Book graphs are cycle 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:

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

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

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: \[\xi_{d}(x_{(k,j)})=\begin{cases} 2j+1,\; \;\; & k=1,\; \; j=1,2, \dots, \frac{n}{2} \\ 2j-n,\; \; \; & k=1,\; j=\frac{n}{2}+1, \dots, n \\ \frac{3n}{2}+2-j, & k=2,\; \; j=1,2, \dots, \frac{n}{2}\\ \frac{5n}{2}+3-j, & k=2,\; j=\frac{n}{2}+1, \dots, n \\ \end{cases}\] \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 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

\[\xi_{12}(x_{(k,j)})=\begin{cases} 1+j\; \; \; & k=1,\ \ j=1,2, \dots, \frac{n}{2} \\ 2j+1-\frac{n}{2}\; \; \; & k=1,\ \ j= \frac{n}{2}+1, \dots, n \\ \frac{n}{2}+2(1+j),\; \; \; & k=2,\ \ j=1,2, \dots, \frac{n}{2}\\ n+2+j,\; \; \; & k=2,\ \ j= \frac{n}{2}+1, \dots, n\\ \end{cases}\] \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 \[\xi_{12}(x_{(k,j)})=\begin{cases} j\; \; \; & k=1,\ \ j=1,2, \dots, \frac{n+1}{2} \\ 2j-\frac{n+3}{2}\; \; \; & k=1,\; \; j=\frac{n+1}{2}+1, \dots, n \\ \frac{n+1}{2}+2j,\; \; \; & k=2,\; \; j= \frac{n}{2}+1, \dots, n \\ n+2+j,\; \; \; & k=2,\ \ j= \frac{n}{2}+1, \dots, n\\ \end{cases}\] \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.