A graph \(\Gamma\) (simple, finite, undirected) with an \(\Omega\)-covering has an \((\alpha,\delta)\)-\(\Omega\)-antimagic labeling if the weights of all subgraphs \(\Omega\) of graph \(\Gamma\) constitute an arithmetic progression with the common difference \(\delta\). Such a~graph is called super \((\alpha,\delta)\)-\(\Omega\)-antimagic if \(\nu(V(\Gamma))= \{ 1,2,3,\dots,|V(\Gamma)|\}\). In the present paper, the cycle coverings of subdivision of fan graphs has been considered and results are proved for several differences.
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~ \(\Omega\)-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 \(\Omega\)-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 \(\delta=0, 1, 2, 3, 4\).
Definition 1. A graph \(F_n\cong P_n+K_1\) is called fan graph obtained by the join of path \(P_n\) and one isolated vertex \(K_1\).
The central vertex, or the hub vertex is of degree \(n\) and path vertices are the other ones. Spokes are the adjacent edges of central vertex and 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 obtainTheorem 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,
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,
