Abstract
Let be a graph and be subgraph of . The graph is said to be - antimagic total graph if there exists a bijective function such that, for all subgraphs isomorphic to , the total weights forms an arithmetic sequence , where and are positive integers and is the number of subgraphs isomorphic to . An - antimagic total labeling is said to be super if the vertex labels are from the set . In this paper, we discuss super --antimagic total labeling for generalized antiprism and a super --antimagic total labeling for toroidal octagonal map.
1. Introduction
All the graphs that we consider in this works are finite, simple, and connected. Let be a graph with vertex set and edge set denoted by and , respectively. For the cardinality of vertex set and edge set, we use the notation and , respectively. For basic definitions and terminology related to graph theory, the readers can see the book by Gross et al. [1].
A graph labeling is a map that sends some of the graph elements (vertices or edges or both) to the set of positive integers. If the domain set of is the set of vertices (edges), then is called vertex (edge) labeling. If the domain set is , then is called total labeling. Let be a graph and be subgraphs of . We say that the graph has an covering if each edge of belongs to at least one of the subgraph , where . If all , , are isomorphic to a graph , then such a covering is called covering of . Suppose that a graph admits an covering. The graph is called antimagic if there exists a bijective function such that, for all subgraphs isomorphic to , the total weights,form an arithmetic sequence , where and are positive integers and is the number of subgraphs isomorphic to . An - antimagic total labeling is said to be super if the vertex labels are from the set . If , then is called - antimagic.
Kotzig and Rosa [2] and Enomoto et al. [3] introduced the concept of edge-magic and super edge-magic labeling. Gutierrez and Llado [4] first studied the (super) magic coverings of a graph . They proved that the cycle and path are super magic for some . The cycle (super) magic behavior of some classes of connected graphs is studied in Llado et al. [5]. They proved that prisms, windmills, wheels, and books are -magic for some . Maryati et al. [6] investigated the -supermagicness of a disjoint union of copies of a graph and showed that the disjoint union of any paths is -supermagic for some and . Maryati et al. [7] and Salman et al. [8] proved that certain families of trees are path-supermagic. Ngurah et al. [9] proved that triangles, chains, ladders, wheels, and grids are cycle-supermagic.
Inaya et al. [10] firstly introduced the concept of -magic decomposition and -antimagic decomposition. They showed that, for any graceful tree with edges, the complete graph admits antimagic decomposition for some and all even differences . They also proved that if any tree with edges admits labeling, then the complete bipartite graph admits an antimagic decomposition for some and having same parity as . The condition on the existence of super magic decomposition of complete partite graph and its copies were given by Lian [11]. The H-supermagic decomposition of antiprisms is described by Hendy in [12] and the H-supermagic decompositions of the lexicographic product of graphs are discussed by Hendy et al. in [13]. In [14], Hendy et al. examined the existence of super magic labeling for toroidal grids and toroidal triangulations. Recently, Fenovcikova et al. [15] proved that wheels are cycle antimagic.
In this paper, we discuss the Super --antimagic total labeling for generalized antiprism and a Super --antimagic total labeling for toroidal octagonal map. We proved that the generalized antiprism admits --antimagic total labeling for and the toroidal octagonal map admits a Super --antimagic total labeling, for .
2. Results on Super --Antimagic Total Covering of Generalized Antiprism
An -sided generalized antiprism is defined as a polyhedron which is composed of parallel copies of some particular -sided polygon and connected by an alternating band of triangles. Figure 1 represents the labeled graph of generalized antiprism . We denote its vertex set and edge set by and , respectively. The vertex set and the edge set of the generalized antiprism can be defined as follows:
The generalized antiprism admits a covering. Let and be the cycles which cover , where and . The cycles and can be defined as
It is easy to observe that and . We first give an upper bound for such that admits a super --antimagic covering.
Theorem 1. Let and be generalized antiprism graph. Then, there is no super --antimagic covering with .
Proof. Suppose that has a super --antimagic covering. Let be a super --antimagic covering and be the set of weights. The minimum weight on cycle is at least which is the sum of the smallest vertex labels () and sum of smallest edge labels (). Thus,On the contrary, the maximum possible -weight is the sum of three largest possible vertex labels, namely, , and three the largest possible edge labels from the set, . Hence, we haveFrom (4) and (5), an upper bound for the parameter can be obtained asThus, we have arrived at the desired result.
Theorem 2. Let ; then, the generalized antiprism admits a super --antimagic total covering.
Proof. Let be a total labeling of generalized antiprism defined as follows:Under the labeling , the weights of 3- cycles areAnd, the weights of 3-cycles areObserve that the weights and of all cycles and are equal, and therefore, the resulting labeling is super - total labeling.
Theorem 3. Let ; then, the generalized antiprism admits a super -antimagic total covering.
Proof. Let be a total labeling of generalized antiprism defined as follows.
For , the label on vertices is defined asFor , the label on vertices is defined asFor , the label on edges is defined asFor , the label on edges is defined asThe label on edges is defined asAnd, the label on edges is defined asUnder the labeling , the weights of 3-cycle areFor , we haveFor , we haveThe weight of 3-cycle areFor , we haveFor , we haveObserve that the weights and form an arithmetic progression with common difference 2 starting from and ending at . This implies that the defined labeling is a super --antimagic total covering.
3. Results on Super --Antimagic Total Covering of Toroidal Octagonal Planner Map
A planar octagonal map is a graph obtained by joining octagons and squares in such a way that they cover the plane. To obtain the toroidal octagonal map, we apply torus identification on octagonal planner map. We denote the toroidal octagonal map with rows and column of octagons by , where . The planar representation of is depicted in Figure 2. The vertex set and the edge set of octagonal planner map can be defined as follows:
From the above sets, we have and . We can cover the toroidal octagonal map by the 8-sided cycles . For and , the vertex set and edge set of 8-sided cycles can be defined as
We start by giving an upper bound for such that admits a super --antimagic covering.
Theorem 4. Suppose admits a super --antimagic covering; then, .
Proof. Suppose admits a super --antimagic covering. Then, the weight on cycle is atleastand the largest weight of is atmostThus, we have
In the next two theorems, we show that toroidal octagonal map admits a super --antimagic covering for .
Theorem 5. Let ; then, the toroidal octagonal map is super --antimagic for .
Proof. Define a total labeling , where as follows:The total labeling labels the vertices from the set and the edges from the set . For and , the weight of cycles under isFor the case , we have weights’ set ; similarly, for cases , we get the weights from the sets , , and , respectively. Hence, the weights of cycles form an arithmetic sequence with difference , and 7.
Theorem 6. Let ; then, the toroidal map is super --antimagic for .
Proof. Let and . We define a total labeling of as follows:The total labeling labels the vertices from the set and edges from the set . This show that is a bijection from set to set . For and , the weights of under the labeling areFor the case , we have weights from the set . Similarly, for cases , we get weights from the sets and , respectively. This showed that weights of the cycles form an arithmetic sequence with difference , and 6.
4. Conclusion
In the present paper first, we constructed an upper bound for the parameter for super --antimagic covering. Secondly, we examined the existence of super --antimagic labeling of generalized antiprism . We showed that, for the generalized antiprism had --antimagic covering for . Thirdly, we constructed an upper bound for the parameter for super --antimagic covering. Finally, we examined the existence of super --antimagic labeling of torodial map . We showed that, for , the torodial octagonal map had --antimagic covering for . We conclude the paper with the following open problems. Open problem 1: find other possible bound for parameter under --antimagic total covering and the corresponding remaining labeling of for generalized antiprism Open problem 2: find other possible bound for parameter under --antimagic total covering and the corresponding remaining labeling of for torodial octagonal map
Data Availability
No data were used to support the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.