Super \((a,d)\)-\(C_3\)-antimagicness of a Corona Graph

1. Introduction

Let \(G\) be a simple graph with vertex set \(V\) and edge set \(E\). An edge-covering of finite and simple graph \(G\) is a family of subgraphs \(H_1, H_2, \dots,H_t\) such that each edge of \(E(G)\) belongs to at least one of the subgraphs \(H_i\), \(i=1, 2, \dots, t\). In this case we say that \(G\) 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 \(f:V(G)\cup E(G) \to \{1,2,\dots, |V(G)|+|E(G)| \}\) such that for each subgraph \(H’\) of \(G\) isomorphic to \(H\), the \(H’\)-weights, $$wt_f(H’)= \sum\limits_{v\in V(H’)} f(v) + \sum\limits_{e\in E(H’)} f(e),$$ 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 \(H\).

The (super) \(H\)-magic graph was first introduced by Gutiérrez and Lladó in [1]. The \((a,d)\)-\(H\)-antimagic labeling was introduced by Inayah et al. [2].

In [3] Bača et al. investigated the super tree-antimagic total labelings of disjoint union of graphs. Bača et al. [4] showed the constructions for \(H\)-antimagicness of Cartesian product of graphs. In [5], authors proved the \(C_n\)-antimagicness of Fan graph for several difference depending on the length of the cycle. In [6, 7, 8] Umar et al. proved the existence of super \((a,1)\)-Tree-antimagicness of Sun graphs, super \((a,d)\)-\(C_n\)-antimagicness of Windmill graphs for several differences and super \((a,d)\)-\(C_4\)-antimagicness of Book graph and their disjoint union.

In this paper, we study the existence of super \((a,d)\)-\(C_3\)-antimagic labeling of a special type of a corona graph.

2. Super Cycle-antimagic labeling of Corona graph

The join of two graphs \(H_1\) and \(H_2\), denoted by \(H_1+H_2\), is the graph where \(V(H_1) \cap V(H_2)= \emptyset\) and each vertex of \(H_1\) is adjacent to all vertices of \(H_2\) [9]. When \(H_1=K_1\), this is the corona graph \(K_1 \odot H_2\). In this paper, we consider a special type of a corona graph.

Let \(K_1\) be a complete graph and \(S_n\) be a star on \(n+1\) vertices. We consider the corona graph \(G= K_1 \odot S_n\), where $$V(G):=\{v_1,v_2,x_1,x_2,\dots,x_n\}$$ and $$E(G):=\{v_1v_2,v_1x_1,v_1x_2,\dots,v_1x_n,v_2x_1,v_2x_2,\dots,v_2x_n\}$$ The corona graph \(G\) is covered by the cycles \(C_3^{(i)}\), \( 1\leq i \leq n\) and the \(C_3^{(i)}\)-weights under a labeling \(h\) is:

2.1. \(C_3\)-Supermagic labeling

Theorem 2.1. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a \(C_3\)-supermagic labeing.

Proof. \(n \equiv 0 (\text{mod}\ 2)\)
The labeling \(h_0\) is defined as: \begin{align*} h_0(v_1)&=1,\\ h_0(v_2)&=\frac{n}{2}+2,\\ h_0(v_1v_2)&= 3n+3,\\ h_0(v_1x_i)&=3n+3-i.\\ \end{align*} \[ h_0(x_{i})= \begin{cases} \frac{n}{2}+2-i \ \ & \ \ \ \ \ \textrm{ if $i = 1,2,\dots, \frac{n}{2}$} \\ \frac{3n+6}{2}-i \ \ & \ \ \ \ \ \textrm{ if $i = \frac{n}{2}+1,\frac{n}{2}+2, \dots, n$} \\ \end{cases} \] \[ h_0(v_2x_{i})= \begin{cases} n+2(1+i) \ \ & \textrm{ if $i = 1,2,\dots, \frac{n}{2}$} \\ 2i+1 \ \ & \textrm{ if $i = \frac{n}{2}+1,\frac{n}{2}+2, \dots, n$} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_0\) and edges receive the labels \(\{n+3, n+4,\dots, 3n+3\}\). Therefore \(h_0\) is a super total labeling.
Using equation (1)

When \(n \equiv 1 \ \ (\text{mod}\ 2)\)
The labeling \(h_0\) is defined as: \begin{align*} h_0(v_i)&=i,\\ h_0(v_1v_2)&= n+3,\\ h_0(x_i)&=n+3-i.\\ \end{align*} For \(i \equiv 0 \) (mod \(2\)) \[ h_0(v_jx_{i})= \begin{cases} n+3 +\frac{i}{2} \ \ & \textrm{ if $j = 1$} \\ \frac{5n+7+i}{2} \ \ & \textrm{ if $j = 2$} \\ \end{cases} \] For \(i \equiv 1\) (mod \(2\)) \[ h_0(v_jx_{i})= \begin{cases} \frac{3(n+2)+i}{2} \ \ & \textrm{ if $j = 1$} \\ \frac{4n+7+i}{2} \ \ & \textrm{ if $j = 2$} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_0\) and edges receive the labels \(\{n+3, n+4,\dots, 3n+3\}\). Therefore \(h_0\) is a super total labeling.
Using equation (1) Equations (2, 3) shows \(wt_{h_0}(C_3^{(i)})\) is independent of \(i\). Hence the corona graph \(G\) admits a \(C_3\)-supermagic labeling. This completes the proof.

2.2. Super \((a, d)\)-\(C_3\)-antimagic labeling

Theorem 2.2. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a super \((a,1)\)-\(C_3\)-antimagic labeing.

Proof. The labeling \(h_1\) is defined as: \begin{align*} h_1(v_i)&=i,\\ h_1(v_1v_2)&= n+3,\\ h_1(v_2x_{i})&= 2n+3+i. \end{align*} \[ h_1(x_{i})= \begin{cases} \frac{i+1}{2}+2 \ \ & \textrm{ if $i \equiv 1$ (mod \ $2$)} \\ \lceil\frac{n}{2}\rceil+ 2 +\frac{i}{2} \ \ & \textrm{ if $i \equiv 0$ (mod $2$)} \\ \end{cases} \] \[ h_1(v_1x_{i})= \begin{cases} \frac{4n+7-i}{2} \ \ & \textrm{ if $i \equiv 1$ (mod \ $2$)} \\ \lceil\frac{n-1}{2}\rceil+ n+4-\frac{i}{2} \ \ & \textrm{ if $i \equiv 0$ (mod $2$)} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_1\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_1\) is a super total labeling.
Using equation (1)

Equation (4) shows \(wt_{h_0}(C_3^{(i)})\) constitute an arithmetic progression with \(a=5(n+3)+1\) and \(d=1\). Hence the corona graph \(G\) admits a super \((a,1)\)-\(C_3\)-antimagic labeling. This completes the proof.

Theorem 2.3. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a super \((a,d)\)-\(C_3\)-antimagic labeing for \(d=3,5\).

Proof. The labeling \(h_d\) is defined as: \begin{align*} h_d(v_i)&=i,\\ h_d(v_1v_2)&= n+3,\\ h_d(x_i)&= 2+i. \end{align*} \[ h_3(v_jx_{i})= \begin{cases} 2n+3+i \ \ & \textrm{ if $j=1$} \\ n+3+i \ \ & \textrm{ if $j=2$} \\ \end{cases} \] \[ h_5(v_jx_{i})= \begin{cases} n+2+2i \ \ & \textrm{ if $j=1$} \\ n+3+2i \ \ & \textrm{ if $j=2$} \\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_d\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_d\) is a super total labeling.
Using equation (1)

Equation (5) shows \(wt_{h_3}(C_3^{(i)})\) constitute an arithmetic progression with \(a=2(2n+7)+3\) and \(d=3\). Hence the corona graph \(G\) admits a super \((a,3)\)-\(C_3\)-antimagic labeling.
Now, for case \(d=5\), Using equation (1) Equation (6) shows \(wt_{h_3}(C_3^{(i)})\) constitute an arithmetic progression with \(a=3(n+6)\) and \(d=5\). Hence the corona graph \(G\) admits a super \((a,5)\)-\(C_3\)-antimagic labeling. This completes the proof.

Theorem 2.4. Let \(G:=K_1\odot S_n\) be a corona graph of \(K_1\) and \(S_n\) and \(n \geq 2\) be an integer then the graph \(G\) admits a super \((a,d)\)-\(C_3\)-antimagic labeing for \(d=2,4\).

Proof. The labeling \(h_d\) is defined as: $$h_d(v_i)=i$$ \[ h_d(x_{i})= \begin{cases} n+3-i \ \ & \textrm{ if $d=2$} \\ 2+i \ \ & \textrm{ if $d=4$} \\ \end{cases} \] The edges are labeled as:
When \(n \equiv 0 \ \ \ (\text{mod} \;2)\)
$$h_d(v_1v_2)= 5\left(\frac{n}{2}\right)+3$$ \[ h_d(v_1x_{i})= \begin{cases} n + 2 + i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n}{2}+1$ }\\ \frac{n}{2}+1+2i \ \ & \textrm{ if $i = \frac{n}{2} +2, \frac{n}{2} +3,…,n$}\\ \end{cases} \] \[ h_d(v_2x_{i})= \begin{cases} \frac{3n}{2}+ 2(1 + i) \ \ & \textrm{ if $i = 1,2,\dots,\frac{n}{2}+1$ }\\ 2n+3+i \ \ & \textrm{ if $i = \frac{n}{2}+2, \frac{n}{2}+3,…,n$}\\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_d\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_d\) is a super total labeling.
Using equation (1)

Equation (7) shows \(wt_{h_2}(C_3^{(i)})\) constitute an arithmetic progression with \(a=6n+15\) and \(d=2\). Hence the corona graph \(G\) admits a super \((a,2)\)-\(C_3\)-antimagic labeling. Using equation (1) Equation (8) shows \(wt_{h_4}(C_3^{(i)})\) constitute an arithmetic progression with \(a=5n+16\) and \(d=4\). Hence the corona graph \(G\) admits a super \((a,4)\)-\(C_3\)-antimagic labeling.
When \(n \equiv 1 \ \ (\text{mod}\; 2)\)
$$h_d(v_1v_2)= 3n+3$$ \[ h_d(v_1x_{i})= \begin{cases} n + 2 + i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n+1}{2}$}\\ \frac{n+1}{2} + 1 + 2i \ \ & \textrm{ if $i = \frac{n+1}{2}+1, \frac{n+1}{2} +2,…,n$}\\ \end{cases} \] \[ h_d(v_2x_{i})= \begin{cases} \frac{n+1}{2}+n+1+2i \ \ & \textrm{ if $i = 1,2,\dots,\frac{n+1}{2}$}\\ 2(n+1)+i \ \ & \textrm{ if $i = \frac{n+1}{2}+1, \frac{n+1}{2}+2,…,n$}\\ \end{cases} \] Clearly, the vertices assume least possible integers \(\{1,2,\dots, n+2\}\) under the labeling \(h_d\) and edges receive labels \(\{n+3, n+4, \dots, 3n+3\}\). Therefore \(h_d\) is a super total labeling.
Using equation (1) Equation (9) shows \(wt_{h_2}(C_3^{(i)})\) constitute an arithmetic progression with \(a=\frac{13n+29}{2}\) and \(d=2\). Hence the corona graph \(G\) admits a super \((a,2)\)-\(C_3\)-antimagic labeling.
Using equation (1) Equation (10) shows \(wt_{h_4}(C_3^{(i)})\) constitute an arithmetic progression with \(a=\frac{11n+31}{2}\) and \(d=4\). Hence the corona graph \(G\) admits a super \((a,4)\)-\(C_3\)-antimagic labeling. This completes the proof.

Competing Interests

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