Book graphs are cycle antimagic

Author(s): Muhammad Awais Umar1, Noshad Ali2, Afshan Tabassum2, Basharat Rehman Ali3
1Government Degree College (B), Sharaqpur Shareef, Pakistan
2Department of Mathematics, NCBA\&E, DHA Campus, Lahore, Pakistan
3Abdus Salam School of Mathematical Sciences, GC University, Lahore.
Copyright © Muhammad Awais Umar, Noshad Ali, Afshan Tabassum, Basharat Rehman Ali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let \(G=(V,E)\) be a finite simple graph with \(v =|V(G)|\) vertices and \(e=|E(G)|\) edges. Further suppose that \(\mathbb{H}:=\{H_1, H_2, \dots, H_t\}\) is a family of subgraphs of \(G\). In case, each edge of \(E(G)\) belongs to at least one of the subgraphs \(H_i\) from the family \(\mathbb{H}\), we say \(G\) admits an edge-covering. When every subgraph \(H_i\) in \(\mathbb{H}\) is isomorphic to a~given graph \(H\), then the graph \(G\) admits an \(H\)-covering. A graph \(G\) admitting \(H\) covering is called an \((a,d)-H\)-antimagic if there is a bijection \(\eta:V\cup E \to \{1,2,\dots, v+e \}\) such that for each subgraph \(H’\) of \(G\) isomorphic to \(H\), the sum of labels of all the edges and vertices belongs to \(H’\) constitutes an arithmetic progression with the initial term \(a\) and the common difference \(d\). For \(\eta(V)= \{ 1,2,3,\dots,v\}\), the graph \(G\) is said to be super \((a,d)-H\)-antimagic and for \(d=0\) it is called \(H\)-supermagic. When the given graph \(H\) is a cycle \(C_m\) then \(H\)-covering is called \(C_m\)-covering and super \((a,d)-H\)-antimagic labeling becomes super \((a,d)-C_m\)-antimagic labeling. In this paper, we investigate the existence of super \((a,d)-C_m\)-antimagic labeling of book graphs \(B_n\), for \(m=4,\ n\geq2\) and for differences \(d=1, 2, 3, \dots,13\).

Keywords: Book graph \(B_n\), super \((a,d)-C_4\)-antimagic.

1. Introduction

Let \(G\) be a finite and simple graph. A family of subgraphs \(H_1, H_2, \dots, H_t\) is defined as an edge-covering of \(G\) such that each edge of \(E(G)\) belongs to at least one of the subgraphs \(H_i\), \(i=1, 2, \dots, t\). Then \(G\) admits an \((H_1, H_2, \dots, H_t)\)-(edge) covering. If every subgraph \(H_i\) is isomorphic to a given graph \(H\), then the graph \(G\) admits an \(H\)-covering. A graph \(G\) admitting an \(H\)-covering is called \((a,d)\)-\(H\)-antimagic if there exists a total labeling \(\eta:V(G)\cup E(G) \to \{1,2,\dots, v+e\}\) such that for each subgraph \(H’\) of \(G\) isomorphic to \(H\), the \(H’\)-weights, $$wt_{\eta}(H’)= \sum\limits_{v\in V(H’)} \eta(v) + \sum\limits_{e\in E(H’)} \eta(e),$$ constitute an arithmetic progression \(a, a+d, a+2d,\dots , a+(t -1)d\), where \(a>0\) and \(d\ge 0\) are two integers and \(t\) is the number of all subgraphs of \(G\) isomorphic to \(H\). Moreover, \(G\) is said to be super \((a,d)\)-\(H\)-antimagic, if the smallest possible labels appear on the vertices. If \(G\) is a super \((a,d)\)-\(H\)-antimagic graph then the corresponding total labeling \(\eta\) is called the super \((a,d)\)-\(H\)-antimagic labeling. For \(d=0\), the super \((a,d)\)-\(H\)-antimagic graph is called \(H\)-supermagic.

The \(H\)-supermagic graph was first introduced by Guti’errez et al. in [1]. They proved that the star \(K_{1,n}\) and the complete bipartite graphs \(K_{n,m}\) are \(K_{1,h}\)-supermagic for some \(h\). They also proved that the path \(P_n\) and the cycle \(C_n\) are \(P_h\)-supermagic for some \(h\). Llad’o et al. [2] investigated \(C_n\)-supermagic graphs and proved that wheels, windmills, books and prisms are \(C_h\)-magic for some \(h\). Some results on \(C_n\)-supermagic labelings of several classes of graphs can be found in [3]. Maryati et al. [4] gave \(P_h\)-supermagic labelings of shrubs, subdivision of shrubs and banana tree graphs. Other examples of \(H\)-supermagic graphs with different choices of \(H\) have been given by Jeyanthi et al. in [5]. Maryati et al. [6] investigated the \(G\)-supermagicness of a disjoint union of \(c\) copies of a graph \(G\) and showed that disjoint union of any paths is \(cP_h\)-supermagic for some \(c\) and \(h\).

The \((a,d)\)-\(H\)-antimagic labeling was introduced by Inayah et al. [7]. In [8] Inayah et al. investigated the super \((a,d)\)-\(H\)-antimagic labelings for some shackles of a connected graph \(H\).

For \(H\cong K_2\), super \((a,d)\)-\(H\)-antimagic labelings are also called super \((a,d)\)-edge-anti-magic total labelings. For further information on super edge-magic labelings, one can see [9, 10, 11, 12].

The super \((a,d)\)-\(H\)-antimagic labeling is related to a super \(d\)-antimagic labeling of type \((1,1,0)\) of a plane graph which is the generalization of a face-magic labeling introduced by Lih [13]. Further information on super \(d\)-antimagic labelings can be found in [14, 15, 16].

In [17], Awais et al. proved the existence of \((a,d)\)-\(C_4\)-antimagic labeling of book graphs \(B_n\) (for difference \(d=0,1\)) and of its disjoint union. In this paper, we study the existence of super \((a,d)\)-\(C_4\)-antimagic labeling of book graphs \(B_n\) for differences \(d=1, 2, 3, \dots,13\) and \(n\geq2\).

2. Super Cycle Antimagic Labeling

In this section, we discussed super \((a,d)\)-\(C_4\)-antimagicness of book graphs for difference \(d=1, 2, 3, \dots,13\).
Let \(K_{1,n}\), \(n\geq2\) be a complete bipartite graph on \(n+1\) vertices. The book graph \(B_n\) is a cartesian product of \(K_{1,n}\) with \(K_2\).\ i.e., \(B_n \cong K_{1,n} \Box K_2\). Clearly book graph \(B_n\) admits \(C_4\)-covering. The book graph \(B_n\) has the vertex set and edge set as $$V(B_n)=\{y_1,y_2\}\cup \cup_{i=1}^{n} \{x_{(1,i)}, x_{(2,i)}\}$$ $$E(B_n)=\cup_{i=1}^{n}\{y_1x_{(1,i)},y_2x_{(2,i)},x_{(1,i)}x_{(2,i)}\}\cup\{y_1y_2\}$$ respectively. It can be noted that \(|V(B_n)|=2(n+1)\) and \(|E(B_n)|= 3n+1\).
Every \(C_4^{(j)}, 1\leq j\leq n \) in \(B_n\) has the vertex set:
\(V(C_4^{(j)})=\{y_1, y_2, x_{(1,j)},x_{(2,j)}\}\)
and the edge set:
\(E(C_4^{(j)})=\{y_1y_2,y_1x_{(1,j)},y_2x_{(2,j)}, x_{(1,j)}x_{(2,j)}\}.\)
Under a total labeling \(\xi\), the \(C_4^{(j)}\)-weights, \(j=1,\dots, n\), would be:
\begin{align}\label{partial_sum1} \nonumber wt_\xi(C_4^{(j)}) &= \sum\limits_{v\in V(C_4^{(j)})} \xi(v) + \sum\limits_{e\in E(C_4^{(j)})} \xi(e).\\ &=\sum_{k=1}^{2}\left(\xi(y_k)+\xi(x_{(k,j)})+ \xi(y_kx_{(k,j)})\right)+ \xi(y_1y_2)+\xi(x_{(1,j)}x_{(2,j)}) \end{align}
(1)

Theorem 1. For any integer \(n\geq 2\), the book graph \(B_n\) admits super \((a,d)\)-\(C_4\)-antimagic labeling for differences \(d=1, 3, \dots, 13\).

Proof. Under a labeling \(\xi\), the set \(\{y_1,y_2, y_1y_2\}\), would be labeled as: \begin{align*} \nonumber \xi(y_k) &= k,\;\; k= 1, 2 \\ \xi(y_1y_2) &= 2(n+1)+1 \end{align*} and therefore the partial sum of \(wt_\xi(C_4^{(j)})\) would be

\begin{equation}\label{partial_sum2} \xi (y_1)+\xi (y_2) +\xi(y_1y_2)= 2(n+3). \end{equation}
(2)
For \(d=1, 3, \dots, 9, 13\) \[\xi_{d}(x_{(k,j)})=\begin{cases} 2j+1,\; \; \; & k=1\\ 2(j+1),\; \; \; & k=2\\ \end{cases}\] \[\xi_{11}(x_{(k,j)})=\begin{cases} 2+j,\; \; \; & k=1\\ n+2+j,\; \; \; & k=2\\ \end{cases}\] \[\xi_{d}(x_{(1,j)}x_{(2,j)})=\begin{cases} 3n+4-j,\; \; \; & d=1\\ 2n+3+j,\; \; \; & d=3,5,7,9\\ 2n+1+3j,\; \; \; & d=11,13\\ \end{cases}\] \[\xi_{d}(y_kx_{(k,j)})=\begin{cases} (k+3)n+4-j,\; \; \; &k=1,2 \; \; \; d=1,3\\ 5n+4-j,\; \; \; & k=1 \; \; \; d=5\\ 3n+3+j,\; \; \; & k=2 \; \; \; d=5\\ (k+2)n+3+j,\; \; \; & k=1, 2 \; \; \; d=7\\ 3n+2j+k+1,\; \; \; & k=1, 2 \; \; \; d=9\\ 3(n+j)-k,\; \; \; & k=1, 2 \; \; \; d=11\\ 3(n+j)+k-1,\; \; \; & k=1, 2 \; \; \;d=13\\ \end{cases}\] where indices \(j\) are taken modulo \(n\).
Clearly \(\xi(V(B_n))=\{1,2, \dots, 2(n+1)\}\).
Therefore \(\xi\) is a super labeling together with \(\xi(E(B_n))=\{2(n+1)+1, 2(n+1)+2,\dots, 5n+3\}\) which shows \(\xi\) is a total labeling.
Using (1) and (2), \(wt_{\xi_{d}}(C_4^{(j)})\) are: \[wt_{\xi_{d}}(C_4^{(j)})=\begin{cases} 14n+21+j,\; \; \; & d=1\\ 13n+20+3j,\; \; \; & d=3\\ 12n+19+5j,\; \; \; & d=5\\ 11n+18+7j,\; \; \; & d=7\\ 10n+17+9j,\; \; \; & d=9\\ 11n+8+11j,\; \; \; & d=11\\ 10n+11+13j,\; \; \; & d=13\\ \end{cases}\] Clearly \(wt_{\xi_{d}}(C_4^{(j)})\) constitutes arithmetic progression and therefore book graphs are super \((a,d)\)-\(C_4\)-antimagic for \(d=1,3, \dots,13\). This completes the proof.

Theorem 2. For any integer \(n\geq 2\), the book graph \(B_n\) admits super \((a,d)\)-\(C_4\)-antimagic labeling for differences \(d=2, 4, \dots, 10\).

Proof. Case \(n \equiv 0\) (mod \(2\))
For \(d=2,4,6,8\) the labeling \(\xi\) for the set \(\{y_1,y_2, y_1y_2\}\), would be labeled as: \begin{align*} \xi_{d}(y_1)&=1\\ \xi_{d}(y_2)&=\frac{n}{2}+2 \end{align*} \begin{equation*} \xi_{d}(y_1y_2)= 2n+3 \end{equation*} and therefore the partial sum of \(wt_\xi(C_4^{(j)})\) would be

\begin{equation}\label{2to8} \xi_{d} (y_1)+\xi_{d} (y_2) +\xi_{d}(y_1y_2)=\frac{5n}{2}+6 \end{equation}
(3)
The remaining set of elements has the labeling \(\xi\) as:
\[\xi_{d}(x_{(k,j)})=\begin{cases} 1+j,\; \; \; & k=1,\ \ j=1,2, \dots, \frac{n}{2}\\ 2j-\frac{n}{2}+1, \; \; \; & k=1,\ j=\frac{n}{2}+1, \dots, n \\ \frac{n}{2}+2(1+j), \; \; \; & k=2,\ \ j=1,2, \dots, \frac{n}{2}\\ n+2+j,\; \; \; & k=2, j=\frac{n}{2}+1, \dots, n \\ \end{cases}\] \[\xi_{d}(x_{(1,j)}x_{(2,j)})=\begin{cases} 2(n+1)+1+j,\; \; \; & d=2 \\ 5n+4-j,\; \; \; & d=4 \\ 4n+3+j,\; \; \; & d=6, 8 \end{cases}\] \[\xi_{d}(y_kx_{(k,j)})=\begin{cases} n(k+3)+4-j,\; \; \; & d=2\\ n(k+1)+3+j,\; \; \; & d=4, 6 \\ 2(n+j)+k+1,\; \; \; & d=8\\ \end{cases}\] For difference \(d=10\) the labeling \(\xi\) is defined as: \begin{align*} \xi_{d}(y_1)&=1\\ \xi_{d}(y_2)&=\frac{3n}{2}+2 \end{align*} \begin{equation*} \xi_{d}(y_1y_2)= 2n+3 \end{equation*} and the partial sum of \(wt_\xi(C_4^{(j)})\) would be:
\begin{equation}\label{even10} \xi_{d} (y_1)+\xi_{d} (y_2) +\xi_{d}(y_1y_2)=\frac{7n}{2}+6 \end{equation}
(4)
\[\xi_{d}(x_{(k,j)})=\begin{cases} 2j+1,\; \;\; & k=1,\; \; j=1,2, \dots, \frac{n}{2} \\ 2j-n,\; \; \; & k=1,\; j=\frac{n}{2}+1, \dots, n \\ \frac{3n}{2}+2-j, & k=2,\; \; j=1,2, \dots, \frac{n}{2}\\ \frac{5n}{2}+3-j, & k=2,\; j=\frac{n}{2}+1, \dots, n \\ \end{cases}\] \begin{align*} \xi_{d}(x_{(1,j)}x_{(2,j)})&=2n+1+3j\\ \xi_{d}(y_kx_{(k,j)})&=2n+(k+1)+3j,\; \; \; k=1, 2 \\ \end{align*} Clearly \(\xi(V(B_n))=\{1,2, \dots, 2(n+1)\}\). Therefore \(\xi\) is a super labeling and together with \(\xi(E(B_n))=\{2(n+1)+1, 2(n+1)+2,\dots, 5n+3\}\) which shows \(\xi\) is a total labeling.
Using (1), (3) and (4), \(wt_\xi(C_4^{(j)})\) are: \[wt_{\xi_{d}}(C_4^{(j)})=\begin{cases} 14n+20+2j,\; \; \; & d=2\\ 13n+19+4j,\; \; \; & d=4\\ 12n+18+6j,\; \; \; & d=6\\ 11n+17+8j,\; \; \; & d=8\\ 11n+15+10j,\; \; \; & d=10\\ \end{cases}\] Therefore \(wt_{\xi_{d}}(C_4^{(j)})\) constitutes arithmetic progression for differences \(d=2,4, \dots,10\) when \(n\equiv 0\) (mod \(2\)).
Case \(n \equiv 1\) (mod \(2\)
For the set \(\{y_1,y_2, y_1y_2\}\), labeling \(\xi\) would be: \begin{align*} \xi_{d}(y_1)&=1\\ \xi_{d}(y_2)&=n+2 \end{align*} \begin{equation*} \xi_{d}(y_1y_2)= 2n+3 \end{equation*} and therefore the partial sum of \(wt_\xi(C_4^{(j)})\) would be
\begin{equation}\label{oddpartialsum} \xi_{d} (y_1)+\xi_{d} (y_2) +\xi_{d}(y_1y_2)= 3(n+2) \end{equation}
(5)
For differences \(d=2,4,6,10\) \[\xi_{d}(x_{(k,j)})=\begin{cases} 2j,\; \; \; & k=1,\ \ j=1,2, \dots, \frac{n+1}{2}\\ 2j-n,\; \; \; & k=1,\ j=\frac{n+1}{2}+1, \dots, n \\ 3\left(\frac{n+1}{2}\right)+2-j,\; \; \; & k=2,\; \; j=1,2, \dots, \frac{n+1}{2}\\ 5\left(\frac{n+1}{2}\right)+1-j,\; \; \; & k=2,\; j=\frac{n+1}{2}+1, \dots, n \\ \end{cases}\] and for differences \(d=8\) \[\xi_{d}(x_{(k,j)})=\begin{cases} n+2-2j,\; \; \; & k=1,\; \; j=1,2, \dots, \frac{n-1}{2}\\ 2(n+1)-2j,\; \; \; & k=1,\; j=\frac{n+1}{2}, \dots, n \\ 3\left(\frac{n+1}{2}\right)+1+j,\; \; \; & k=2,\ \ j=1,2, \dots, \frac{n-1}{2}\\ \frac{n+1}{2}+2+j,\; \; \; & k=2,\ j=\frac{n+1}{2}, \dots, n\\ \end{cases}\] For differences \(d=2, 4, \dots, 10\), the set of edges has the labeling \(\xi\) defined as: \[\xi_{d}(x_{(1,j)}x_{(2,j)})=\begin{cases} 5n+4-j,\; \; \; & d=2\\ 4n+3+j,\; \; \; & d= 4,6\\ 2n+3+3j,\; \; \; & d=8, 10 \end{cases}\] \[\xi_{d}(y_kx_{(k,j)})=\begin{cases} n(k+1)+3+j,\; \; \; & k=1,2,\; \; d=2, 4\\ 2(n+j)+k+1,\; \; \; & k=1,2,\; \; d=6\\ 2n+k+3j,\; \; \; & d=8, 10\\ \end{cases}\] Clearly \(\xi(V(B_n))=\{1,2, \dots, 2(n+1)\}\). Therefore \(\xi\) is a super labeling together with \(\xi(E(B_n))=\{2(n+1)+1, 2(n+1)+2,\dots, 5n+3\}\) which shows \(\xi\) is a total labeling. Using (1) and (5), \(wt_\xi(C_4^{(j)})\) are: \[wt_{\xi_{d}}(C_4^{(j)})=\begin{cases} \frac{27n+33}{2}+2j,\; \; \; & d=2\\ \frac{25n+31}{2}+4j,\; \; \; & d=4\\ \frac{23n+29}{2}+6j,\; \; \; & d=6\\ \frac{21n+27}{2}+8j,\; \; \; & d=8\\ \frac{19n+25}{2}+4j,\; \; \; & d=10\\ \end{cases}\] Therefore \(wt_{\xi_{d}}(C_4^{(j)})\) constitute arithmetic progression for differences \(d=2,4, \dots, 10\) when \(n \equiv 1\) (mod \(2\)).
Hence book graphs are super \((a,d)\)-\(C_4\)-antimagic for \(d=2,4, \dots,10\). This completes the proof.

Theorem 3. For any integer \(n\geq 2\), the book graph \(B_n\) admits super \((a,12)\)-\(C_4\)-antimagic labeling.

Proof. Case \(n \equiv 0\) (mod \(2\))
Under a labeling \(\xi\), the set \(\{y_1,y_2, y_1y_2\}\), would be labeled as: \begin{align*} \xi_{12}(y_1)&=1\\ \xi_{12}(y_2)&=\frac{n+4}{2} \end{align*} \begin{equation*} \xi_{12}(y_1y_2)= 2n+3 \end{equation*} and therefore the partial sum of \(wt_\xi(C_4^{(j)})\) would be

\begin{equation}\label{even12sum} \xi_{12} (y_1)+\xi_{d} (y_2) +\xi_{d}(y_1y_2)=\frac{5n+12}{2} \end{equation}
(6)
\[\xi_{12}(x_{(k,j)})=\begin{cases} 1+j\; \; \; & k=1,\ \ j=1,2, \dots, \frac{n}{2} \\ 2j+1-\frac{n}{2}\; \; \; & k=1,\ \ j= \frac{n}{2}+1, \dots, n \\ \frac{n}{2}+2(1+j),\; \; \; & k=2,\ \ j=1,2, \dots, \frac{n}{2}\\ n+2+j,\; \; \; & k=2,\ \ j= \frac{n}{2}+1, \dots, n\\ \end{cases}\] \begin{align*} \xi_{12}(y_kx_{(k,j)})&=2(n+k)+3j-1\; \; \; k=1,2\\ \xi_{12}(x_{(1,j)}x_{(2,j)})&=2(n+1)+3j\; \; \; \end{align*} where indices \(j\) are taken modulo \(n\).
Case \(n \equiv 1\) (mod \(2\))
Under a labeling \(\xi\), the set \(\{y_1,y_2, y_1y_2\}\), would be labeled as: \begin{equation*} \xi_{12}(y_k)= \frac{3}{2}(n-1)+2k \end{equation*} \begin{equation*} \xi_{12}(y_1y_2)= 2n+3 \end{equation*} and therefore the partial sum of \(wt_\xi(C_4^{(j)})\) would be
\begin{equation}\label{odd12sum} \xi_{12} (y_1)+\xi_{d} (y_2) +\xi_{d}(y_1y_2)=5n+6 \end{equation}
(7)
\[\xi_{12}(x_{(k,j)})=\begin{cases} j\; \; \; & k=1,\ \ j=1,2, \dots, \frac{n+1}{2} \\ 2j-\frac{n+3}{2}\; \; \; & k=1,\; \; j=\frac{n+1}{2}+1, \dots, n \\ \frac{n+1}{2}+2j,\; \; \; & k=2,\; \; j= \frac{n}{2}+1, \dots, n \\ n+2+j,\; \; \; & k=2,\ \ j= \frac{n}{2}+1, \dots, n\\ \end{cases}\] \begin{align*} \xi_{12}(y_kx_{(k,j)})&=2n+k+3j\; \; \; k=1,2\\ \xi_{12}(x_{(1,j)}x_{(2,j)})&=2n+3(1+j)\;\;\; \end{align*} where indices \(j\) are taken modulo \(n\). Clearly \(\xi(V(B_n))=\{1,2, \dots, 2(n+1)\}\). Therefore \(\xi\) is a super labeling together with \(\xi(E(B_n))=\{2(n+1)+1, 2(n+1)+2,\dots, 5n+3\}\) which shows \(\xi\) is a total labeling. Using (1), (6) and (7), \(wt_\xi(C_4^{(j)})\) are: \[wt_{\xi_{12}}(C_4^{(j)})=\begin{cases} 3(3n+5)+12j\; \; \; & n\equiv 0 (mod 2) \\ \frac{23n+25}{2}+12j\; \; \; & n\equiv 1 (mod 2) \end{cases}\] Hence book graphs are super \((a,12)\)-\(C_4\)-antimagic. This completes the proof.

Acknowledgment

The author wishes to express his profound gratitude to the reviewers for their useful comments on the manuscript.

Author Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Competing Interests

The author(s) do not have any competing interests in the manuscript.

References:

  1. Gutiérrez, A., & Lladó, A. (2005). Magic coverings. Journal of Combinatorial Mathematics and Combinatorial Computing, 55(2005), 43-56. [Google Scholor]
  2. Lladó, A., & Moragas, J. (2007). Cycle-magic graphs. Discrete Mathematics, 307(23), 2925-2933.[Google Scholor]
  3. Ngurah, A. A. G., Salman, A. N. M., & Susilowati, L. (2010). H-supermagic labelings of graphs. Discrete Mathematics, 310(8), 1293-1300.[Google Scholor]
  4. Maryati, T. K., Baskoro, E. T., & Salman, A. N. M. (2008). \(P_h\)-(super)magic labelings of some trees. Journal of Combinatorial Mathematics and Combinatorial Computing, 65, 198-204. [Google Scholor]
  5. Jeyanthi, P., & Selvagopal, P. (2010). More classes of H-supermagic graphs. International Jornal of Algorithms, Computing and Mathematics, 3(1), 93-108.[Google Scholor]
  6. Maryati, T. K., Salman, A. N. M., & Baskoro, E. T. (2013). Supermagic coverings of the disjoint union of graphs and amalgamations. Discrete Mathematics, 313(4), 397-405.[Google Scholor]
  7. Inayah, N., Salman, A. N. M., & Simanjuntak, R. (2009). On \((a, d)\)-\(H\)-antimagic coverings of graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 71, 273-281.[Google Scholor]
  8. Inayah, N., Simanjuntak, R., Salman, A. N. M., & Syuhada, K. I. A. (2013). Super \((a, d)\)-\(H\)-antimagic total labelings for shackles of a connected graph \(H\). Australasian Journal of Combinatorics, 57, 127-138. [Google Scholor]
  9. Wallis, W. D. (2001). Magic graphs. Birkhauser, New York.[Google Scholor]
  10. Bača, M., Lin, Y. Q., Muntaner-Batle, F. A., & Rius-Font, M. (2009). Strong labelings of linear forests. Acta Mathematica Sinica, English Series, 25(12), 1951-1964. [Google Scholor]
  11. Bača, M., & Miller, M. (2008). Super edge-antimagic graphs: A wealth of problems and some solutions. Universal-Publishers. [Google Scholor]
  12. Figueroa-Centeno, R. M., Ichishima, R., & Muntaner-Batle, F. A. (2001). The place of super edge-magic labelings among other classes of labelings. Discrete Mathematics, 231(1-3), 153-168. [Google Scholor]
  13. Lih, K. W. (1983). On magic and consecutive labelings of plane graphs. Utilitas Math, 24, 165-197. [Google Scholor]
  14. Bača, M., Miller, M., Phanalasy, O., & Semaničová-Feňovčíková, A. (2010). Super d-antimagic labelings of disconnected plane graphs. Acta Mathematica Sinica, English Series, 26(12), 2283-2294. [Google Scholor]
  15. Bača, M., Brankovic, L., & Semaničová-Feňovčíková, A. (2011). Labelings of plane graphs containing Hamilton path. Acta Mathematica Sinica, English Series, 27(4), 701-714.[Google Scholor]
  16. Bača, M., Numan, M., & Shabbir, A. (2013). Labelings of type (1, 1, 1) for toroidal fullerenes. Turkish Journal of Mathematics, 37(6), 899-907. [Google Scholor]
  17. Umar, M. A., Javed, M. A., Hussain, M., & Ali, B. R. (2018).Super \((a,d)\)-\(C_4\)-antimagicness of book graphs. Open Journla of Mathematical Sciences, 2, 115-121. [Google Scholor]