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 . Gutiérrez and Lladó, 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 [8–14]. 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 [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 [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 ’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 . Thus, the graph is a -supermagic and super --antimagic. This completes the proof.
Data Availability
No such data were used in preparing the present manuscript.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The research was supported by the National Natural Science Foundation of China (Grant nos. 11971142, 11871202, 61673169, 11701176, 11626101, and 11601485).