Abstract
The idea of super -edge-antimagic labeling of graphs had been introduced by Enomoto et al. in the late nineties. This article addresses super -edge-antimagic labeling of a biparametric family of pancyclic graphs. We also present the aforesaid labeling on the disjoint union of graphs comprising upon copies of and different trees. Several problems shall also be addressed in this article.
1. Introduction
A graph is a combination of two different sets, one is the set of vertices and the other is the set of connections between these vertices, termed as set of edges . A graph can either be connected or comprises upon connected parts known as graphs’ components. The nonempty and simple graphs shall be considered here only all the way, consisting of , the set of vertices, and , the set of edges, having and . In this case, the graph is called a -graph. Additionally, [1] can be cited for the comprehension of the graph theoretic terminologies.
A labeling is a function from the set of integers onto the components of a graph under certain constraints. The labeling is said to be total if it covers all components of the graph. If the labeling covers or only in the domain, then it is termed to be the vertex or edge labeling, respectively. The two important categories of labeling are magic and antimagic. The equal or unequal edge/vertex weights point towards, respectively, the magic and antimagic types of labeling.
Throughout the article, the abbreviations given in Table 1 are used.
Definition 1. On a -graph , a bijection from is the notion of labeling if we keep the restriction upon the edge weights , , which generates a consecutive integer sequence, with being the minimum edge weight and being the common difference. is notioned as an graph, with the existence of such labeling.
Definition 2. If the smallest labels are allocated to the points (vertices) of the -graph in an labeling, then this labeling is addressed as labeling. And, , in his scenario, is referred to be an graph.
The edge weight (minimum) (Definitions 1 and 2) becomes a constant at , , and is called magic constant or magic sum of .
Definition 3. A pancyclic graph is a graph that contains the cycles of all orders up to .
The notion of magic labeling was highlighted by Sedlacek in the early sixties [2]. Later, Ringel and Hartsfield capitulated the idea of antimagic labeling with respect to vertex-sums of graphs in [3]. The idea of magic valuations of graphs had been brought by Kotzig and Rosa [4] which was indeed the graphs’ labeling (introduced by Ringel et. al. in the nineties [5]). Enomoto and Llado introduced the idea of labeling of graphs using the term super edge-magic labeling in [6]. In the early century, Bertault and Simanjuntak brought out the graphs’ labeling [7]. The following notable and handy conjectures are included in the vicinity of graphs’ labeling.
Conjecture 1 (see [4]). Every tree admits anlabeling.
Conjecture 2 (see [6]). Every tree admits anlabeling.
In the support of Conjecture 2, certain classes of trees have been sorted out by scientists. For trees having maximum of seventeen vertices, in [8], this conjecture has been verified. For instance, the labeling on a class of trees termed as -trees can be observed in [9]. Similarly, labeling on various classes consisting of subdivisions of trees can be seen in [10, 11]. Some derivations on vertex-antimagicness of regular graphs have been discussed in [12]. In [13], the same labeling for the union of unicyclic graphs and isolated vertices has been provided. Enomoto et al. proved [6] that if a -graph (simple) is , it implies . In addition, they derived that is only if either or is equal to 1. It is derived in [14] that the combination of graphs in the form of union of and is if or . For only odd values of , is concreted to be [6]. The cycle of order 3 and cycle of order are proven to be for even values of . The generalized prism is proven to be for all odd values of in [15]. In [16, 17], labeling of maximum symmetric generalization of prism and special networks with magic constant has been exhibited, respectively. An extremely important result on graphs is as follows.
Lemma 1 (see [15]). A-graphisif and only if there exists a bijective functionsuch that the setconsists ofconsecutive integers. In such a case,extends to anlabeling ofwith magic constant, whereand.
In Lemma 1, the sum is defined as edge sum for each edge . This lemma shall be used frequently in our findings, while it keeps this enough to allot the labels to merely the vertices of the network to capacitate the graph to be , where the edge-sums (consecutive) belong to . The given result is extremely relevant with regard to .
Lemma 2 (see [18]). A simple graphpossesses anlabelingpossesses anlabeling.
2. Main Results
In this section, we shall address our main findings. In Section 2.1, we define an labeling on a pancyclic family of graphs, namely, Usmanian pancyclic graph . In Section 2.2, we design labeling on various disjoint unions of graphs comprising upon copies of and various trees/forests. Throughout, represents , for , whereas and are the representations of odd and even natural numbers, respectively.
2.1. Usmanian Pancyclic Graph
In computer science, there is a similar importance of the networks having no cycles and networks having a range of cycles. The importance is similar for a network containing cycles of all lengths from one to the number of systems connected within. In this situation, the role of programmers becomes prominent to avoid hackers halting of data as there is a closed path between any two arbitrary computers corresponding to such networks. The first kind of network is termed as a tree (connected and acyclic) and later is known as pancyclic network (connected and containing every order’s cycle). This section deals with a family of pancyclic graphs denoted by , which is biparametric, and reveals that such complex structures are . We shall first introduce this structure as Definition 4.
Definition 4. We are defining the Usmanian pancyclic graph, denoted by , being a graph with and, having the construction as follows ( being the order of the cycle and being the number of cycles):(1)For even : For :(i)For ,(ii)For ,(iii)For ,(iv)For , For :(i)For ,(ii)For ,(iii)For ,(2)For odd : For :(i)For ,(ii)For ,(iii)For , For :(i)For : Define as follows:(ii)For ,(iii)For ,
Theorem 1. The pancyclic graph is admitting , and .
Proof. We discuss here two cases as per Definition 4. For :(i)For : We define a labeling as follows: The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers . As per Lemma 1, is extendable to an labeling of with magic constant .(ii)For : We define a labeling as follows: The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers . In the light of Lemma 1, constitutes an labeling of admitting .(iii)For : We define a labeling as follows:With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence set . is extendible to labeling of , according to Lemma 1, with . For :(i)For : The labeling is being defined as follows: The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence . As per Lemma 1, is extendable towards labeling of with .(ii)For : Define a labeling as the following function:With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence set . constitutes labeling of , according to Lemma 1, admitting .
Theorem 2. The pancyclic graph is with magic sum , for all and .
Proof. We discuss here two cases. For :(i)For : Define a labeling as follows: The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence . is extendable to labeling of having (as per Lemma 1).(ii)For : We construct a labeling as follows: The edge-sums’ set generated as per the labeling design constitutes a natural numbers’ sequence . is extendable to labeling of , by Lemma 1, with the admittance of .(iii)For : We are going to construct a labeling as follows: With the abovementioned scheme, the edge-sums being generated forms a consecutive integer sequence set . Once again, extends to a labeling of having by Lemma 1. For :(i)For : We define a labeling as follows: The edge-sums’ set generated as per the labeling design constitutes a consecutive sequence of positive integers . Now, is extendable to labeling of [15] with according to Lemma 1.(ii)For : A labeling is being defined as follows: The edge-sums’ set generated as per the labeling design constitutes a consecutive natural numbers’ sequence . Now, is extendable to labeling of , according to Lemma 1 having .(iii)For : The labeling is constructed as follows: With the abovementioned scheme, the edge-sums being generated form a consecutive integer sequence set . is extendable to an labeling of , under the light of Lemma 1, admitting .A direct outcome of Lemma 2 is as follows.
Theorem 3. The pancyclic graph is , for all and .
2.2. Labeling of Disjoint Union of with Trees
It is a well-known fact that the graph is not [6], and work is still in progress in order to determine if its disjoint copies are . In this section, we shall provide an labeling of disjoint copies of with various trees in the form of several results. This will give a support to researchers to carry out their work to determine the aforesaid labeling of the disjoint copies of . Throughout this section, the union will represent a disjoint union of graphs only.
Theorem 4. For odd , the graph acquires an labeling admitting .
Proof. Consider a graph with vertex and edge sets:If and , we sketch a labeling as follows:The edge-sums’ sets constituted by the abovementioned design generates a consecutive sequence of integer . Under the shadow of Lemma 1, accredits an labeling of having .
Theorem 5. For odd , the graph acquires an labeling having .
Proof. Consider the graph , for odd , with the following vertex-edge connections:Here, order is and size is . Now, we design a labeling as follows:The edge-sums’ set constituted by the scheme generates a sequence consisting of consecutive integer . Under the shadow of Lemma 1, constitutes an labeling of with magic sum .
Theorem 6. For odd , the graph acquires an labeling having .
Proof. Let , where is odd, having vertex and edge sets interlinked:We have and . A labeling is being defined as follows:The edge-sums’ set constituted by the scheme generates a sequence consisting of consecutive integer . Under the shadow of Lemma 1, constitutes an of the graph with magic constant .
Theorem 7. For odd , possesses labeling having .
Proof. With taken odd, consider , having vertex-edge connections:Here, we have and . A labeling function is being defined as follows:The edge-sums’ set constituted by the scheme generates a sequence consisting of consecutive integer . Lemma 1 implies that extends to an labeling of with .
Theorem 8. For odd , possesses an labeling with .
Proof. Consider the graph with vertex-edge connections as follows:Then, and . Again, a labeling is being defined as follows:The edge-sums’ set constituted by the scheme generates a sequence consisting of consecutive integer . Under the shadow of Lemma 1, constitutes to an labeling of admitting .
The following results are direct consequences of Lemma 2, from Theorems 4–8.
Theorem 9. For odd , the graph admits an labeling.
Theorem 10. For odd , the graph admits an labeling.
Theorem 11. For odd , the graph admits an labeling.
Theorem 12. For odd , admits an labeling.
Theorem 13. For odd , admits an labeling.
2.3. Examples and Proposed Open Problems
An labeling of the graph is being presented in Figure 1(a), corresponding to the parameters and . Furthermore, Figure 1(b) presents an labeling of corresponding to and . Here, it can be observed that the value of the magic constant is perfect as per our depiction in Theorem 1.
(a)
(b)
Figures 2(a) and 2(b) illustrate Theorem 2 by providing and labeling of the graph .
(a)
(b)
Figures 3–7 are the illustrations of Theorems 4–8, respectively, for particular values of the parameters involved.
The open problems related to Section 2.2 are as follows:(i)For even , determine any labeling of (ii)For even , determine any labeling of (iii)For even , determine any labeling of (iv)For even , determine any labeling of (v)For even , determine any labeling of
3. Conclusion
In this article,(i)We have obtained and labeling of a pancyclic class of graphs, namely, Usmanian pancyclic graph, denoted by .(ii)We have exhibited the existence of and labeling on disjoint copies of with various trees. Specifically, and , whereas itself is not . These obtained results open a new direction for researchers to derive labeling of disjoint copies of .(iii)A few open problems have also been proposed for future work in this area.
Data Availability
The whole data are included within this article. However, the reader may contact the corresponding author for more details on the data.
Conflicts of Interest
The authors declare no conflicts of interest.