# Orientable Group Distance Magic Labeling of Directed Graphs

Authors: Wasim Ashraf; Hani Shaker; Muhammad Imran
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

A directed graphG is said to have the orientable group distance magic labeling if there exists an abelian group ℋ and one-one map ℓ from the vertex set of G to the group elements, such that ∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=μ for all x∈V, where NGx is the open neighborhood of x, and μ∈ℋ is the magic constant; more specifically, such graph is called orientable ℋ-distance magic graph. In this study, we prove directed antiprism graphs are orientable ℤ2n, ℤ2×ℤn, and ℤ3×ℤ6m-distance magic graphs. This study also concludes the orientable group distance magic labeling of direct product of the said directed graphs.


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## Body

## 1. Introduction

Graph labeling is a process of assigning the labels by elements from certain set to the vertices or edges or both subject to certain conditions. For any graphG of order n, the distance magic labeling (also called sigma labeling) is defined as a bijection λ:VG⟶1,2,3,…,n, such that for every x∈V,(1)wx=∑y∈NGxλy=k,where NGx is the neighborhood of vertex x defined as the set of vertices adjacent to x, wx is the weight of each vertex of the graph G, and k is the positive integer called magic constant [1, 2]. Motivated from the idea of distance magic labeling, in 2013 [3, 4], the group distance magic labeling (GDML) was introduced by Dalibor Froncek.For a given graphG of order n and an abelian group ℋ of order n, the group distance magic labeling is a one-one map ℓ:VG⟶ℋ, such that for every x∈V,(2)wx=∑y∈NGxλy=μ,where μ∈ℋ. Generally, we can say that elements of an abelian group are used to assign the labels to the vertices of the graph G. In 2015, Marcin Anholcer et al. did some remarkable work on GDML of direct product of graphs [5]. Motivated from the idea of group distance magic labeling, in 2017, the orientable group distance magic labeling (OGDML) was introduced by Brayn Freyberg. OGDML defining manner is the same as GDML, but the difference is in computing weight of vertices, that is, weight of a vertex is defined as ∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y. In [6, 7], the authors provided many important results regarding the existence and nonexistence of OGDML of many graphs which include the existence of OGDML of complete graph on odd vertices, the existence of OGDML of Cartesian product of Cm and Cn in case of m,n≥3, and nonexistence of OGDML of a graph with order n≡2mod4 having only odd degree vertices. In [8], the strong product of Cm and Cn is proved as orientable ℤmn-DM graph for d=gcdm,n=3,5, or 6 if d2∣m and d2∣n; moreover, it is also proved that the graph Cm⊠Cn is orientable ℤmn-DM graph if mn≡2 (mod 4) or if m≡n≡2 (mod 4). Sylwia Cichacz in [7] proved G×H is orientable ℤ2nt-DM graph, where H is the circulant graph C2n1,3,5,…,2n/2−1 and G is the Eulerian graph of order t. They proved Kn,n is orientable ℤ2n-DM graph iff n is even. According to them, the complete K-partite graph Kn1,n2,…,nk (1≤n1≤n2≤…≤nk and n=n1+n2+…+nk is odd) is orientable ℤn-DM graph if n2≥2. If H=Km1,m2 with m=m1+m2, then H is orientable ℤm-DML if m≅2 (mod 4). If H=Km1,m2,m3 with m=m1+m2+m3, then H is orientable ℤm-DML for all m1,m2,m3. Recently, in [9], the authors provided the necessary and sufficient conditions for lexicographic product of Km and Kn¯ being orientable ℤmn-distance magic. They also proved that prism graph of order 2n is not an orientable ℤ2n-distance magic graph.In this study, we target the antiprism family of graphs for finding the orientable group distance magic labeling with respect to the modulo group and the product of modulo groups. We present the orientableℤ2n and ℤ2×ℤn-distance magic labeling for the antiprism. We also provide ℤ2n and ℤ2×ℤn-distance magic labeling for the direct product of the antiprism graphs. The direct product G×H of graphs G and H is a graph with vertex set VG×VH and edge set as follows:(3)EG×H=u,vu′,v′u,v∈VG,u′,v′∈VH,uu′∈EG,vv′∈EH,that is, any two vertices u,v and u′,v′ are the adjacent in G×H if and only if u is adjacent to u′ in G and v is adjacent to v′ in H [6].


## 2. Main Results

### 2.1. Orientable Group Distance Magic Labeling of Antiprism Graph

We provide the OGDML of directed antiprism graph of order2n in the following theorems.Before we provide our main results, the vertex and edge representations of directed antiprism graphAn are given as(4)VAn=xi,yi|0≤i≤n−1,EAn=xixi−1,yiyi−1,xiyi,yixi−1|1≤i≤n−1∪x0xn−1,y0yn−1,x0y0,y0xn−1.In the representation of edge set, any edgeuivj is directed and direction is from ui to uj, as shown in Figure 1.Figure 1
Directed antiprismA4.Theorem 1.
LetG≅An, where An is the directed antiprism graph and ℤ2n be the module 2n group; then, G admits orientable ℤ2n-distance magic labeling.LetAn be the directed antiprism graph, and the orientation is anticlockwise. We know that An is a 4-regular graph of order 2n. We use the vertex and edge representations as given in the start of this section. Consider ℓ⟶:VG⟶ℤ2n that is defined as follows:Case (i). Ifn is even, then the labeling of each vertex of graph An is given as(5)ℓ⟶xi=2i,for0≤i≤n−1,ℓ⟶yj=2j+1,for0≤j≤n−1.Case (ii). Ifn is odd, then the labeling of each vertex of graph An is given as(6)ℓ⟶xi=2i+1,for0≤i≤n−1,ℓ⟶yj=2j,for0≤j≤n−1.Underℓ⟶, An is orientable ℤ2n-distance magic graph with magic constant(7)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=2n−6.Theorem 2.
LetG≅An, where An is the directed antiprism graph, where n even. Then, G admits orientable ℤ2×ℤn-distance magic labeling.We use already provided vertex and edge representation ofAn and define ℓ⟶:VG⟶ℤ2×ℤn as follows:(8)ℓ⟶xi=0,i,for0≤i≤n−1,ℓ⟶yj=1,j,for0≤j≤n−1.Underℓ⟶, An is orientable ℤ2×ℤn-distance magic graph with magic constant:(9)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,n−3.Remark: forn odd, ℤ2n≅ℤ2×ℤn, so Theorem 1 covers the case for odd n.Theorem 3.
LetG≅An, where An is the directed antiprism graph, such that n=9m,m≥1 and m≠3k,k≥1. Then, G admits orientable ℤ3×ℤ6m-distance magic labeling.We use already provided vertex and edge representation ofAn and define ℓ⟶:VG⟶ℤ3×ℤ6m as follows:(10)ℓ⟶xi=i−1mod3,2imod6m,for1+9t≤i≤3+9t,t≥0,i−2mod3,2imod6m,for4+9t≤i≤6+9t,t≥0,i−3mod3,2imod6m,for7+9t≤i≤9+9tmodn,t≥0.(11)ℓ⟶yj=j+2mod3,2j+1mod6m,for,9t≤j≤2+9t,t≥0,j+4mod3,2j+1mod6m,for,3+9t≤j≤5+9t,t≥0,j+6mod3,2j+1mod6m,for,6+9t≤i≤8+9t,t≥0.Underℓ⟶, An is orientable ℤ3×ℤ6m-distance magic graph with magic constant:(12)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=1,6m−1.


### 2.2. Group Distance Magic Labeling of Direct Product of Antiprism Graphs

Now, we provide OGDML for the direct product of antiprism graphs in following theorems.Theorem 4.
LetG≅Am and H≅An, where Am and An be the directed antiprism graphs such that m≤n and ℤ4mn be the module group of order 4mn. Then, the graph G×H admits the orientable ℤ4mn-distance magic labeling for all m,n≥3.We use the following vertex and edge representative ofAm and An as(13)VAm=xi,yi|0≤i≤m−1,EAm=xixi−1,yiyi−1,xiyi,yixi−1|1≤i≤m−1∪x0xm−1,y0ym−1,x0y0,y0xm−1,VAn=xi′,yi′|0≤i≤n−1,EAn=xixi−1,yiyi−1,xiyi,yixi−1|1≤i≤n−1∪x0xn−1,y0yn−1,x0y0,y0xn−1.Each edge is written in strict ordering manner as it shows the direction of the edge as well. By using the definition of the direct product, we have the following vertex representative ofAm×An,(14)VAm×An=xi,yi,xj′,yj′|0≤i≤m−1,0≤j≤n−1.Let us defineℓ⟶:VAm×An⟶ℤ4mn as follows:(15)ℓ⟶xi,xj′=4ni+2j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶xi,yj′=4ni+2j+n,for0≤i≤m−1,0≤j≤n−1,ℓ⟶yi,xj′=4n−1+22ni−j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶yi,yj′=4n−1+2n2i−1−j,for0≤i≤m−1,0≤j≤n−1.Then, underℓ⟶, Am×An is orientable ℤ4mn-distance magic graph with magic constant:(16)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,form=3ormis even,3nm−1,form=5,9,n≥5,8nm−6,form=7,11,n≥7,4nm−12,form≥13,mis odd,n≥13,Theorem 5.
LetG≅Am and H≅An, where Am and An be the directed antiprism graphs, such that m≤n and ℤ2×ℤ2mn be the module group of order 4mn. Then, the graph G×H admits the orientable ℤ2×ℤ2mn-distance magic labeling for all m,n≥3.We use the vertex and edge representation ofAn given in Theorem 4 and define ℓ⟶:VAm×An⟶ℤ2×ℤ2mn as follows:(17)ℓ⟶xi,xj′=0,2ni+j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶xi,yj′=0,n2i+1+j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶yi,xj′=1,2n1+i−1−j,for0≤i≤m−1,0≤j≤n−1ℓ⟶yi,yj′=1,n1+2i−1−j,for0≤i≤m−1,0≤j≤n−1.Then, underℓ⟶, Am×An is orientable ℤ2×ℤ2mn-distance magic graph with magic constant:(18)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,0,form=3ormis even,0,32nm−1,form=5,9n≥5,0,4nm−6,form=7,11n≥7,0,2nm−12,form≥13,mis oddn≥13.Theorem 6.
LetG≅A3 and H≅An, where A3 and An be the directed antiprism graphs, such that n=3m, m≥1, and m≠3k, k≥1. ℤ3×ℤ4n be the module group of order 12n. Then, the graph G×H admits the orientable ℤ3×ℤ4n-distance magic labeling.We use the vertex and edge representation ofAn given in Theorem 4 and define ℓ⟶:VA3×An⟶ℤ3×ℤ4n as follows:(19)ℓ⟶xi,xj′=i,2j,for0≤i≤2,0≤j≤n−1,ℓ⟶xi,yj′=i,2n+2ni+jmod4n,for0≤i≤2,0≤j≤n−1,ℓ⟶yi,xj′=i,22n+2ni−j−1mod4n,for0≤i≤2,0≤j≤n−1,ℓ⟶yi,yj′=i,2n+2ni−j−1mod4n,for0≤i≤2,0≤j≤n−1.Then, underℓ⟶, Am×An is orientable ℤ2×ℤ2mn-distance magic graph with magic constant:(20)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,0.


### 2.3. Open Problems

(1)
Figure out all the abelian groupsℋ for which antiprism is orientable ℋ-distance magic graph(2)
Figure out all the abelian groupsℋ for which the direct product of antiprisms is orientable ℋ-distance magic graph(3)
Does there exist any abelian groupsℋ for which antiprism and direct product of antiprism is not orientable ℋ-distance magic?


## 2.1. Orientable Group Distance Magic Labeling of Antiprism Graph

We provide the OGDML of directed antiprism graph of order2n in the following theorems.Before we provide our main results, the vertex and edge representations of directed antiprism graphAn are given as(4)VAn=xi,yi|0≤i≤n−1,EAn=xixi−1,yiyi−1,xiyi,yixi−1|1≤i≤n−1∪x0xn−1,y0yn−1,x0y0,y0xn−1.In the representation of edge set, any edgeuivj is directed and direction is from ui to uj, as shown in Figure 1.Figure 1
Directed antiprismA4.Theorem 1.
LetG≅An, where An is the directed antiprism graph and ℤ2n be the module 2n group; then, G admits orientable ℤ2n-distance magic labeling.LetAn be the directed antiprism graph, and the orientation is anticlockwise. We know that An is a 4-regular graph of order 2n. We use the vertex and edge representations as given in the start of this section. Consider ℓ⟶:VG⟶ℤ2n that is defined as follows:Case (i). Ifn is even, then the labeling of each vertex of graph An is given as(5)ℓ⟶xi=2i,for0≤i≤n−1,ℓ⟶yj=2j+1,for0≤j≤n−1.Case (ii). Ifn is odd, then the labeling of each vertex of graph An is given as(6)ℓ⟶xi=2i+1,for0≤i≤n−1,ℓ⟶yj=2j,for0≤j≤n−1.Underℓ⟶, An is orientable ℤ2n-distance magic graph with magic constant(7)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=2n−6.Theorem 2.
LetG≅An, where An is the directed antiprism graph, where n even. Then, G admits orientable ℤ2×ℤn-distance magic labeling.We use already provided vertex and edge representation ofAn and define ℓ⟶:VG⟶ℤ2×ℤn as follows:(8)ℓ⟶xi=0,i,for0≤i≤n−1,ℓ⟶yj=1,j,for0≤j≤n−1.Underℓ⟶, An is orientable ℤ2×ℤn-distance magic graph with magic constant:(9)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,n−3.Remark: forn odd, ℤ2n≅ℤ2×ℤn, so Theorem 1 covers the case for odd n.Theorem 3.
LetG≅An, where An is the directed antiprism graph, such that n=9m,m≥1 and m≠3k,k≥1. Then, G admits orientable ℤ3×ℤ6m-distance magic labeling.We use already provided vertex and edge representation ofAn and define ℓ⟶:VG⟶ℤ3×ℤ6m as follows:(10)ℓ⟶xi=i−1mod3,2imod6m,for1+9t≤i≤3+9t,t≥0,i−2mod3,2imod6m,for4+9t≤i≤6+9t,t≥0,i−3mod3,2imod6m,for7+9t≤i≤9+9tmodn,t≥0.(11)ℓ⟶yj=j+2mod3,2j+1mod6m,for,9t≤j≤2+9t,t≥0,j+4mod3,2j+1mod6m,for,3+9t≤j≤5+9t,t≥0,j+6mod3,2j+1mod6m,for,6+9t≤i≤8+9t,t≥0.Underℓ⟶, An is orientable ℤ3×ℤ6m-distance magic graph with magic constant:(12)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=1,6m−1.


## 2.2. Group Distance Magic Labeling of Direct Product of Antiprism Graphs

Now, we provide OGDML for the direct product of antiprism graphs in following theorems.Theorem 4.
LetG≅Am and H≅An, where Am and An be the directed antiprism graphs such that m≤n and ℤ4mn be the module group of order 4mn. Then, the graph G×H admits the orientable ℤ4mn-distance magic labeling for all m,n≥3.We use the following vertex and edge representative ofAm and An as(13)VAm=xi,yi|0≤i≤m−1,EAm=xixi−1,yiyi−1,xiyi,yixi−1|1≤i≤m−1∪x0xm−1,y0ym−1,x0y0,y0xm−1,VAn=xi′,yi′|0≤i≤n−1,EAn=xixi−1,yiyi−1,xiyi,yixi−1|1≤i≤n−1∪x0xn−1,y0yn−1,x0y0,y0xn−1.Each edge is written in strict ordering manner as it shows the direction of the edge as well. By using the definition of the direct product, we have the following vertex representative ofAm×An,(14)VAm×An=xi,yi,xj′,yj′|0≤i≤m−1,0≤j≤n−1.Let us defineℓ⟶:VAm×An⟶ℤ4mn as follows:(15)ℓ⟶xi,xj′=4ni+2j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶xi,yj′=4ni+2j+n,for0≤i≤m−1,0≤j≤n−1,ℓ⟶yi,xj′=4n−1+22ni−j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶yi,yj′=4n−1+2n2i−1−j,for0≤i≤m−1,0≤j≤n−1.Then, underℓ⟶, Am×An is orientable ℤ4mn-distance magic graph with magic constant:(16)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,form=3ormis even,3nm−1,form=5,9,n≥5,8nm−6,form=7,11,n≥7,4nm−12,form≥13,mis odd,n≥13,Theorem 5.
LetG≅Am and H≅An, where Am and An be the directed antiprism graphs, such that m≤n and ℤ2×ℤ2mn be the module group of order 4mn. Then, the graph G×H admits the orientable ℤ2×ℤ2mn-distance magic labeling for all m,n≥3.We use the vertex and edge representation ofAn given in Theorem 4 and define ℓ⟶:VAm×An⟶ℤ2×ℤ2mn as follows:(17)ℓ⟶xi,xj′=0,2ni+j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶xi,yj′=0,n2i+1+j,for0≤i≤m−1,0≤j≤n−1,ℓ⟶yi,xj′=1,2n1+i−1−j,for0≤i≤m−1,0≤j≤n−1ℓ⟶yi,yj′=1,n1+2i−1−j,for0≤i≤m−1,0≤j≤n−1.Then, underℓ⟶, Am×An is orientable ℤ2×ℤ2mn-distance magic graph with magic constant:(18)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,0,form=3ormis even,0,32nm−1,form=5,9n≥5,0,4nm−6,form=7,11n≥7,0,2nm−12,form≥13,mis oddn≥13.Theorem 6.
LetG≅A3 and H≅An, where A3 and An be the directed antiprism graphs, such that n=3m, m≥1, and m≠3k, k≥1. ℤ3×ℤ4n be the module group of order 12n. Then, the graph G×H admits the orientable ℤ3×ℤ4n-distance magic labeling.We use the vertex and edge representation ofAn given in Theorem 4 and define ℓ⟶:VA3×An⟶ℤ3×ℤ4n as follows:(19)ℓ⟶xi,xj′=i,2j,for0≤i≤2,0≤j≤n−1,ℓ⟶xi,yj′=i,2n+2ni+jmod4n,for0≤i≤2,0≤j≤n−1,ℓ⟶yi,xj′=i,22n+2ni−j−1mod4n,for0≤i≤2,0≤j≤n−1,ℓ⟶yi,yj′=i,2n+2ni−j−1mod4n,for0≤i≤2,0≤j≤n−1.Then, underℓ⟶, Am×An is orientable ℤ2×ℤ2mn-distance magic graph with magic constant:(20)∑y∈NG+xℓ⟶y−∑y∈NG−xℓ⟶y=0,0.


## 2.3. Open Problems

(1)
Figure out all the abelian groupsℋ for which antiprism is orientable ℋ-distance magic graph(2)
Figure out all the abelian groupsℋ for which the direct product of antiprisms is orientable ℋ-distance magic graph(3)
Does there exist any abelian groupsℋ for which antiprism and direct product of antiprism is not orientable ℋ-distance magic?


## 3. Conclusion

Orientable group distance magic labeling (OGDML) links directed graph theory with groups. This fact leads us to define the relation between groupℤ2n and antiprism graph of order 2n through OGDML. Our results also provide the association of ℤ2×ℤn and ℤ3×ℤ6m with antiprisms in OGDML perspective. We also extend our work from OGDML of antiprism graphs to OGDML of direct product of antiprism graph and prove that it is orientable ℤ4mn to ℤ2×ℤ2mn and ℤ3×ℤ4n-distance magic. At the end, we proposed some open problems.



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