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
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 . Gutiérrez and Lladó, in , defined the -supermagic graphs for , , , and for some subgraph . Jeyanthi and Selvagopal  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 . Inayah et al.  introduced -antimagic graphs with weights forming an arithmetic progression. She also proved some bounds for and for general graphs and fans. Susanto  derived bound for cycle-antimagic labeling of disjoint union of cycles. Recent results on -antimagic labeling of graphs can be seen in [8–14]. Also, in , Baca et al. discussed the tree-antimagicness of disconnected graphs. Recently, authors in  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 : a generalization of a face-magic labeling introduced by Lih . Baca et al. proved -antimagic labeling of type for toroidal fullerenes in , while, in , Baca et al. proved labeling for plane graphs containing Hamiltonian paths. For more details, we refer [20–29] 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 . -supermagic labeling of (i) edge amalgamation of 2-connected simple graph is proved by Jeyanthi and Selvagopal in  and of (ii) by Salman and Maryati in . 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
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 ’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