then $G$ admits an $H$-covering. Suppose $G$ admits an $H$-covering. A total labeling $f:V\left(G\right)\cup E\left(G\right)\to \{1,2,3,\dots ,\left|V\left(G\right)\right|+\left|E\left(G\right)\right|\}$ is called an $H$-magic labeling of $G$ if there exists a positive integer $k_f$ (called the magic constant) such that for every subgraph $H^{\prime }$ of $G$ isomorphic to $H$, ${\sum }_{v\in V\left(H^{\prime }\right)}f\left(v\right)+{\sum }_{e\in E\left(H^{\prime }\right)}f\left(e\right)=k_f$. A graph that admits such a labeling is called $H$-magic. An $H$-magic labeling $f$ is called an $H$-$E$-supermagic labeling if $f\left(E\left(G\right)\right)=\{1,2,3,\dots ,\left|E\left(G\right)\right|\}$. A graph that admits an $H$-$E$-supermagic labeling is called an $H$-$E$-supermagic graph. The sum of all vertex and edge labels on $H$ (under a labeling $f$) is denoted by $\sum f\left(H\right)$.

The notion of $H$-magic labeling was introduced by Gutierrez and Lladó [1] in 2005. They proved that the star graph $K_{1,n}$ and the complete bipartite graphs $K_{m,n}$ are $K_{1,h}$-supermagic for some $h$. They also proved that the paths $P_n$ and the cycles $C_n$ are $P_h$-supermagic for some $h$.

In 2007, Llado and Moragas [2] studied some $C_n$-supermagic graphs. They proved that the wheel $W_n$, the windmill $W(r,k)$, the subdivided wheel $W_n(r,k)$ and the graph obtained by joining two end vertices of any number of internally disjoint paths of length $p\ge 2$ are $C_h$-supermagic for some $h$. Maryati et al. [3] studied some $P_h$-supermagic trees. They proved that shrubs, balanced subdivision of shrubs and banana trees are $P_h$-supermagic for some $h$.

MacDougall et al. [4] and Swaminathan and Jeyanthi [5] introduced different labelings with the same name super vertex-magic total labeling. To avoid confusion Marimuthu and Balakrishnan [6] call a vertex magic total labeling is $E$-super if $f\left(E\left(G\right)\right)=\{1,2,\dots ,\left|E\left(G\right)\right|\}$. Note that the smallest labels are assigned to the edges. A graph $G$ is called $E$-super vertex magic if it admits $E$-super vertex labeling. There are many graphs that have been proved to be $E$-super vertex magic; see for instance [7,8] and [9]. For further information about $H$-$E$-supermagic graphs, see [10].

In this paper, we study $C_m$-$E$-supermagic labeling of some families of connected graphs such as generalized books, generalization of a graph obtained by joining of a star $K_{1,n}$ with one isolated vertex found in [11], generalized fans, generalized friendship graphs and generalized grids.

2 $C_m$-$E$-Supermagic graphs

A bookgraph $B_n=K_{1,n}\times K_2$, is defined as follows: $$V\left(B_n\right)=\{u_1,u_2\}\cup \{v_i,w_i:1\le i\le n\}\text{and}$$$$E\left(B_n\right)=\{u_1,u_2\}\cup \{u_1{w}_i,u_2{v}_i,v_i{w}_i:1\le i\le n\}.$$ We define the generalized bookgraph $B_{n,m}$ with vertex and edge sets by $$V\left(B_{n,m}\right)=\{u_i:1\le i\le m-2\}\cup \{v_i,w_i:1\le i\le n\}\text{and}$$$$E\left(B_{n,m}\right)=\{u_i{u}_{i+1}:for1\le i\le m-3\}\cup \{u_i{v}_j:1\le j\le n,i=m-2\}\cup \{u_1{w}_i:1\le i\le n\}\cup \{v_i{w}_i:1\le i\le n\}.$$ The following theorem shows that the generalized bookgraph $B_{n,m}$ is $C_m$-$E$-supermagic.

Theorem 2.1

For any integer $n\ge 2$ and $m\ge 4$ and$m\ne 6$ the generalized book graph $B_{n,m}$ is$C_m$-$E$-supermagic.

Proof

We define a total labeling $g:V\left(B_{n,m}\right)\cup E\left(B_{n,m}\right)\to \{1,2,3,\dots ,2m+5(n-1)\}$ as given below. $$g\left(x\right)=\left\{\begin{array}{cc}3n+(m-3)+i,\hfill & \text{if}x=u_i,\text{for}1\le i\le m-2,\hfill \\ 3n+(2m-5)+i,\hfill & \text{if}x=v_i,\text{for}1\le i\le n,\hfill \\ 5n+2(m-2)-i,\hfill & \text{if}x=w_i,\text{for}1\le i\le n.\hfill \end{array}\right.$$The case of odd n: $$g\left(x\right)=\left\{\begin{array}{cc}i,\hfill & \text{if}x=u_i{u}_{i+1},\text{for}1\le i\le m-3,\hfill \\ (m-3)+j,\hfill & \text{if}x=u_i{v}_j,\text{for}1\le j\le nandi=m-2,\hfill \\ 3n+(m-1)-2i,\hfill & \text{if}x=u_1{w}_i,\text{for}1\le i\le \lceil \frac{n}{2}\rceil ,\hfill \\ 4n+(m-1)-2i,\hfill & \text{if}x=u_1{w}_i,\text{for}\lceil \frac{n}{2}\rceil +1\le i\le n,\hfill \\ \frac{1}{2}(3n+2m-7+2i),\hfill & \text{if}x=v_i{w}_i,\text{for}1\le i\le \lceil \frac{n}{2}\rceil ,\hfill \\ \frac{1}{2}(n+2m-7+2i),\hfill & \text{if}x=v_i{w}_i,\text{for}\lceil \frac{n}{2}\rceil +1\le i\le n.\hfill \end{array}\right.$$The case of even n: $$g\left(x\right)=\left\{\begin{array}{cc}\frac{n}{2}+1,\hfill & \text{if}x=u_1{u}_2,\hfill \\ n+i,\hfill & \text{if}x=u_i{u}_{i+1},\text{for}2\le i\le m-3,\hfill \\ j,\hfill & \text{if}x=u_i{v}_j,\text{for}1\le j\le \frac{n}{2}andi=m-2,\hfill \\ j+1,\hfill & \text{if}x=u_i{v}_j,\text{for}\frac{n}{2}+1\le j\le nandi=m-2,\hfill \\ 3n+(m-1)-2i,\hfill & \text{if}x=u_1{w}_i,\text{for}1\le i\le \frac{n}{2},\hfill \\ 4n+(m-2)-2i,\hfill & \text{if}x=u_1{w}_i,\text{for}\frac{n}{2}+1\le i\le n,\hfill \\ \frac{1}{2}(3n+2m-6+2i),\hfill & \text{if}x=v_i{w}_i,\text{for}1\le i\le \frac{n}{2},\hfill \\ \frac{1}{2}(n+2m-6+2i),\hfill & \text{if}x=v_i{w}_i,\text{for}\frac{n}{2}+1\le i\le n.\hfill \end{array}\right.$$Let us show that $g$ is a $C_m$-$E$-supermagic labeling.

For this, let $C_m^{\left(i\right)}$, $1\le i\le n$ be the subcycle of the graph $B_{n,m}$ with $$V\left(C_m^{\left(i\right)}\right)=\{u_j:1\le j\le m-2\}\cup \{v_i,w_i\}\text{and}$$$$E\left(C_m^{\left(i\right)}\right)=\{u_j{u}_{j+1}:1\le j\le m-3\}\cup \{u_1{w}_i,v_i{w}_i\}\cup \{u_k{v}_i:k=m-2\}.$$ For each $i,1\le i\le n$, $k=m-2$, we have $$\sum _{j=1}^{k}g\left(u_j\right)+g\left(v_i\right)+g\left(w_i\right)+\sum _{j=1}^{k-1}g\left(u_j{u}_{j+1}\right)+g\left(u_k{v}_i\right)+g\left(u_1{w}_i\right)+g\left(v_i{w}_i\right)=\left\{\begin{array}{cc}\frac{1}{2}(37n+35)+(m-4)[(3n+16)+2(m-5)],\hfill & \text{for odd n},\hfill \\ (19n+17)+(m-4)[4n+15+2(m-5)],\hfill & \text{for even n}.\hfill \end{array}\right.$$

This completes the proof. □

Consider the graph $G\cong K_{1,n}+K_1$, where $V\left(G\right)=\{c_1,c_2\}\cup \{v_j:1\le j\le n\}$ and $E\left(G\right)=\left\{c_1{c}_2\right\}\cup \{c_i{v}_j:i=1,2;1\le j\le n\}$.

We define the generalization $G_{n,m}$ of the graph $G$ given above as follows: $$V\left(G_{n,m}\right)=\{c_i:1\le i\le m-1\}\cup \{v_i:1\le i\le n\}\text{and}$$$$E\left(G_{n,m}\right)=\{c_i{c}_{i+1}:1\le i\le m-2\}\cup \{c_1{v}_i:1\le i\le n\}\cup \{c_j{v}_i:j=m-1,1\le i\le n\}.$$ Now we prove that the graph $G_{n,m}$ is $C_m$-$E$-supermagic.

Theorem 2.2

For any positive integer $n$ and $m\ge 3$,$m\ne 4$ the graph$G_{n,m}$ is$C_m$-$E$-supermagic.

Proof

Define a total labeling $h:V\left(G_{n,m}\right)\cup E\left(G_{n,m}\right)\to \{1,2,3,\dots ,2m+3(n-1)\}$ as follows: $$h\left(x\right)=\left\{\begin{array}{cc}2n+(m-2)+i,\hfill & \text{if}x=c_i,\text{for}1\le i\le m-1,\hfill \\ 2n+(2m-3)+i,\hfill & \text{if}x=v_i,\text{for}1\le i\le n.\hfill \end{array}\right.$$The case of odd $n$: $$h\left(x\right)=\left\{\begin{array}{cc}i,\hfill & \text{if}x=c_i{c}_{i+1},\text{for}1\le i\le m-2,\hfill \\ \frac{1}{2}(n+(2m-3)+2i),\hfill & \text{if}x=c_1{v}_i,\text{for}1\le i\le \frac{n-1}{2},\hfill \\ \frac{1}{2}(2i+(2m-3)-n),\hfill & \text{if}x=c_1{v}_i,\text{for}i=\frac{n+1}{2},\frac{n+3}{2},\dots ,n,\hfill \\ 2n+(m-1)-2i,\hfill & \text{if}x=c_j{v}_i,\text{for}1\le i\le \frac{n-1}{2},j=m-1,\hfill \\ 3n+(m-1)-2i,\hfill & \text{if}x=c_j{v}_i,\text{for}i=\frac{n+1}{2},\frac{n+3}{2},\dots ,n,j=m-1.\hfill \end{array}\right.$$The case of even $n$: $$h\left(x\right)=\left\{\begin{array}{cc}\frac{n}{2}+1,\hfill & \text{if}x=c_1{c}_2,\hfill \\ n+i,\hfill & \text{if}x=c_i{c}_{i+1},\text{for}i=2,3,4,\dots ,m-2,\hfill \\ \frac{1}{2}(2n+3-i),\hfill & \text{if}x=c_1{v}_i,\text{for}i=1,3,5,\dots ,n-1,\hfill \\ \frac{1}{2}(n+2-i),\hfill & \text{if}x=c_1{v}_i,\text{for}i=2,4,6,\dots ,n,\hfill \\ \frac{1}{2}(3n+(2m-3)-i),\hfill & \text{if}x=c_j{v}_i,\text{for}i=1,3,5,\dots ,n-1,j=m-1,\hfill \\ \frac{1}{2}(4n+2(m-1)-i),\hfill & \text{if}x=c_j{v}_i,\text{for}i=2,4,6,\dots ,n,j=m-1.\hfill \end{array}\right.$$To show that $h$ is a $C_m$-$E$-supermagic labeling of $G_{n,m}$, Let $C_m^{\left(i\right)}$, for $1\le i\le n$ be the subcycle of $G_{n,m}$ with $$V\left(C_m^{\left(i\right)}\right)=\{c_i:1\le i\le m-1\}\cup \{v_i,1\le i\le n\}\text{and}$$$$E\left(G_{n,m}\right)=\{c_i{c}_{i+1}:1\le i\le m-2\}\cup \{c_1{v}_i,c_j{v}_i\}:j=\{m-1,1\le i\le n\}.$$ For each $i,1\le i\le n$, $k=m-1$, we have $$h\left(v_i\right)+\sum _{j=1}^{k}h\left(c_j\right)+\sum _{j=1}^{k-1}h\left(c_j{c}_{j+1}\right)+h\left(c_1{v}_i\right)+h\left(c_k{v}_i\right)=\left\{\begin{array}{cc}\frac{1}{2}(17n+25)+(m-3)[(2n+13)+(m-4)2],\hfill & \text{for odd n},\hfill \\ (9n+12)+(m-3)[(3n+12)+(m-4)2],\hfill & \text{for even n}.\hfill \end{array}\right.$$ So $G_{n,m}$ is a $C_m$-$E$-supermagic graph. □

To generalize the ladder $P_m\times P_2$, we can always substitute $P_2$ with any path $P_n,n\ge 3$. The resulting graph is the grid $P_m\times P_n$ with $mn$ vertices and $(m-1)n+(n-1)m$ edges.

Now we define grid graph $G\cong P_m\times P_n$ as $V\left(G\right)=\{u_{i,j}:1\le i\le m,1\le j\le n\}$ and $E\left(G\right)=\{u_{i,j}{u}_{i+1,j}:1\le i\le m-1,1\le j\le n\}\cup \{u_{i,j}{u}_{i,j+1}:1\le i\le m,1\le j\le n-1\}$. Now we prove that the graph $G\cong P_m\times P_n$ is $C_4$-$E$-supermagic.

Theorem 2.3

For any positive integer $m\ge 3$ and $n=6,7$ the grid $G\cong P_m\times P_n$ is$C_4$-$E$-supermagic.

Proof

We define a total labeling $h:V\left(G\right)\cup E\left(G\right)\to \{1,2,3,\dots ,3mn-n-m\}$.

Label the edges of $G$ in the following way: $$h\left(u_{i,j}{u}_{i+1,j}\right)=mn+1-(m+n)+i-(j-1)(m-1),\text{for}1\le i\le m-1\text{and}1\le j\le n$$For even $m$ and $1\le i\le m$, $$h\left(u_{i,j}{u}_{i,j+1}\right)=\left\{\begin{array}{cc}2mn+1-(m+n)-j-\frac{1}{2}(i-1)(n-1),\hfill & \text{for odd}i\text{and}1\le j\le n-1,\hfill \\ \frac{1}{2}(3mn+2-m-2n)-j-\frac{1}{2}(i-2)(n-1),\hfill & \text{for even}i\text{and}1\le j\le n-1.\hfill \end{array}\right.$$For odd $m$ and $1\le i\le m$, $$h\left(u_{i,j}{u}_{i,j+1}\right)=\left\{\begin{array}{cc}\frac{1}{2}(3mn+1-m-n)-j-\frac{1}{2}(i-1)(n-1),\hfill & \text{for odd}i\text{and}1\le j\le n-1,\hfill \\ (2mn+1-m+n)-j-\frac{1}{2}(i-2)(n-1),\hfill & \text{for even}i\text{and}1\le j\le n-1.\hfill \end{array}\right.$$To label the vertices of $G$, we consider two cases depending on the values of $n$.

The case of $n=6$:

For $1\le i\le m$, $$h\left(u_{i,j}\right)=\left\{\begin{array}{cc}2mn-(m+n)+i+\frac{1}{2}(j-2)m,\hfill & \text{for}j=2,4,6,\hfill \\ \frac{1}{2}[j+(4n+2)]m+\frac{3}{2}(m-4)+\frac{1}{2}[i-(m-1)],\hfill & \text{if m is even for odd}i\text{and}j=1,3,5,\hfill \\ \frac{1}{2}[j+4n]m+2(m-3)+\frac{1}{2}(i-m),\hfill & \text{if m is even for even}i\text{and}j=1,3,5.\hfill \end{array}\right.$$For $1\le i\le m$, $$h\left(u_{i,j}\right)=\left\{\begin{array}{cc}\frac{1}{2}(j+4n)m+2(m-3)+\frac{1}{2}[i-(m-1)],\hfill & \text{if m is odd for odd}i\text{and}j=1,3,5,\hfill \\ \frac{1}{2}[j+(4n+2)]m+\frac{3}{2}(m-4)+\frac{1}{2}[i-(m-1)],\hfill & \text{if m is odd for even}i\text{and}j=1,3,5.\hfill \end{array}\right.$$

Let $C_4^{(i,j)}$, $1\le i\le m-1$ and $1\le j\le n-1$ be the subcycle of $G$ with $$V\left(C_4^{(i,j)}\right)=\{u_{i,j},u_{i+1,j},u_{i,j+1},u_{i+1,j+1}\},$$ $$E\left(C_4^{(i,j)}\right)=\{u_{ij}{u}_{i+1,j},u_{i,j}{u}_{i,j+1},u_{i+1,j}{u}_{i+1,j+1},u_{i,j+1}{u}_{i+1,j+1}\}.$$

It can be checked that for each $1\le i\le m-1$ and $1\le j\le n-1$, $$\sum h\left(\underset{4}{\overset{(i,j)}{C}}\right)=\left\{\begin{array}{cc}14nm-5m-38,\hfill & \text{for even}m,\hfill \\ 14nm-5m-35,\hfill & \text{for odd}m.\hfill \end{array}\right.$$

The case of $n=7$: $$h\left(u_{i,j}\right)=\left\{\begin{array}{cc}2mn-(m+n)+i+\frac{1}{2}(j-2)m,\hfill & \text{for}1\le i\le m\text{and}j=2,4,6,\hfill \\ 2mn-(m+n)+i+\frac{1}{2}(j+5)m,\hfill & \text{for}1\le i\le m\text{and}j=1,3,5,7.\hfill \end{array}\right.$$

Let $C_4^{(i,j)}$, $1\le i\le m-1$ and $1\le j\le n-1$ be the subcycle of $G$ with $$V\left(C_4^{(i,j)}\right)=\{u_{i,j},u_{i+1,j},u_{i,j+1},u_{i+1,j+1}\},$$ $$E\left(C_4^{(i,j)}\right)=\{u_{i,j}{u}_{i+1,j},u_{i,j}{u}_{i,j+1},u_{i+1,j}{u}_{i+1,j+1},u_{i,j+1}{u}_{i+1,j+1}\}.$$It can be checked that for each $1\le i\le m-1$ and $1\le j\le n-1$, $$\sum h\left(\underset{4}{\overset{(i,j)}{C}}\right)=\left\{\begin{array}{cc}14nm-6m-45,\hfill & \text{for even}m,\hfill \\ 14nm-6m-42,\hfill & \text{for odd}m.\hfill \end{array}\right.$$

Hence $G$ is $C_4$-$E$-supermagic. □

Open Problem 2.4

Determine $C_4$-$E$-supermagic labeling of $P_m\times P_n$ for the remaining cases of $m$ and $n$.

Let us define the friendship graph as follows:

The friendship graph $F_n$ is a set of $n$ triangles having a common center vertex and otherwise disjoint.

Let $c$ denote the center vertex. For the $i$th triangle, let $x_i$ and $y_i$ denote the other two vertices.

We define the generalized friendship graph $G{F}_n^m$ with vertex and edge sets by $$V\left(G{F}_n^m\right)=\{x_{i,j}:1\le i\le n,1\le j\le m-1\}\cup \left\{c\right\}\text{and}$$$$E\left(G{F}_n^m\right)=\{x_{i,j}{x}_{i,j+1}:1\le i\le n,1\le j\le m-2\}\cup \{x_{i,j}c:1\le i\le n,j=m-2\}$$

The following result is interesting because it characterizes $C_m$-$E$-supermagicness of generalized friendship graph $G{F}_n^m$.

Theorem 2.5

For any integer $m\ge 3$, the generalized friendship graph $G{F}_n^m$ is$C_m$-$E$-supermagic.

Proof

Suppose that a bijection $h:V\left(G{F}_n^m\right)\cup E\left(G{F}_n^m\right)\to \{1,2,3,\dots ,2nm+1-n\}$ is $C_m$-$E$-supermagic total labeling.

We label the vertices of $G{F}_n^m$ in the following way:

The case of odd $n$,

For $i=1,2,3,\dots ,n$ and $j=1,2,3,\dots ,m-1$, $$h\left(u\right)=\left\{\begin{array}{cc}mn+i+\frac{1}{2}(j-1)(2n+1),\hfill & \text{if}u=x_{i,j},\text{for}j=1,3,\hfill \\ mn+n(j-1)+i+1,\hfill & \text{if}u=x_{i,j},\text{for}j=5,7,\dots ,m-1,\hfill \\ nm+2(n+1)-i+(j-2)n,\hfill & \text{if}u=x_{i,j},\text{for}j=2,4,\dots ,m-1.\hfill \\ mn+n+1,\hfill & \text{if}u=c.\hfill \end{array}\right.$$The edge labeling for $x_{i,j},x_{i,j+1}$, is given by,

For odd $m$, and for $i=1,2,3,\dots ,n$. $$h\left(u\right)=\left\{\begin{array}{cc}nj-n+i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=1,3,5,\dots ,m-2,\hfill \\ nj+1-i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=2,4,6,\dots ,m-3.\hfill \end{array}\right.$$For even $m$, and for $i=1,2,3,\dots ,n$ $$h\left(u\right)=\left\{\begin{array}{cc}nj+1-i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=1,3,5,\dots ,m-3,\hfill \\ nj-n+i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=2,4,6,\dots ,m-2.\hfill \end{array}\right.$$The edge labeling for $x_{i,j}c$, is given by, $$h\left(u\right)=\left\{\begin{array}{cc}\frac{1}{2}[2mn-3n+2i+1],\hfill & \text{if}u=x_{i,j}c,\text{for}1\le i\le \frac{n-1}{2},\text{for}j=1,\hfill \\ \frac{1}{2}[2mn-5n+2i+1],\hfill & \text{if}u=x_{i,j}c,\text{for}\frac{n+1}{2}\le i\le n,\text{for}j=1.\hfill \end{array}\right.$$The edge labeling for $x_{i,j+1}c$, is given by, $$h\left(u\right)=\left\{\begin{array}{cc}mn+1-2i,\hfill & \text{if}u=x_{i,j+1}c,\text{for}1\le i\le \frac{n-1}{2},\text{for}j=m-2,\hfill \\ mn+n+1-2i,\hfill & \text{if}u=x_{i,j+1}c,\text{for}\frac{n+1}{2}\le i\le n,\text{for}j=m-2.\hfill \end{array}\right.$$The case of even $n$,

We label the vertices of $G{F}_n^m$ in the following way: $$h\left(u\right)=\left\{\begin{array}{cc}nm+i,\hfill & \text{if}u=x_{i,j},\text{for}1\le i\le \frac{n}{2}andj=1,\hfill \\ nm+i+1,\hfill & \text{if}u=x_{i,j},\text{for}\frac{n}{2}+1\le i\le n\text{and}j=1,\hfill \\ nm+n(j-1)+i+1,\hfill & \text{if}u=x_{i,j},\text{for}j=3,5,7,\dots ,m-1\text{and}1\le i\le n,\hfill \\ nm+nj+2-i,\hfill & \text{if}u=x_{i,j},\text{for}j=2,4,6,\dots ,m-1\text{and}1\le i\le n,\hfill \\ nm+n,\hfill & \text{if}u=c,\text{for}n=2,\hfill \\ \frac{1}{2}(2nm+n+2),\hfill & \text{if}u=c,\text{for}n>2.\hfill \end{array}\right.$$The edge labeling for $x_{i,j},x_{i,j+1}$ is given by,

For odd $m$ and for $i=1,2,3,\dots ,n$, $$h\left(u\right)=\left\{\begin{array}{cc}nj-n+i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=1,3,5,7,\dots ,m-2,\hfill \\ nj+1-i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=2,4,6,\dots ,m-3.\hfill \end{array}\right.$$

For even $m$ and for $i=1,2,3,\dots ,n$ $$h\left(u\right)=\left\{\begin{array}{cc}nj+1-i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=1,3,5,7,\dots ,m-3,\hfill \\ nj-n+i,\hfill & \text{if}u=x_{i,j}{x}_{i,j+1},\text{for}j=2,4,6,\dots ,m-2.\hfill \end{array}\right.$$

The edge labeling for $x_{i,j}c$, is given by, $$h\left(u\right)=\left\{\begin{array}{cc}\frac{1}{2}[2mn-3n+2i],\hfill & \text{if}u=x_{i,j}c,\text{for}1\le i\le \frac{n}{2},\text{for}j=1,\hfill \\ \frac{1}{2}[2mn-5n+2i],\hfill & \text{if}u=x_{i,j}c,\text{for}\frac{n}{2}+1\le i\le n,\text{for}j=1.\hfill \end{array}\right.$$The edge labeling for $x_{i,j+1}c$, is given by, $$h\left(u\right)=\left\{\begin{array}{cc}nm+2-2i,\hfill & \text{if}u=x_{i,j+1}c,\text{for}1\le i\le \frac{n}{2},\text{for}j=m-2,\hfill \\ nm+(n+1)-2i,\hfill & \text{if}u=x_{i,j+1}c,\text{for}\frac{n}{2}+1\le i\le n,\text{for}j=m-2.\hfill \end{array}\right.$$For each $i,1\le i\le n,k=m-1$, we have $$\sum _{j=1}^{k}h\left(x_{i,j}\right)+h\left(c\right)+\sum _{j=1}^{k-1}h(x_{i,j},x_{i,j+1})+h\left(x_{i,k}c\right)+h\left(x_{i,1}c\right)=\left\{\begin{array}{cc}\frac{1}{2}(33n+9)+(m-3)[2mn+5n+2],\hfill & \text{for odd n},\hfill \\ (16n+5)+(m-3)[2mn+2+5n],\hfill & \text{for even n}.\hfill \end{array}\right.$$

This completes the proof. □

The generalized fan graph denoted by $F_{n,m}$ is a graph with $$V\left(F_{n,m}\right)=\{c,x_i:1\le i\le n\}\cup \{v_{i,j}:1\le i\le m-3,1\le j\le \lfloor \frac{n}{2}\rfloor \}\text{and}$$ $E\left(F_{n,m}\right)=\{x_i{x}_{i+1}:1\le i\le n-1\}\cup \{v_{i,j}{x}_k:i=1,1\le j\le \lfloor \frac{n}{2}\rfloor ,k=2j\}\cup \{v_{i+1,j}{v}_{i,j}:1\le i\le m-4,1\le j\le \lfloor \frac{n}{2}\rfloor \}\cup \{c{v}_{i,j}:i=m-3,1\le j\le \lfloor \frac{n}{2}\rfloor \}\cup \{c{x}_i:i=1,3,5,\dots ,n\}$. Note that the vertices $v_{i,j}$ are introduced between the vertices $x_k^{\prime }s(k=2j,1\le j\le \lfloor \frac{n}{2}\rfloor )$ and $c$.

Our next result shows that the generalized fan $F_{n,m}$ is $C_m$-$E$-supermagic for $n=3$ and $m\ge 3$, $m\ne 4$ (see Fig. 1).

Fig. 1 A $C_5$-$E$ supermagic labeling of $F_{3,5}$.

Theorem 2.6

For any integer $m\ge 3$,$m\ne 4$, the generalized fan $F_{3,m}$ is$C_m$-$E$-supermagic.

Proof

Define a total labeling $h:V(F_3,m)\cup E\left(F_{3,m}\right)\to \{1,2,3,\dots ,2m+3\}$ as follows:

We label the vertices of $F_{3,m}$ in the following way: $$h\left(u\right)=\left\{\begin{array}{cc}m+6,\hfill & \text{if}u=c,\hfill \\ (m+2)+\frac{1}{2}(i+1),\hfill & \text{if}u=x_i,\text{for}i=1,3,\hfill \\ m+i+3,\hfill & \text{if}u=x_i,\text{for}i=2.\hfill \end{array}\right.$$For the cases $m\ge 4$, we introduce additional vertices $v_i$ between $x_2$ and $c$.

The vertex labeling of $v_i$ and edge labeling are defined as follows: $$h\left(u\right)=\left\{\begin{array}{cc}m+6+i,\hfill & \text{if}u=v_i,\text{for}1\le i\le m-3,\hfill \\ 4,\hfill & \text{if}u=v_1{x}_2,\hfill \\ 5+i,\hfill & \text{if}u=v_i{v}_{i+1},\text{for}1\le i\le m-4,\hfill \\ 5+i,\hfill & \text{if}u=c{v}_i,\text{for}i=m-3,\hfill \\ i,\hfill & \text{if}u=x_i{x}_{i+1},\text{for}i=1,2,\hfill \\ 4,\hfill & \text{if}u=c{x}_i,\text{for}i=2,\hfill \\ 6-i,\hfill & \text{if}u=c{x}_i\text{for}i=1,3.\hfill \end{array}\right.$$For $i=1,2$, let $C_m^{\left(i\right)}$ be the subcycle of $F_{3,m}$ with $$V\left(C_m^{\left(i\right)}\right)=\{c,x_i,x_{i+1}\}\cup \{v_i:1\le i\le m-3\}\text{and}$$$$E\left(C_m^i\right)=\left\{x_i{x}_{i+1}\right\}\cup \left\{v_1{x}_2\right\}\cup \{v_k{v}_{k+1}:1\le k\le m-4\}\cup \{c{v}_r:r=m-3\}\cup \{c{x}_j:j=2i-1\}.$$ It can be checked that for $i=1,2$, $r=m-3$, and $j=2i-1$, $$\sum \left(\underset{m}{\overset{\left(i\right)}{C}}\right)=h\left(c\right)+h\left(x_i\right)+h\left(x_{i+1}\right)+\sum _{k=1}^{r}h\left(v_k\right)+h\left(x_i{x}_{i+1}\right)+h\left(c{x}_j\right)+\sum _{k=1}^{r-1}h\left(v_k{v}_{k+1}\right)$$$$+h\left(c{v}_r\right)+h\left(v_1{x}_2\right)$$$$=33+(m-3)(12+2m).$$ Hence $F_{3,m}$ is a $C_m$-$E$-supermagic. □

Open Problem 2.7

Determine $C_m$-$E$-supermagic labeling of generalized fan graphs $F_{n,m}$ for the remaining cases of $m$ and $n$.