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).
(a)
(b)
Notice that when and is odd, . In other words, the chain graph with string , where , is super edge-magic when and is odd. Based on this fact and previous results, we propose the following open problems.
Open Problem 1. Let be an integer. For , decide if there exists a super edge-magic labeling of . Further, for any even integer , find the super edge-magic deficiency of .
Next, we investigate the super edge-magic deficiency of the chain graph with string , where . is a graph of order and size We define the vertex and edge sets of as follows: : : : : : , where and , and : : : ∪ : Hence, the cut vertices of are , , and , Notice that has string , if at least one of is , and its string is , if for every
Theorem 5. For any integers and , .
Proof. Define a bijective function as follows:Under the labeling , it can be checked that and . Also, it can be checked that , , and . By Lemma 1, can be extended to a super edge-magic labeling of with the magic constant . Hence, .
Open Problem 2. For any integers and , find the super edge-magic deficiency of with string , where .
Next, we study the edge-magic deficiency of ladder and chain graphs whose blocks are combination of and with some strings. In [6], Figueroa-Centeno et al. proved that the ladder is super edge-magic for any odd and suspected that is super edge-magic for any even . Here, we can prove that is edge-magic for any by showing its edge-magic deficiency is zero. The result is presented in Theorem 6.
Theorem 6. For any integer , .
Proof. Let and : be the vertex set and edge set, respectively, of . It is easy to verify that the labeling is a bijection and, for every , Thus, for every
Theorem 7. Let and be integers.(a)If is an even integer and with string , then .(b)If is an odd integer and with string , then .
Proof. (a) First, we introduce a constant as follows: , if is odd and , if is even. Next, we define as a graph with : : : : : , where and , and : : : The cut vertices of are , , and ,
Next, define a bijection as follows:where , , , , , and It is easy to verify that, for every edge ,
(b) We define as graph with , where and , and . Under this definition, the cut vertices of are , , and , Next, we define a bijection , where for all . It can be checked that is an edge-magic labeling of with the magic constant
Open Problem 3. Let and be integers.(a)If is an odd integer, find the super edge-magic deficiency of with string (b)If is an even integer, find the super edge-magic deficiency of with string
Theorem 8. Let and be integers.(a)If is an even integer and with string , then .(b)If is an odd integer and with string , then .
Proof. (a) First, we define as a graph with : : : : : : : , where , , and , and : ∪ ∪ : : :
Next, define a bijection as follows:where , , , , , , and are defined as in the proof of Theorem 7. It can be checked that, for every edge , Hence
An illustration of the labeling defined in the proof of Theorem 8 is given in Figure 1(b).
(b) We define as graph with , where , , and , and . It can be checked that defined by , for all , is an edge-magic labeling of with the magic constant
Open Problem 4. Let and be integers.(a)If is an odd integer, find the edge-magic deficiency of with string (b)If is an even integer, find the edge-magic deficiency of with string
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
The first author has been supported by “Hibah Kompetensi 2016” (018/SP2H/P/K7/KM/2016) from the Directorate General of Higher Education, Indonesia.