Antimagicness of subdivided fans

1. Introduction

Let \(\Gamma=(V(\Gamma),E(\Gamma))\) be a~finite simple and undirected graph with a~family of subgraphs \(\Omega_1, \Omega_2, \dots, \Omega_t\) such that every element of \(E(\Gamma)\) belongs to \(\Omega_i\cong \Omega,\ i=1, 2, \dots, t\), then \(\Gamma\) admits an~{\it \(\Omega\)-covering}. An \(\Omega\)-covered graph \(\Gamma\) with \(\nu\) is called an \((\alpha, \delta)\)-\(\Omega\)-antimagic if \(wt_\nu(\Omega)=\{\alpha,\alpha+\delta,\dots,\alpha+(t-1)\delta\}\) where the associated {\it \(\Omega\)-weights} denoted by \(wt_\nu(\Omega)\) are defined as $$wt_\nu(\Omega) = \sum\limits_{v\in V(\Omega)} \nu(v) + \sum\limits_{e\in E(\Omega)} \nu(e).$$ and \(\alpha>0\) and \(\delta\ge 0\) are two integers, \(t\) is the number of \(\Omega_i\cong \Omega\). For a total labeling \(\nu\) to be super we require \(\nu(V(\Gamma))= \{ 1,2,\dots,|V(\Gamma)|\}\).

The results about \(\Omega\)-(super)magic graphs with \(\Omega\) as cycle, path and tree can be studied in [1, 2, 3, 4, 5, 6, 7].

Inayah et al. [8] introduced the \((\alpha,\delta)\)-\(\Omega\)-antimagic labeling. We refer [9, 10, 11] for some results on super \((\alpha,\delta)\)-\(\Omega\)-antimagic labeling. In [11], Lih proved that \(F_n\) is \(C_3\)-supermagic for every \(n\) except \(n \equiv 2\) (mod 4). In [12], Ngurah et al. proved that \(F_n\) is \(C_3\)-supermagic for every \(n\ge 2\). In the present paper, we proved the super \((\alpha,\delta)\)-\(C_{r+2k+3}\)-antimagic labelings of subdivided fans for differences \(\delta=0, 1, 2, 3, 4\).

2. Preliminaries

In this section, we give basic definitions of concepts concerning a subdivided fan \(F_n(r,k)\).

Definition 1. A graph \(F_n\cong P_n+K_1\) is called {\it fan} graph obtained by the join of {\it path} \(P_n\) and one isolated vertex \(K_1\).

The {\it central vertex}, or the {\it hub vertex} is of degree \(n\) and {\it path vertices} are the other ones. {\it Spokes} are the adjacent edges of central vertex and {\it path edges} are the remaining edges. \begin{eqnarray*}V(F_n)&=&\{c\}\cup \{x_1, x_2, \dots, x_{n}\},\\ E(F_n)&=& \{x_1x_2,x_2x_3, \dots,x_{n-1}x_n\}\cup\{ cx_1,cx_2,\dots,cx_{n}\}. \end{eqnarray*}

Definition 2. The subdivided fan \(F_n(r,k)\) is the graph obtained from a fan \(F_n\) by inserting \(r \geq 1\) new vertices \(\{v_1^{(i)}, \dots, v_r^{(i)}\}\) into each path edge \(x_ix_{i+1}, 1\leq i\leq n-1\), denoted by \(P_{x_ix_{i+1}}\)-vertices and by inserting \(k \geq 1\) new vertices \(\{w_1^{(i)}, \dots, w_k^{(i)}\}\) into every spoke \(cx_i, 1 \leq i \leq n\), denoted by \(S^{(i)}\)-vertices.

\begin{eqnarray*} E(P_{x_ix_{i+1}})&=&\{x_iv_1^{(i)},v_2^{(i)}v_3^{(i)}, \dots, v_{r-1}^{(i)}v_{r}^{(i)}, v_r^{(i)}x_{i+1}, 1 \leq i \leq n-1\},\\ E(S^{(i)})&=&\{cw_1^{(i)},w_2^{(i)}w_3^{(i)}, \dots, w_{k-1}^{(i)}w_{k}^{(i)}, w_k^{(i)}x_{i}, 1 \leq i \leq n\}.\end{eqnarray*} Let \(C_{r+2k+3}^{(i)}\) be the \(i^{\text{th}}\)-subcycle. For the weight of \(i^{\text{th}}\)-subcycle \(C_{r+2k+3}^{(i)}\), we obtain where indices \(i\) are taken modulo \(n\).

3. Main results

In this section, we introduce the super \((\alpha,\delta)\)-\(C_{r+2k+3}\)-antimagic labelings of subdivided fans for differences \(d=0, 1, 2, 3, 4\).

Theorem 1. Let \(r, k \geq 1\) and \(n\ge3\) be positive integers. The subdivided fan \(F_n(r,k)\) is super \((\alpha, \delta)\)-\(C_{r+2k+3}\)-antimagic for difference \(\delta=0, 1, 4\).

Proof. The total labeling \(\psi_{\delta}\) is defined as: $$\psi_{\{\delta\}}(c)= 1$$ \[\psi_{\{0,4\}}(x_{i})=\begin{cases} \lceil\frac{n}{2}\rceil+2-\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ n+2-\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2) \end{cases}\] \[\psi_{\{1\}}(x_{i})=\begin{cases} 1+\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 1+\lceil\frac{n}{2}\rceil+\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2) \end{cases}\] \[\psi_{\{0,1\}}(cw_1^{(i)})=\begin{cases} 2(n-1)(r+1)+2(nk+1)+\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 2(n-1)(r+1)+2(nk+1)+\lceil\frac{n}{2}\rceil+\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2) \end{cases}\] $$\psi_{\{4\}}(cw_1^{(i)})=2n(r+k)+(3n-2r+1)-i.$$ For \(\delta=0,1,4\) \begin{eqnarray*}\psi_{\delta}(V(P_{x_ix_{i+1}}))&=&\{(n-1)j+2+i:1 \leq i \leq n-1, 1 \leq j \leq r\} \\ \psi_{\delta}(E(P_{x_ix_{i+1}}))&=&\{(n-1)(2r+2-j)+n(k+1)+2-i:1 \leq j \leq r+1\}\\ \psi_{\delta}(V(S^{(i)}))&=&\{ r(n-1)+1+nj+i:1 \leq i \leq n, 1 \leq j \leq k \}\\ \psi_{\delta}(E(S^{(i)}) \setminus \{cw_1^{(i)}\})&=&\{2n(r+k)+(3n-2r+1)-nj-i:1 \leq j \leq k\}\end{eqnarray*} where indices \(i\) are taken modulo \(n\). Evidently \(\psi_{\delta}\) is a super labeling as \(V(F_n(rk)))=\{1, 2, \dots, n(k+r+1)-r+1\}\). The spoke vertices are labeled with the numbers \(n+2,n+3 , \dots, n+2k+1\) and the path edge vertices are labeled with \(n+2k+2, n+2k+3, \dots, n(k+r+1)-r+1\). Clearly,

According to (1) and (2), we obtain Equation (3) shows that all \(C_{r+2k+3}^{(i)}\)-weights are independent of \(i\). According to (1) and (2), we obtain Equation (4) shows that all \(C_{r+2k+3}^{(i)}\)-weight consists of consecutive integers. According to (1) and (2), we obtain Equation (5) shows that all \(C_{r+2k+3}^{(i)}\)-weight constitute an arithmetic progression with common difference \(\delta=4\). This completes the proof.

Theorem 2. Let \(r, k \geq 1\) and \(n\ge3\) be positive integers. The subdivided fan \(F_n(r,k)\) is super \((\alpha, \delta)\)-\(C_{r+2k+3}\)-antimagic for difference \(\delta=2,3,5\).

Proof. The total labeling \(\psi_{\delta}\) is defined as: \begin{align*} \psi_{\{\delta\}}(c)&= 1\\ \psi_{\{\delta\}}(x_i)&= 2i\\ \psi_{\{\delta\}}(v_r)&= 2i+1 \end{align*} $$\psi_{\{2\}}(cw_1^{(i)})=2n(r+k)+(3n-2r+1)-i$$ \[\psi_{\{3\}}(cw_1^{(i)})=\begin{cases} 2n(r+k)+(3n-2r+1)-\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 2n(r+k)+(3n-2r+1)-\lceil\frac{n}{2}\rceil-\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2)\\ \end{cases}\] \[\psi_{\{5\}}(cw_1^{(i)})=\begin{cases} 2\{r(n-1)+n(k+1)\}+\frac{i+1}{2}, & {\textrm if}\ i\equiv 1\ ({\textrm mod}\ 2)\\ 2\{r(n-1)+n(k+1)\}+\lceil\frac{n}{2}\rceil+\frac{i}{2}, & {\textrm if}\ i\equiv 2\ ({\textrm mod}\ 2)\\ \end{cases}\] For \(\delta=2,3,5\) \begin{eqnarray*}\psi_{\delta}(V(P_{x_ix_{i+1}}))&=&\{n+(n-1)j+1+i:1 \leq i \leq n-1, 1 \leq j \leq r-1\} \\ \psi_{\delta}(E(P_{x_ix_{i+1}}))&=&\{(n-1)(2r+2-j)+n(k+1)+2-i:1 \leq j \leq r+1\}\\ \psi_{\delta}(V(S^{(i)}))&=&\{ r(n-1)+1+nj+i:1 \leq i \leq n, 1 \leq j \leq k \}\\ \psi_{\delta}(E(S^{(i)}) \setminus \{cw_1^{(i)}\})&=&\{2n(r+k)+(3n-2r+1)-nj-i:1 \leq j \leq k\} \end{eqnarray*} where indices \(i\) are taken modulo \(n\). Evidently \(\psi_{\delta}\) is a super labeling as \(V(F_n(rk)))=\{1, 2, \dots, n(k+r+1)-r+1\}\). The spoke vertices are labeled with the numbers \(n+2,n+3 , \dots, n+2k+1\) and the path edge vertices are labeled with \(n+2k+2, n+2k+3, \dots, n(k+r+1)-r+1\). Clearly,

According to (1) and (6), we obtain Equation (7) shows that all \(C_{r+2k+3}^{(i)}\)-weights constitute an arithmetic progression with common difference \(\delta=2\). According to (1) and (6), we obtain Equation (8) shows that all \(C_{r+2k+3}^{(i)}\)-weights constitute an arithmetic progression with common difference \(\delta=3\). According to (1) and (6), we obtain Equation (9) shows that all \(C_{r+2k+3}^{(i)}\)-weights constitute an arithmetic progression with common difference \(\delta=5\).

Concluding remarks

An \(\Omega\)-covering graphs is the extension of the edge-antimagic labeling and generalizes the structure for \(\Omega\)-antimagic labeling. Several results concerning \(\Omega\)-antimagic labelings for different families of graphs are proved and available in literature. In the present manuscript, the super \((\alpha,\delta)\)-\(C_n\)-antimagicness of subdivided fans has been considered for few of differences. One can work to extend the labeling for further differences greater than \(5\).

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.