International Journal of Mathematics and Mathematical Sciences

Volume 2019, Article ID 8780329, 7 pages

https://doi.org/10.1155/2019/8780329

## On -Supermagic Labelings of -Shadow of Paths and Cycles

^{1}CGANT Research Group, University of Jember, Indonesia^{2}Department of Mathematics, University of Jember, Indonesia^{3}Department of Mathematics Education, University of Jember, Indonesia^{4}Department of Elementary School Teacher Education, University of Jember, Indonesia^{5}Combinatorial and Applied Mathematics Research Group, University of Tadulako, Indonesia

Correspondence should be addressed to Dafik

Received 9 October 2018; Accepted 3 January 2019; Published 3 February 2019

Academic Editor: Vladimir V. Mityushev

Copyright © 2019 Ika Hesti Agustin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.