International Journal of Mathematics and Mathematical Sciences

Volume 2017, Article ID 5156974, 6 pages

https://doi.org/10.1155/2017/5156974

## New Results on the (Super) Edge-Magic Deficiency of Chain Graphs

^{1}Department of Civil Engineering, Universitas Merdeka Malang, Jl. Taman Agung No. 1, Malang 65146, Indonesia^{2}School of Computing, Telkom University, Jl. Telekomunikasi No. 1, Bandung 40257, Indonesia

Correspondence should be addressed to Adiwijaya

Received 3 February 2017; Accepted 29 May 2017; Published 12 July 2017

Academic Editor: Dalibor Froncek

Copyright © 2017 Ngurah Anak Agung Gede and Adiwijaya. 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

Let be a graph of order and size . An* edge-magic labeling* of is a bijection such that is a constant for every edge . An edge-magic labeling of with is called a* super edge-magic labeling*. Furthermore, the* edge-magic deficiency* of a graph , , is defined as the smallest nonnegative integer such that has an edge-magic labeling. Similarly, the* super edge-magic deficiency* of a graph , , is either the smallest nonnegative integer such that has a super edge-magic labeling or if there exists no such integer . In this paper, we investigate the (super) edge-magic deficiency of chain graphs. Referring to these, we propose some open problems.

#### 1. Introduction

Let be a finite and simple graph, where and are its vertex set and edge set, respectively. Let and be the number of the vertices and edges, respectively. In [1], Kotzig and Rosa introduced the concepts of edge-magic labeling and edge-magic graph as follows: an* edge-magic labeling* of a graph is a bijection such that is a constant, called the* magic constant* of , for every edge of . A graph that admits an edge-magic labeling is called an* edge-magic graph*. A* super edge-magic labeling* of a graph is an edge-magic labeling of with the extra property that . A* super edge-magic graph* is a graph that admits a super edge-magic labeling. These concepts were introduced by Enomoto et al. [2] in 1998.

In [1], Kotzig and Rosa introduced the concept of edge-magic deficiency of a graph. They define the* edge-magic deficiency* of a graph , , as the smallest nonnegative integer such that is an edge-magic graph. Motivated by Kotzig and Rosa’s concept of edge-magic deficiency, Figueroa-Centeno et al. [3] introduced the concept of super edge-magic deficiency of a graph. The* super edge-magic deficiency* of a graph , , is defined as the smallest nonnegative integer such that is a super edge-magic graph or if there exists no such .

A* chain graph* is a graph with blocks such that, for every , and have a common vertex in such a way that the block-cut-vertex graph is a path. We will denote the chain graph with blocks by If , we will write as If, for every , for a given graph , then is denoted by -path. Suppose that are the consecutive cut vertices of The* string* of is -tuple , where is the distance between and , We will write as , if .

For any integer , let . Let and be the graphs obtained from the ladder by adding a single diagonal and two diagonals in each rectangle of , respectively. Thus, , , and . and are called triangle ladder and diagonal ladder, respectively.

Recently, the author studied the (super) edge-magic deficiency of -path, , and -path with some strings. Other results on the (super) edge-magic deficiency of chain graphs can be seen in [4]. The latest developments in this area can be found in the survey of graph labelings by Gallian [5]. In this paper, we further investigate the (super) edge-magic deficiency of chain graphs whose blocks are combination of and and and , as well as the combination of and . Additionally, we propose some open problems related to the (super) edge-magic deficiency of these graphs. To present our results, we use the following lemmas.

Lemma 1 (see [6]). *A graph is a super edge-magic graph if and only if there exists a bijective function such that the set consists of consecutive integers.*

Lemma 2 (see [2]). *If is a super edge-magic graph, then .*

#### 2. Main Results

For , let , where when is odd and when is even. Thus is a chain graph with and when is odd, or when is even. By Lemma 2, it can be checked that is not super edge-magic when and is even and when and is odd. As we can see later, when and is odd, is super edge-magic. Next, we investigate the super edge-magic deficiency of . Our first result gives its lower bound. This result is a direct consequence of Lemma 2, so we state the result without proof.

Lemma 3. *Let be an integer. For any integer , *

Notice that the lower bound presented in Lemma 3 is sharp. We found that when is odd, the chain graph with particular string has the super edge-magic deficiency equal to its lower bound as we state in Theorem 4. First, we define vertex and edge sets of as follows.

, for : : : , for , when is odd, and , for , when is even.

Theorem 4. *Let be an integer and with string when is odd or when is even, where . For any odd integer ,*

*Proof. *First, we define as a graph with vertex set , where , , and edge set Under this definition, , , are the cut vertices of .

Next, for and , define the labeling , where when is even or when is odd, as follows:where

Under the vertex labeling , it can be checked that no labels are repeated, , , is a set of consecutive integers, and the largest vertex label used is when is even or when is odd. Also, it can be checked that when is odd.

Next, label the isolated vertices in the following way.*Case ** Is Odd*. In this case, we denote the isolated vertices with and set .*Case ** Is Even*. In this case, we denote the isolated vertices with and set and

By Lemma 1, can be extended to a super edge-magic labeling of with the magic constant when is even or when is odd. Based on these facts and Lemma 3, we have the desired result.

An example of the labeling defined in the proof of Theorem 4 is shown in Figure 1(a).