Abstract
A simple graph is said to be an -covering if every edge of belongs to at least one subgraph isomorphic to . A bijection is an (a,d)--antimagic total labeling of if, for all subgraphs isomorphic to , the sum of labels of all vertices and edges in form an arithmetic sequence where , are two fixed integers and is the number of all subgraphs of isomorphic to . The labeling is called super if the smallest possible labels appear on the vertices. A graph that admits (super) --antimagic total labeling is called (super) --antimagic. For a special , the (super) --antimagic total labeling is called -(super)magic labeling. A graph that admits such a labeling is called -(super)magic. The -shadow of graph , , is a graph obtained by taking copies of , namely, , and then joining every vertex in , , to the neighbors of the corresponding vertex in . In this paper we studied the -supermagic labelings of where are paths and cycles.
1. Introduction
Graph theory is a branch of discrete mathematics that has been grown rapidly. There are many applications of graph theory in other fields such as computer science, physics, chemistry, biology, engineering, and sociology [1]. A graph is a pair of two sets, i.e., and . These two sets, respectively, represent a vertex set of and an edge set of . The number of vertices in is denoted by and the number of edges in is denoted by . Other basic terminologies about graph theory that are not mentioned in this paper can be seen in [2]. Note that all graphs considered in this paper are simple, finite, and undirected. By notation with integers we mean .
One of important topics in graph theory is graph labeling. A graph labeling can be defined as a mapping from some set of graph elements to a set of positive integers. A graph labeling whose domain is vertex set or edge set is called a vertex labeling or an edge labeling, respectively. Moreover, if domain is both vertex set and edge set, then we call such labelings as a total labeling.
An edge-covering of a graph is a collection of subgraphs such that every edge of belongs to at least one of subgraphs , . In this case, is said to be an -(edge) covering. If every subgraph is isomorphic to a given graph , then is said to be an -covering.
For a graph admitting an -covering, an --antimagic total labeling of is a bijection such that, for all subgraphs isomorphic to , the -weight, which is defined by , forms an arithmetic sequence where , are two fixed integers and is the number of all subgraphs of isomorphic to . The labeling is called super if the smallest possible labels appear on the vertices. A graph that admits (super) --antimagic total labeling is called (super)--antimagic. For a special , the (super) --antimagic total labeling is called -(super)magic labeling. A graph that admits -(super)magic labeling is called -(super)magic.
The notion of super --antimagic total labeling was firstly introduced by Inayah, Salman, and Simanjuntak [3]. In 2013, Inayah et al. [4] studied super --antimagic total labeling of a shackle graph . Dafik et al. [5] introduced a generalized shackle of graph denoted by . They showed the existence of super --antimagic total labeling of when . Furthermore, Dafik et al. [6, 7] studied about -super antimagicness of disconnected graphs as well as constructions of -antimagic graphs using smaller edge antimagic graphs. More results about super --antimagic total labeling can be seen in [8–11].
The notion -supermagic labeling was firstly introduced by Gutiérrez and Lladó [12]. In their paper, they investigated star-(super)magic and path-(super)magic labelings of some classes of connected graphs. Maryati et al. [13] studied -supermagic labeling of some classes of trees, i.e., shrubs and banana trees. For more results about -supermagic labeling can be seen in [14].
In this paper, we investigate the -supermagic labeling of graphs, namely, -shadow of graphs which is a generalization of a shadow graph introduced by [15]. The -shadow of graph denoted by is a graph obtained by taking copies of , namely, , and then joining every vertex in , , to the neighbors of the corresponding vertex in . We have proved that admits -supermagic labelings for some classes of graph , namely, paths and cycles.
2. Main Results
2.1. -Supermagic Labeling of
In this part, we present the -supermagic labeling of -shadow of paths. Let be the -shadow of paths with vertex set , and edge set , , , . Next, we will show the existence of -supermagic labeling of in the following theorem.
Theorem 1. is -supermagic for any integer , and .
Proof (let ). Define a total labeling . In constructing the total labeling , we distinguish between the vertices labeling and the edges labeling. First, label every vertex of in the following way. with To label the edges of , first, let . Next, label every edge as follows: with It can be checked that . For , let be sub--shadow of paths with , and , , , . It can be shown that Furthermore, it can be shown that and , , , . By combining these pieces of information, we obtain Let be the magic constant. For the value of , consider the following cases: (i)For and (), it can be verified that .(ii)For and (), it can be verified that .(iii)For , and (), it can be verified that .(iv)For , and (), it can be verified that . Therefore, is -supermagic for each , , and .
For an illustration, we give an example of -supermagic labeling of in Figure 1.
2.2. -Supermagic Labeling of
In this part, we focus on the -supermagic labeling of -shadow of cycles. Let be the -shadow of cycles with vertex set and edge set , , , . Next, the -supermagic labeling of will be shown in the following theorem.
Theorem 2. is -supermagic for any integer , , and .
Proof (let ). Define a total labeling in the following way: (i)Label the sets as follows: with (ii)Label the sets . To label these sets, first, let . Next, label the sets in the following way: with It can be seen that . Let be sub--shadow of cycles with , and , , , . It can be checked that Furthermore, for , it can be shown that , and , , , . By combining these pieces of information, we have Let be the magic constant. For the value of , consider the following cases: (i)For and (), it can be verified that .(ii)For and (), it can be verified that .(iii)For , and (), it can be verified that .(iv)For , and (), it can be verified that . Hence, is -supermagic for each , , and .
For an illustration, an example of -supermagic labeling of can be seen in Figure 2.
3. Conclusion
We have studied the -supermagic labeling of one type of graph operation, namely, -shadow of graphs. In this paper, we have only shown the existence of -supermagic labeling of -shadow of paths and cycles. Meanwhile, the -supermagic labeling of -shadow of other classes of graphs is still widely open. Therefore, we propose open problems as follows.
Open Problem 1. Determine -supermagic labelings of -shadow of other classes of graphs.
Open Problem 2. Determine -supermagic labelings of -shadow of any connected graph.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
We gratefully acknowledge the support from CGANT University of Jember of year 2019.