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 ,