# SuperH-Antimagic Total Covering for Generalized Antiprism and Toroidal Octagonal Map

Authors: Amir Taimur; Gohar Ali; Muhammad Numan; Adnan Aslam; Kraidi Anoh Yannick
Journal: Journal of Mathematics
 (2021)

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## Abstract

LetG be a graph and H⊆G be subgraph of G. The graph G is said to be a,d-H antimagic total graph if there exists a bijective function f:VH∪EH⟶1,2,3,…,VH+EH such that, for all subgraphs isomorphic to H, the total H weights WH=WH=∑x∈VHfx+∑y∈EHfy forms an arithmetic sequence a,a+d,a+2d,…,a+n−1d, where a and d are positive integers and n is the number of subgraphs isomorphic to H. An a,d-H antimagic total labeling f is said to be super if the vertex labels are from the set 1,2,…,|VG. In this paper, we discuss super a,d-C3-antimagic total labeling for generalized antiprism and a super a,d-C8-antimagic total labeling for toroidal octagonal map.


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## Body

## 1. Introduction

All the graphs that we consider in this works are finite, simple, and connected. LetG be a graph with vertex set and edge set denoted by VG and EG, respectively. For the cardinality of vertex set and edge set, we use the notation VG and EG, 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 mapf that sends some of the graph elements (vertices or edges or both) to the set of positive integers. If the domain set of f is the set of vertices (edges), then f is called vertex (edge) labeling. If the domain set is VG∪EG, then f is called total labeling. Let G be a graph and H1,H2,…,Hk be subgraphs of G. We say that the graph G has an H1,H2,…,Hk covering if each edge of G belongs to at least one of the subgraph Hi, where 1≤i≤k. If all Hi, i=1,2,…,k, are isomorphic to a graph H, then such a covering is called H covering of G. Suppose that a graph G admits an H covering. The graph G is called a,dH antimagic if there exists a bijective function f:VH∪EH⟶1,2,3,…,VH+EH such that, for all subgraphs isomorphic to H, the total H weights,(1)WH=WK=∑x∈VKfx+∑y∈EKfy,form an arithmetic sequence a,a+d,a+2d,…,a+n−1d, where a and d are positive integers and n is the number of subgraphs isomorphic to H. An a,d-H antimagic total labeling f is said to be super if the vertex labels are from the set 1,2,…,|VG. If d=0, then H is called a,d-H 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 H (super) magic coverings of a graph G. They proved that the cycle Cn and path Pn are Pm super magic for some m. 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 Cm-magic for some m. Maryati et al. [6] investigated the G-supermagicness of a disjoint union of c copies of a graph G and showed that the disjoint union of any paths is cPm-supermagic for some c and m. 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 H-magic decomposition and H-antimagic decomposition. They showed that, for any graceful tree T with n edges, the complete graph K2n+1 admits a,d−T antimagic decomposition for some a and all even differences 0≤d≤n+1. They also proved that if any tree T with n edges admits α labeling, then the complete bipartite graph Kn,n admits an a,d−T antimagic decomposition for some a and d having same parity as n. The condition on the existence of C2k super magic decomposition of complete n 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 a,d−H 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 Supera,d-C3-antimagic total labeling for generalized antiprism and a Super a,d-C8-antimagic total labeling for toroidal octagonal map. We proved that the generalized antiprism Ars admits a,d-C3-antimagic total labeling for d=0,1 and the toroidal octagonal map Osr admits a Super a,d-C8-antimagic total labeling, for d=1,2,…,7.


## 2. Results on Supera,d-C3-Antimagic Total Covering of Generalized Antiprism Ars

Anr-sided generalized antiprism Ars is defined as a polyhedron which is composed of s parallel copies of some particular r-sided polygon and connected by an alternating band of triangles. Figure 1 represents the labeled graph of generalized antiprism Ars. We denote its vertex set and edge set by VArs and EArs, respectively. The vertex set and the edge set of the generalized antiprism Ars can be defined as follows:(2)VArs=xij,for0≤i≤r−1,0≤j≤s−1,EArs=xijxi+1j,for0≤i≤r−1,0≤j≤s−1∪xijxij+1,for0≤i≤r−1,0≤j≤s−2∪xijxi+1j+1,for0≤i≤r−1,0≤j≤s−2.Figure 1
Generalized antiprismArs.The generalized antiprismArs admits a C3 covering. Let zij and fij be the C3 cycles which cover Ars, where 0≤i≤r−1 and 0≤j≤s−2. The cycles zij and fij can be defined as(3)zij=xijxi+1jxi+1j+1xij,for0≤i≤r−1,0≤j≤s−2,fij=xijxi+1j+1xij+1xij,for0≤i≤r−1,0≤j≤s−2.It is easy to observe thatVArs=rs and EArs=3rs−2r. We first give an upper bound for d such that Ars admits a super a,d-C3-antimagic covering.Theorem 1.
Letr,s≥3 and Ars be generalized antiprism graph. Then, there is no super a,d-C3-antimagic covering with d≥6.Proof.
Suppose thatArs has a super a,d-C3-antimagic covering. Let f:VArs∪EArs⟶1,2,3,…,4rs−2r be a super a,d-C3-antimagic covering and a3,a3+d,a3+2d,…,a3+2rs−2r−1d be the set of C3 weights. The minimum weight on cycle C3 is at least 12+3rs which is the sum of the smallest vertex labels (1,2,3) and sum of smallest edge labels (rs+1,rs+2,rs+3). Thus,(4)a3≥12+3rs.
On the contrary, the maximum possibleC3-weight is the sum of three largest possible vertex labels, namely, rs−2,rs−1,rs, and three the largest possible edge labels from the set, 4rs−2r−2,4rs−2r−1,4rs−2r. Hence, we have(5)a3+2rs−2r−1d≤15rs−6r−6.
From (4) and (5), an upper bound for the parameter d can be obtained as(6)d≤12rs−16r−182rs−2r−1,d≤6−4r+62rs−2r−1,d≤6.
Thus, we have arrived at the desired result.Theorem 2.
Letr,s≥3; then, the generalized antiprism Ars admits a super 9rs−3r+4,o-C3-antimagic total covering.Proof.
Letϕ:VArs∪EArs⟶1,2,3,…,4rs−2r be a total labeling of generalized antiprism Ars defined as follows:(7)ϕxij=jr+1+i,for0≤i≤r−1,0≤j≤s−1,ϕxijxi+1j=2s−jr−i,for0≤i≤r−1,0≤j≤s−1,ϕxijxij+1=3s−2−jr+r−i,for0≤i≤r−1,0≤j≤s−2,ϕxijxi+1j+1=4s−3−jr+r−i,for0≤i≤r−2,0≤j≤s−2,4s−3−jr+1,fori=r−1,0≤j≤s−2.
Under the labelingϕ, the weights of 3- cycles zij are(8)Wzij=ϕxij+ϕxi+1j+ϕxi+1j+1+ϕxijxi+1j+ϕxi+1jxi+1j+1+ϕxijxi+1j+1,Wzij=9rs−3r+4,for0≤i≤r−1,0≤j≤s−2.
And, the weights of 3-cyclesfij are(9)Wfij=ϕxij+ϕxi+1j+1+ϕxij+1+ϕxijxi+1j+1+ϕxi+1j+1xij+1+ϕxij+1xij,Wfij=9rs−3r+4,for0≤i≤r−1,0≤j≤s−2.
Observe that the weightsWzij and Wfij of all cycles zij and fij are equal, and therefore, the resulting labeling is super 9rs−3r+4,0-C3 total labeling.Theorem 3.
Letr,s≥3; then, the generalized antiprism Ars admits a super 7rs+4,2-antimagic total covering.Proof.
Letχ:VArs∪EArs⟶1,2,3,…,4rs−2r be a total labeling of generalized antiprism Ars defined as follows.
Forj=even, the label on vertices xij is defined as(10)χxij=1+i,for0≤i≤r−1,j=0,j+1r,fori=0,2≤j≤s−1,jr+i,for1≤i≤r−1,2≤j≤s−1.
Forj=odd, the label on vertices xij is defined as(11)χxij=jr+1,fori=0,1≤j≤s−1,j+1r+1−i,for1≤i≤r−1,1≤j≤s−1.
Forj=even, the label on edges xijxi+1j is defined as(12)χxijxi+1j=rs+1+i,for0≤i≤r−1,j=0,rs+j+1r,fori=0,2≤j≤s−1,rs+jr+i,for1≤i≤r−1,2≤j≤s−1.
Forj=odd, the label on edges xijxi+1j is defined as(13)χxijxi+1j=rs+jr+1,fori=0,1≤j≤s−1,rs+j+1r+1−i,for1≤i≤r−1,1≤j≤s−1.
The label on edgesxijxij+1 is defined as(14)χxijxij+1=3s−2r+1+i,for0≤i≤r−1,j=0,3s−1−jr,fori=0,1≤j≤s−1,3s−2−jr+i,for1≤i≤r−1,1≤j≤s−1.
And, the label on edgesxijxi+1j+1 is defined as(15)χxijxi+1j+1=3rs+jr−i,for0≤i≤r−1,0≤j≤s−2.
Under the labelingχ, the weights of 3-cycle zij are(16)Wzij=χxij+χxi+1j+χxi+1j+1+χxijxi+1j+χxi+1jxi+1j+1+χxijxi+1j+1.
Forj=even, we have(17)Wzij=7rs+8+2i,for0≤i≤r−2,j=0,7rs+4,fori=r−1,j=0,7rs+4jr+2r+2,fori=0,2≤j≤s−2,7rs+4jr+2+2i,for1≤i≤r−1,2≤j≤s−2.
Forj=odd, we have(18)Wzij=7rs+4jr+4,fori=0,1≤j≤s−2,7rs+4jr+2r+4−2i,for1≤i≤r−1,1≤j≤s−2.
The weight of 3-cyclefij are(19)Wfij=χxij+χxi+1j+1+χxij+1+χxijxi+1j+1+χxi+1j+1xij+1+χxij+1xij.
Forj=even, we have(20)Wfij=7rs+2r+4,fori=0,j=0,7rs+4r+4−2i,for1≤i≤r−1,j=0,7rs+4jr+4r+2−2i,for0≤i≤r−1,2≤j≤s−2.
Forj=odd, we have(21)Wfij=7rs+4jr+4r+2,fori=0,1≤j≤s−2,7rs+4jr+2r+2+2i,for1≤i≤r−1,1≤j≤s−2.
Observe that the weightsWzij and Wfij form an arithmetic progression with common difference 2 starting from 7rs+4,7rs+6 and ending at 11rs−4r+2. This implies that the defined labeling is a super 7rs+4,2-C3-antimagic total covering.


## 3. Results on Supera,d-C8-Antimagic Total Covering of Toroidal Octagonal Planner Map Osr

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 withr rows and s column of octagons by Osr, where s,r≥2. The planar representation of Osr is depicted in Figure 2. The vertex set VOsr and the edge set EOsr of octagonal planner map Osr can be defined as follows:(22)VOsr=uij,vij,wij,xij;0≤i≤r−1and0≤j≤s−1,EOsr=uijvij,wijxij;0≤i≤r−1and0≤j≤s−1∪wijuij−1;1≤i≤s−1and0≤j≤r−1∪wi0uis−1;0≤i≤r−1∪vijwi+1j+1;0≤i≤r−1and0≤j≤s−2∪vin−1wi+10;0≤i≤r−1∪vijxi+1j;0≤i≤r−1and0≤j≤s−1∪uijxij;0≤i≤r−1and0≤j≤s−1.Figure 2
Toroidal octagonal map identificationOsrFrom the above sets, we haveVOsr=4rs and EOsr=6rs. We can cover the toroidal octagonal map Osr by the 8-sided cycles C8,ij. For 0≤j≤s−1 and 0≤i≤r−1, the vertex set and edge set of 8-sided cycles C8,ij can be defined as(23)VC8,ij=wij,uij−1,vij−1,wi+1j,xi+1j,vij,uij,xij;0≤i≤r−1,1≤j≤s−1,EC8,ij=wijuij−1,uij−1vij−1,vij−1wi+1j,wi+1jxi+1j,vijxi+1j,uijvij,xijuij,xijwij;0≤i≤r−1,1≤j≤s−1,VC8,i0=wi0,uis−1,vis−1,wi+10,xi+10,vi0,ui0,xi0;0≤i≤r−1,,EC8,i0=wi0uis−1,uis−1vis−1,vis−1wi+10,wi+10xi+10,vi0xi+10,ui0vi0,xi0ui0,xi0wi0;0≤i≤s−1.We start by giving an upper bound ford such that Osr admits a super a,d-C8-antimagic covering.Theorem 4.
SupposeOsr admits a super a,d-C8-antimagic covering; then, d≤80.Proof.
SupposeOsr admits a super a,d-C8-antimagic covering. Then, the weight on cycle C8 is atleast(24)∑i=18i+∑i=184rs+i=32rs+72,and the largest weight of C8 is atmost(25)∑i=184rs+1−i+∑i=1810rs+1−i=112rs−56.
Thus, we have(26)a+rs−1d≤112rs−56,rs−1d≤112rs−56−32rs−72,d≤80rs−128rs−1,d≤80.In the next two theorems, we show that toroidal octagonal mapOsr admits a super a,d-C8-antimagic covering for d=1,2,…7.Theorem 5.
Letr,s≥2; then, the toroidal octagonal map Osr is super a,d-C8-antimagic for d∈1,3,5,7.Proof.
Define a total labelingφd:VOsr∪EOsr⟶1,2,3,…,VOsr+EOsr , where d∈1,3,5,7 as follows:(27)φduij=jr+1+i,0≤i≤r−1,0≤j≤s−1,φdvij=rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,φdxij=3rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,φdwij=2rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,φduijvij=4rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,φxijwij=5mn++2jm+1+2i,0≤i≤r−1,0≤j≤s−1,φ1vijxi+1j=φ3vijxi+1j=8rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,φ5vijxi+1j=φ7vijxi+1j=8rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,φduijwij+1=5rs+2s−1−jr+2r−2i,0≤i≤r−1,0≤j≤s−1,φ1vijwij+1=7rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,φ3vijwij+1=φ5vijwij+1=φ7vijwij+1=7rs+rj+1+i,0≤i≤r−1,0≤j≤s−1,φ1xijuij=φ3xijuij=φ5xijuij=9rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,φ7xijuij=9rs+jr+1+i,0≤i≤r−1,0≤j≤s−1.
The total labelingφd labels the vertices uij,vij,wij,xij from the set 1,2,…,4rs and the edges from the set 4rs+1,4rs+2,…,10rs. For 0≥i≥r−1 and 0≥j≥s−1, the weight of cycles C8,ij under φd is(28)WdC8,ij=φduij−1+φdvij−1+φduij−1vij−1+φdwi+1j+φdwij+φdwi+1jvij−1+φdxi+1j+φdxi+1jwi+1j+φdvij+φdvijxi+1j+φduij+φduijvij+φdxij+φdxijuij+φdxijwij+φdwijuij−1,WdC8,ij=68rs+2r+10+jr+i,ford=1,67rs+r+11+3jr+3i,ford=3,66rs+r+12+5jr+5i,ford=5,65rs+13+7jr+7i,ford=7.
For the cased=1, we have weights’ set 68rs+2r+10,68rs+2r+11,…,69rs+2r+9; similarly, for cases d=3,5,7, we get the weights from the sets 67rs+r+11,67rs+2r+12,…,70rs+r+8, 66rs+r+12,66rs+r+17,…,71rs+r+7, and 65rs+r+13,65rs+r+20,…,72rs+r+5, respectively. Hence, the weights of cycles C8,ij form an arithmetic sequence with difference 1,3,5, and 7.Theorem 6.
Letr,s≥2; then, the toroidal map Osr is super a,d-C8-antimagic for d∈2,4,6.Proof.
Letd∈2,4,6 and 0≤i≤r−1,0≤j≤s−1. We define a total labeling ϕd of Osr as follows:(29)ϕduij=jr+1+i,0≤i≤r−1,0≤j≤s−1,ϕdvij=rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,ϕdxij=3rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,ϕdwij=2rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,ϕduijvij=8rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,ϕ2xijwij=φ4xijwij=9rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,ϕ6xijwij=9rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,ϕ2vijxi+1j=φ4vijxi+1j=6rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,ϕ6vijxi+1j=6rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,ϕduijwij+1=4rs+jr+1+i,0≤i≤r−1,0≤j≤s−1,ϕdvijwij+1=5rs+rj+1+i,0≤i≤r−1,0≤j≤s−1,ϕ2xijuij=7rs+s−1−jr+r−i,0≤i≤r−1,0≤j≤s−1,ϕ4xijuij=ϕ6xijuij=7rs+jr+1+i,0≤i≤r−1,0≤j≤s−1.
The total labelingϕd labels the vertices uij,vij,wij,xij from the set 1,2,…,4rs and edges from the set 4rs+1,4rs+2,…,10rs. This show that φd is a bijection from set VOsr∪EOsr to set 1,2,…,10rs. For 1≥i≥l and i≥j≥k, the weights of C8,ij under the labeling ϕd are(30)WdC8,ij=ϕduij−1+ϕdvij−1+ϕduij−1vij−1+ϕdwi+1j+ϕdwij+ϕdwi+1jvij−1+ϕdxi+1j+ϕdxi+1jwi+1j+ϕdvij+ϕdvijxi+1j+ϕduij+ϕduijvij+ϕdxij+ϕdxijuij+ϕdxijwij+ϕdwijuij−1,WdC8,ij=75rs−4r+8+2jr+2i,ford=2,74rs−4r+9+4jr+4i,ford=4,73rs−4r+12+6jr+6i,ford=6.
For the cased=2, we have weights from the set 75rs−4r+8,75rs−4r+10,…,77rs−4r+6. Similarly, for cases d=4,6, we get weights from the sets 74rs−4r+9,74rs−4r+13,…,78rs−4r+5 and 73rs−4r+12,73rs−4r+18,…,79rs−4r+6, respectively. This showed that weights of the cycles C8,ij form an arithmetic sequence with difference 2,4, and 6.


## 4. Conclusion

In the present paper first, we constructed an upper bound for the parameterd for super a,d-C3-antimagic covering. Secondly, we examined the existence of super a,d-C3-antimagic labeling of generalized antiprism Ars. We showed that, for r,s≥3 the generalized antiprism Ars had a,d-C3-antimagic covering for d∈0,2. Thirdly, we constructed an upper bound for the parameter d for super a,d-C8-antimagic covering. Finally, we examined the existence of super a,d-C8-antimagic labeling of torodial map Osr. We showed that, for m,n≥2, the torodial octagonal map Osr had a,d-C8-antimagic covering for d∈1,2,3,4,5,6,7. We conclude the paper with the following open problems.Open problem 1: find other possible bound for parameterd under a,d-C3-antimagic total covering and the corresponding remaining labeling of d for generalized antiprism ArsOpen problem 2: find other possible bound for parameterd under a,d-C8-antimagic total covering and the corresponding remaining labeling of d for torodial octagonal map Osr



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