Research Article | Open Access

Yijun Xiong, Huajun Wang, Muhammad Awais Umar, Yu-Ming Chu, Basharat Rehman Ali, Maria Naseem, "*H*-Coverings of Path-Amalgamated Ladders and Fans", *Mathematical Problems in Engineering*, vol. 2020, Article ID 5846014, 7 pages, 2020. https://doi.org/10.1155/2020/5846014

*H*-Coverings of Path-Amalgamated Ladders and Fans

**Academic Editor:**Francisco R. Villatoro

#### Abstract

Let be a connected, simple graph with finite vertices and edges . A family of subgraphs such that for all , , for some is an edge-covering of . If , , then has an -covering. Graph with -covering is an --antimagic if a bijection exists and the sum over all vertex-weights and edge-weights of forms a set . The labeling is super for and graph is -supermagic for . This manuscript proves results about super -antimagic labeling of path amalgamation of ladders and fans for several differences.

#### 1. Introduction and Preliminaries

Let be a connected, finite, and simple graph [1, 2]. An edge-covering of is a family such that for all , , for some . If , , then has a -covering. Graph with -covering is an --antimagic if a one-to-one correspondence ,

For , the labeling becomes super --antimagic and it would be -supermagic for . GutiÃ©rrez and LladÃ³, in [3], defined the -supermagic graphs for , , , and for some subgraph . Jeyanthi and Selvagopal [4] proved -supermagic results for 2-connected graphs, -polygonal snake, and one point union of -disjoint paths. -path, , book, and ladder-related graphs are -supermagic proved in [5]. Inayah et al. [6] introduced -antimagic graphs with weights forming an arithmetic progression. She also proved some bounds for and for general graphs and fans. Susanto [7] derived bound for cycle-antimagic labeling of disjoint union of cycles. Recent results on -antimagic labeling of graphs can be seen in [8â€“14]. Also, in [12], Baca et al. discussed the tree-antimagicness of disconnected graphs. Recently, authors in [15] discussed the super -tree-antimagicness of Sun graphs. The (super) -antimagic labeling is also related to a (super) -antimagic labeling of type of a plane graph [16]: a generalization of a face-magic labeling introduced by Lih [17]. Baca et al. proved -antimagic labeling of type for toroidal fullerenes in [18], while, in [19], Baca et al. proved labeling for plane graphs containing Hamiltonian paths. For more details, we refer [20â€“29] and the references therein. In the present article, we have studied super -antimagic labeling of path-amalgamation of ladder and fan for several differences, where is isomorphic to cycles , , , and -amalgamation of two cycles and .

#### 2. Path Amalgamation

In [30, 31], authors defined vertex amalgamation of isomorphic copies of , denoted with , for . Maryati et al. proved is -supermagic in [32]. -supermagic labeling of (i) edge amalgamation of 2-connected simple graph is proved by Jeyanthi and Selvagopal in [4] and of (ii) by Salman and Maryati in [33]. In this paper, we extend this idea and use path graph instead of a vertex to define a path amalgamation for two graphs ladders and fans. A ladder graph is defined as the Cartesian product of with , i.e., . A fan graph is defined as the join of with an isolated vertex , i.e., .

*Definition 1. *Let and be ladder and fan graph for . A path amalgamation of ladder and fan denoted by is obtained by identifying path in ladder and fan.

The vertex set of is and edge set is .

Figure 1 depicts -amalgamation of ladder and fan .

*Definition 2. *Let and be two cycles on 3 and 4 vertices, respectively. By identifying one edge of both cycles and , we obtain -amalgamation denoted by .

Figure 2 depicts -amalgamation of cycles and .

In Section 3, we will study the existence of super --antimagic labeling of path amalgamation of ladder and fan, where is , , and for several differences.

#### 3. Super -Antimagic Labeling of Path Amalgamation

From Figures 1 and 2, it is clear that path amalgamation has -coverings by subgraphs , where , , , and .

The set of vertices and edges in subgraphs are

Let be the total labeling of , then would be

Theorem 1. *Let be amalgamation of ladder and fan and be a positive integer. Then, posses a super --antimagic labeling for and .*

*Proof. *Consider the total labeling as follows:Since, , is a super labeling for and . Therefore, is a total labeling.

Using (3) and (4) and ,All the above equations show an arithmetic progression with differences , respectively. This completes the proof.

Theorem 2. *Let be amalgamation of ladder and fan and be a positive integer. Then, possesses a super --antimagic labeling for and --antimagic labeling for .*

*Proof. *The subgraphs contain the subgraphs ; therefore, differences depends upon differences obtained in Theorem 1. We will use the labeling from Theorem 1. Let denote a vertex element in and denote and edge element in ; then, the total labeling is defined as follows:Since is a super total labeling, therefore is super total. For differences , the total labeling of is obtained as follows:For differences , the total labeling of is obtained as follows:Using (5) and (6) and ,All the above equations show an arithmetic progression with required differences and . This completes the proof.

#### 4. -Antimagic Coverings for Disjoint Union

Theorem 3. *Let path amalgamation of ladder and fan admit -supermagic labeling for . Then, the disjoint union of arbitrary number of copies of , i.e., also admits a -supermagic labeling for and super --antimagic for a positive integer.*

*Proof. *Let be a positive integer. In our proof, , denotes a vertex or an edge in the copy of the path amalgamation of ladder and fan, denoted by , corresponding to in , i.e., . In the same way, let , be the subgraph in the copy of corresponding to the subgraph in . The total labeling of is defined as follows:First, we will prove that vertices of use integers from 1 up to under the labeling , where is number of vertices in graph , i.e.,Secondly, for edges of with , under the labeling ,From equations (14) and (15), it is clear the labeling is a total labeling, since is a bijection between the integers and the vertices and edges of .

Under the total labeling , the -weights, would bewhere is the number of â€™s in :As every , , is isomorphic to the cycle , it holdsThus, for the -weights for , we obtainIt is easy to see that the set of all -weights for consists of the same integers and -weight consists of consecutive integers in