# OnH-Supermagic Labelings of m-Shadow of Paths and Cycles

Authors: Ika Hesti Agustin; F. Susanto; Dafik; R. M. Prihandini; R. Alfarisi; I. W. Sudarsana
Journal: International Journal of Mathematics and Mathematical Sciences
 (2019)

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## Abstract

A simple graphG=(V,E) is said to be an H-covering if every edge of G belongs to at least one subgraph isomorphic to H. A bijection f:V∪E→{1,2,3,…,V+E} is an (a,d)-H-antimagic total labeling of G if, for all subgraphs H′ isomorphic to H, the sum of labels of all vertices and edges in H′ form an arithmetic sequence {a,a+d,…,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is calledsuper if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called (super) (a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits such a labeling is called H-(super)magic. The m-shadow of graph G, Dm(G), is a graph obtained by taking m copies of G, namely, G1,G2,…,Gm, and then joining every vertex u in Gi, i∈{1,2,…,m-1}, to the neighbors of the corresponding vertex v in Gi+1. In this paper we studied the H-supermagic labelings of Dm(G) where G are paths and cycles.


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## Body

## 1. Introduction

Graph theory is a branch of discrete mathematics that has been grown rapidly. There are many applications of graph theory in other fields such as computer science, physics, chemistry, biology, engineering, and sociology [1]. A graph G is a pair of two sets, i.e., V(G) and E(G). These two sets, respectively, represent a vertex set of G and an edge set of G. The number of vertices in G is denoted by VG and the number of edges in G is denoted by EG. Other basic terminologies about graph theory that are not mentioned in this paper can be seen in [2]. Note that all graphs considered in this paper are simple, finite, and undirected. By notation [a,b] with a,b integers we mean {x∈N∣a≤x≤b}.One of important topics in graph theory is graph labeling. A graph labeling can be defined as a mapping from some set of graph elements to a set of positive integers. A graph labeling whose domain is vertex set or edge set is called a vertex labeling or an edge labeling, respectively. Moreover, if domain is both vertex set and edge set, then we call such labelings as a total labeling.Anedge-covering of a graph G is a collection of subgraphs H1,H2,…,Hk such that every edge of G belongs to at least one of subgraphs Hi, i∈[1,k]. In this case, G is said to be an (H1,H2,…,Hk)-(edge) covering. If every subgraph Hi is isomorphic to a given graph H, then G is said to be an H-covering.For a graphG admitting an H-covering, an (a,d)-H-antimagic total labeling of G is a bijection f:V(G)∪E(G)→{1,2,…,VG+EG} such that, for all subgraphs H′ isomorphic to H, the H′-weightwt(H′), which is defined by wt(H′)=∑v∈V(H′)f(v)+∑uv∈E(H′)f(uv), forms an arithmetic sequence {a,a+d,…,(k-1)d} where a>0, d≥0 are two fixed integers and k is the number of all subgraphs of G isomorphic to H. The labeling f is calledsuper if the smallest possible labels appear on the vertices. A graph that admits (super) (a,d)-H-antimagic total labeling is called(super)(a,d)-H-antimagic. For a special d=0, the (super) (a,0)-H-antimagic total labeling is called H-(super)magic labeling. A graph that admits H-(super)magic labeling is called H-(super)magic.The notion of super(a,d)-H-antimagic total labeling was firstly introduced by Inayah, Salman, and Simanjuntak [3]. In 2013, Inayahet al. [4] studied super (a,d)-H-antimagic total labeling of a shackle graph shack(H,v,k). Dafiket al. [5] introduced a generalized shackle of graph denoted by gshack(H,K⊂H,k). They showed the existence of super (a,d)-H-antimagic total labeling of gshack(H,K⊂H,k) when K≅K2. Furthermore, Dafiket al. [6, 7] studied about H-super antimagicness of disconnected graphs as well as constructions of H-antimagic graphs using smaller edge antimagic graphs. More results about super (a,d)-H-antimagic total labeling can be seen in [8–11].The notionH-supermagic labeling was firstly introduced by Gutiérrez and Lladó [12]. In their paper, they investigatedstar-(super)magic andpath-(super)magic labelings of some classes of connected graphs. Maryatiet al. [13] studied Ph-supermagic labeling of some classes of trees, i.e., shrubs and banana trees. For more results about H-supermagic labeling can be seen in [14].In this paper, we investigate theH-supermagic labeling of graphs, namely, m-shadow of graphs which is a generalization of a shadow graph introduced by [15]. The m-shadow of graph G denoted by Dm(G) is a graph obtained by taking m copies of G, namely, G1,G2,…,Gm, and then joining every vertex u in Gi, i∈[1,m-1], to the neighbors of the corresponding vertex v in Gi+1. We have proved that Dm(G) admits H-supermagic labelings for some classes of graph G, namely, paths and cycles.


## 2. Main Results

### 2.1.Dm(Pt)-Supermagic Labeling of Dm(Pn)

In this part, we present theH-supermagic labeling of m-shadow of paths. Let Dm(Pn) be the m-shadow of paths with vertex set V(Dm(Pn))={vij∣i∈[1,n], j∈[1,m]} and edge set E(Dm(Pn))={vijvi+1j∣i∈[1,n-1], j∈[1,m]}∪{vijvi+1j+1∣i∈[1,n-1], j∈[1,m-1]}∪{vi+1jvij+1∣i∈[1,n-1], j∈[1,m-1]}. Next, we will show the existence of H-supermagic labeling of Dm(Pn) in the following theorem.Theorem 1.
D m ( P n ) is Dm(Pt)-supermagic for any integer m≥2, n≥4 and t∈[3,n-1].Proof (letG≅Dm(Pn)).
Define a total labelingf:V(G)∪E(G)→{1,2,…,VG+EG}. In constructing the total labeling f, we distinguish between the vertices labeling and the edges labeling. First, label every vertex of Dm(Pn) in the following way. (1)fvij=γ1i+j-1n,forj∈1,m with (2)γ1i=i-kt+k-1nt+1forn≡0modt;i=k,t+k,2t+k,…,n-t+k;k∈1,ti-kt+k-1n-rt+1+1forn≡rmodt,r∈1,t-1;i=k,t+k,2t+k,…,n-r+k;k∈1,ri-kt+k-1n-rt+r+1forn≡rmodt,r∈1,t-1;i=k,t+k,2t+k,…,n-r-t+k;k∈r+1,tTo label the edges of Dm(Pn), first, let q=n-1. Next, label every edge as follows: (3)fvijvi+1j=γ2i+mn+j-1n-1,forj∈1,mfvijvi+1j+1=γ3i+mn+mn-1+j-1n-1,forj∈1,m-1fvi+1jvij+1=γ2i+mn+mn-1+m-1n-1+j-1n-1,forj∈1,m-1with (4)γ2i=-i-kt-1+t-kn-1t-1forq≡0modt-1;i=k,t+k-1,2t-1+k,…,n-t+k;k∈1,t-1-i-kt-1+s-k+1n-s-1t-1+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-1-s+k;k∈1,s-i-kt-1+t-kn-s-1t-1+sn-s-1t-1+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-s-t+k;k∈s+1,t-1γ3i=i-kt-1+k-1n-1t-1+1forq≡0modt-1;i=k,t+k-1,2t-1+k,…,n-t+k;k∈1,t-1i-kt-1+k-1n-s-1t-1+1+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-1-s+k;k∈1,si-kt-1+k-1n-s-1t-1+s+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-s-t+k;k∈s+1,t-1It can be checked that f(V(DmPn))∈[1,mn]. For z∈[1,n-t+1], let Dm(Pt(z)) be sub-m-shadow of paths with V(Dm(Pt(z)))={vij∣i∈[z,z+t-1], j∈[1,m]} and E(Dm(Pt(z)))={vijvi+1j∣i∈[z,z+t-2], j∈[1,m]}∪{vijvi+1j+1∣i∈[z,z+t-2], j∈[1,m-1]}∪{vi+1jvij+1∣i∈[z,z+t-2], j∈[1,m-1]}. It can be shown that (5)fvij=fvi+tj-1,fori∈1,n-tandj∈1,mfvijvi+1j=fvi+t-1jvi+tj+1,fori∈1,n-tandj∈1,mfvijvi+1j+1=fvi+t-1jvi+tj+1-1,fori∈1,n-tandj∈1,m-1fvi+1jvij+1=fvi+tjvi+t-1j+1+1,fori∈1,n-tandj∈1,m-1Furthermore, it can be shown that V(Dm(Pt(z)))∩V(Dm(Pt(z+1)))={vij∣i∈[z+1,z+t-1],j∈[1,m]} and E(Dm(Pt(z)))∩E(Dm(Pt(z+1)))={vijvi+1j∣i∈[z+1,z+t-2], j∈[1,m]}∪{vijvi+1j+1∣i∈[z+1,z+t-2], j∈[1,m-1]}∪{vi+1jvij+1∣i∈[z+1,z+t-2], j∈[1,m-1]}. By combining these pieces of information, we obtain (6)wtDmPtz=∑i=zz+t-1∑j=1mfvij+∑i=zz+t-2∑j=1mfvijvi+1j+∑i=zz+t-2∑j=1m-1fvijvi+1j+1+∑i=zz+t-2∑j=1m-1fvi+1jvij+1=∑i=z+1z+t∑j=1mfvij-m+∑i=z+1z+t-1∑j=1mfvijvi+1j+m+∑i=z+1z+t-1∑j=1m-1fvijvi+1j+1-m-1+∑i=z+1z+t-1∑j=1m-1fvi+1jvij+1+m-1=wtDmPtz+1 Let wt(Dm(Pt(1)))=c be the magic constant. For the value of c, consider the following cases: (i)
Forn≡0(modt) and q≡0(mod(t-1)), it can be verified that c=-8ntm+2nt+8ntm2-3t+8mt-(9/2)m2t+8mn+3-(15/2)m-2n-(15/2)nm2+(9/2)m2.(ii)
Forn≡0(modt) and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=-(9/2)m2t-8ntm+(3/2)smt+8mt+2nt-st+8ntm2-3t-2n+8mn-(15/2)nm2-(3/2)sm+3-(3/2)s2m+(9/2)m2+s2-(15/2)m+s.(iii)
Forn≡r(modt),r∈[1,t-1], and q≡0(mod(t-1)), it can be verified that c=2nt-3t+8mt+8ntm2+(1/2)trm-8ntm-(9/2)m2t+(9/2)m2+3-2n+8mn-(1/2)r2m-(15/2)m-(15/2)nm2.(iv)
Forn≡r(modt),r∈[1,t-1], and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=-st+2nt-(9/2)m2t+8ntm2+(1/2)trm-8ntm+(3/2)smt-3t+8mt-(3/2)sm-(15/2)nm2+8mn+(9/2)m2+s-2n+3-(3/2)s2m+s2-(1/2)mr2-(15/2)m.
Therefore,Dm(Pn) is Dm(Pt)-supermagic for each m≥2, n≥4, and t∈[3,n-1].For an illustration, we give an example ofH-supermagic labeling of Dm(Pn) in Figure 1.Figure 1
D 3 ( P 3 )-supermagic labeling of D3(P7) with magic constant c = 697.


### 2.2.Dt(Cn)-Supermagic Labeling of Dm(Cn)

In this part, we focus on theH-supermagic labeling of m-shadow of cycles. Let Dm(Cn) be the m-shadow of cycles with vertex set V(Dm(Cn))={vij∣i∈[1,n],j∈[1,m]} and edge set E(Dm(Cn))={vijvi+1j∣i∈[1,n-1], j∈[1,m]}∪{vnjv1j∣j∈[1,m]}∪{vijvi+1j+1∣i∈[1,n-1], j∈[1,m-1]}∪{vnjv1j+1∣j∈[1,m-1]}∪{vi+1jvij+1∣i∈[1,n-1], j∈[1,m-1]}∪{v1jvnj+1∣j∈[1,m-1]}. Next, the H-supermagic labeling of Dm(Cn) will be shown in the following theorem.Theorem 2.
D m ( C n ) is Dt(Cn)-supermagic for any integer m≥4, n≥3, and t∈[3,m-1].Proof (letG≅Dm(Cn)).
Define a total labelingf:V(G)∪E(G)→{1,2,…,VG+EG} in the following way: (i)
Label the sets{vij∣i∈[1,n],j∈[1,m]}∪{vijvi+1j∣i∈[1,n-1],j∈[1,m]}∪{vnjv1j∣j∈[1,m]} as follows: (7)fvij=γ1j+i-1m,fori∈1,nfvijvi+1j=γ2j+mn+i-1m,fori∈1,n-1fvnjv1j=γ2j+mn+mn-1with(8)γ1j=j-kt+k-1mt+1form≡0modt;j=k,t+k,2t+k,…,m-t+k;k∈1,tj-kt+k-1m-rt+1+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r+k;k∈1,rj-kt+k-1m-rt+r+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r-t+k;k∈r+1,tγ2j=-j-kt+t-k+1mtform≡0modt;j=k,t+k,2t+k,…,m-t+k;k∈1,t-j-kt+r-k+1m-rt+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r+k;k∈1,r-j-kt+t-k+1m-rt+rm-rt+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r-t+k;k∈r+1,t(ii)
Label the sets{vijvi+1j+1∣i∈[1,n-1],j∈[1,m-1]}∪{vnjv1j+1∣j∈[1,m-1]}∪{vi+1jvij+1∣i∈[1,n-1],j∈[1,m-1]}∪{v1jvnj+1∣j∈[1,m-1]}. To label these sets, first, let q=m-1. Next, label the sets in the following way: (9)fvijvi+1j+1=γ3j+2mn+i-1m-1,fori∈1,n-1fvnjv1j+1=γ3j+2mn+m-1n-1fvi+1jvij+1=γ4j+2mn+m-1n+i-1m-1,fori∈1,n-1fv1jvnj+1=γ4j+2mn+m-1n+m-1n-1with(10)γ3j=j-kt-1+k-1m-1t-1+1forq≡0modt-1;j=k,t+k-1,2t-1+k,…,m-t+k;k∈1,t-1j-kt-1+k-1m-s-1t-1+1+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-1-s+k;k∈1,sj-kt-1+k-1m-s-1t-1+s+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-s-t+k;k∈s+1,t-1γ4j=-j-kt-1+t-km-1t-1forq≡0modt-1;j=k,t+k-1,2t-1+k,…,m-t+k;k∈1,t-1-j-kt-1+s-k+1m-s-1t-1+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-1-s+k;k∈1,s-j-kt-1+t-km-s-1t-1+sm-s-1t-1+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-s-t+k;k∈s+1,t-1It can be seen that f(V(Dm(Cn)))∈[1,mn]. Let Dt(Cn(1)),Dt(Cn(2)),…,Dt(Cn(m-t+1)) be sub-m-shadow of cycles with V(Dt(Cn(z)))={vij∣i∈[1,n], j∈[z,z+t-1] and E(Dt(Cn(z)))={vijvi+1j∣i∈[1,n-1], j∈[z,z+t-1]∪{vnjv1j∣j∈[z,z+t-1]∪{vijvi+1j+1∣i∈[1,n-1], j∈[z,z+t-2]∪{vnjv1j+1∣j∈[z,z+t-2]∪{vi+1jvij+1∣i∈[1,n-1], j∈[z,z+t-2]∪{v1jvnj+1∣j∈[z,z+t-2]. It can be checked that (11)fvij=fvij+t-1,fori∈1,nandj∈1,m-tfvijvi+1j=fvij+tvi+1j+t+1,fori∈1,n-1andj∈1,m-tfvnjv1j=fvnj+tv1j+t+1,forj∈1,m-tfvijvi+1j+1=fvij+t-1vi+1j+t-1,fori∈1,n-1andj∈1,m-tfvnjv1j+1=fvnj+t-1v1j+t-1,forj∈1,m-tfvi+1jvij+1=fvi+1j+t-1vij+t+1,fori∈1,n-1andj∈1,m-tfv1jvnj+1=fv1j+t-1vnj+t+1,forj∈1,m-tFurthermore, for z∈[1,m-t+1], it can be shown that V(Dt(Cn(z)))∩V(Dt(Cn(z+1)))={vij∣i∈[1,n], j∈[z+1,z+t-1]} and E(Dt(Cn(z)))∩E(Dt(Cn(z+1)))={vijvi+1j∣i∈[1,n-1], j∈[z+1,z+t-1]}∪{vnjv1j∣j∈[z+1,z+t-1]}∪{vijvi+1j+1∣i∈[1,n-1], j∈[z+1,z+t-2]}∪{vnjv1j+1∣j∈[z+1,z+t-2]}∪{vi+1jvij+1∣i∈[1,n-1], j∈[z+1,z+t-2]}∪{v1jvnj+1∣j∈[z+1,z+t-2]}. By combining these pieces of information, we have (12)wtDtCnz=∑i=1n∑j=zz+t-1fvij+∑i=1n-1∑j=zz+t-1fvijvi+1j+∑j=zz+t-1fvnjv1j+∑i=1n-1∑j=zz+t-2fvijvi+1j+1+∑j=zz+t-2fvnjv1j+1+∑i=1n-1∑j=zz+t-2fvi+1jvij+1+∑j=zz+t-2fv1jvnj+1=∑i=1n∑j=z+1z+tfvij-n+∑i=1n-1∑j=z+1z+tfvijvi+1j+n-1+∑j=z+1z+tfvnjv1j+1+∑i=1n-1∑j=z+1z+t-1fvijvi+1j+1-n-1+∑j=z+1z+t-1fvnjv1j+1-1+∑i=1n-1∑j=z+1z+t-1fvi+1jvij+1+n-1+∑j=z+1z+t-1fv1jvnj+1+1=wtDtCnz+1Let wt(Dt(Cn(1)))=c be the magic constant. For the value of c, consider the following cases: (i)
Form≡0(modt) and q≡0(mod(t-1)), it can be verified that c=8mn2t-2tn2+2nt-n-6mn2+2n2.(ii)
Form≡0(modt) and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=8mn2t-2tn2+nts+2nt-n-ns-6mn2-ns2+2n2.(iii)
Form≡r(modt),r∈[1,t-1], and q≡0(mod(t-1)), it can be verified that c=8mn2t+2nt+nrt-2tn2-n-6mn2+2n2-nr2.(iv)
Form≡r(modt),r∈[1,t-1], and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=8mtn2-2tn2+nts+2nt+nrt-ns2+2n2-nr2-n-6mn2-ns.
Hence,Dm(Cn) is Dt(Cn)-supermagic for each m≥4, n≥3, and t∈[3,m-1].For an illustration, an example ofH-supermagic labeling of Dm(Cn) can be seen in Figure 2.Figure 2
D 3 ( C 3 )-supermagic labeling of D6(C3) with magic constant c = 954.


## 2.1.Dm(Pt)-Supermagic Labeling of Dm(Pn)

In this part, we present theH-supermagic labeling of m-shadow of paths. Let Dm(Pn) be the m-shadow of paths with vertex set V(Dm(Pn))={vij∣i∈[1,n], j∈[1,m]} and edge set E(Dm(Pn))={vijvi+1j∣i∈[1,n-1], j∈[1,m]}∪{vijvi+1j+1∣i∈[1,n-1], j∈[1,m-1]}∪{vi+1jvij+1∣i∈[1,n-1], j∈[1,m-1]}. Next, we will show the existence of H-supermagic labeling of Dm(Pn) in the following theorem.Theorem 1.
D m ( P n ) is Dm(Pt)-supermagic for any integer m≥2, n≥4 and t∈[3,n-1].Proof (letG≅Dm(Pn)).
Define a total labelingf:V(G)∪E(G)→{1,2,…,VG+EG}. In constructing the total labeling f, we distinguish between the vertices labeling and the edges labeling. First, label every vertex of Dm(Pn) in the following way. (1)fvij=γ1i+j-1n,forj∈1,m with (2)γ1i=i-kt+k-1nt+1forn≡0modt;i=k,t+k,2t+k,…,n-t+k;k∈1,ti-kt+k-1n-rt+1+1forn≡rmodt,r∈1,t-1;i=k,t+k,2t+k,…,n-r+k;k∈1,ri-kt+k-1n-rt+r+1forn≡rmodt,r∈1,t-1;i=k,t+k,2t+k,…,n-r-t+k;k∈r+1,tTo label the edges of Dm(Pn), first, let q=n-1. Next, label every edge as follows: (3)fvijvi+1j=γ2i+mn+j-1n-1,forj∈1,mfvijvi+1j+1=γ3i+mn+mn-1+j-1n-1,forj∈1,m-1fvi+1jvij+1=γ2i+mn+mn-1+m-1n-1+j-1n-1,forj∈1,m-1with (4)γ2i=-i-kt-1+t-kn-1t-1forq≡0modt-1;i=k,t+k-1,2t-1+k,…,n-t+k;k∈1,t-1-i-kt-1+s-k+1n-s-1t-1+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-1-s+k;k∈1,s-i-kt-1+t-kn-s-1t-1+sn-s-1t-1+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-s-t+k;k∈s+1,t-1γ3i=i-kt-1+k-1n-1t-1+1forq≡0modt-1;i=k,t+k-1,2t-1+k,…,n-t+k;k∈1,t-1i-kt-1+k-1n-s-1t-1+1+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-1-s+k;k∈1,si-kt-1+k-1n-s-1t-1+s+1forq≡smodt-1,s∈1,t-2;i=k,t+k-1,2t-1+k,…,n-s-t+k;k∈s+1,t-1It can be checked that f(V(DmPn))∈[1,mn]. For z∈[1,n-t+1], let Dm(Pt(z)) be sub-m-shadow of paths with V(Dm(Pt(z)))={vij∣i∈[z,z+t-1], j∈[1,m]} and E(Dm(Pt(z)))={vijvi+1j∣i∈[z,z+t-2], j∈[1,m]}∪{vijvi+1j+1∣i∈[z,z+t-2], j∈[1,m-1]}∪{vi+1jvij+1∣i∈[z,z+t-2], j∈[1,m-1]}. It can be shown that (5)fvij=fvi+tj-1,fori∈1,n-tandj∈1,mfvijvi+1j=fvi+t-1jvi+tj+1,fori∈1,n-tandj∈1,mfvijvi+1j+1=fvi+t-1jvi+tj+1-1,fori∈1,n-tandj∈1,m-1fvi+1jvij+1=fvi+tjvi+t-1j+1+1,fori∈1,n-tandj∈1,m-1Furthermore, it can be shown that V(Dm(Pt(z)))∩V(Dm(Pt(z+1)))={vij∣i∈[z+1,z+t-1],j∈[1,m]} and E(Dm(Pt(z)))∩E(Dm(Pt(z+1)))={vijvi+1j∣i∈[z+1,z+t-2], j∈[1,m]}∪{vijvi+1j+1∣i∈[z+1,z+t-2], j∈[1,m-1]}∪{vi+1jvij+1∣i∈[z+1,z+t-2], j∈[1,m-1]}. By combining these pieces of information, we obtain (6)wtDmPtz=∑i=zz+t-1∑j=1mfvij+∑i=zz+t-2∑j=1mfvijvi+1j+∑i=zz+t-2∑j=1m-1fvijvi+1j+1+∑i=zz+t-2∑j=1m-1fvi+1jvij+1=∑i=z+1z+t∑j=1mfvij-m+∑i=z+1z+t-1∑j=1mfvijvi+1j+m+∑i=z+1z+t-1∑j=1m-1fvijvi+1j+1-m-1+∑i=z+1z+t-1∑j=1m-1fvi+1jvij+1+m-1=wtDmPtz+1 Let wt(Dm(Pt(1)))=c be the magic constant. For the value of c, consider the following cases: (i)
Forn≡0(modt) and q≡0(mod(t-1)), it can be verified that c=-8ntm+2nt+8ntm2-3t+8mt-(9/2)m2t+8mn+3-(15/2)m-2n-(15/2)nm2+(9/2)m2.(ii)
Forn≡0(modt) and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=-(9/2)m2t-8ntm+(3/2)smt+8mt+2nt-st+8ntm2-3t-2n+8mn-(15/2)nm2-(3/2)sm+3-(3/2)s2m+(9/2)m2+s2-(15/2)m+s.(iii)
Forn≡r(modt),r∈[1,t-1], and q≡0(mod(t-1)), it can be verified that c=2nt-3t+8mt+8ntm2+(1/2)trm-8ntm-(9/2)m2t+(9/2)m2+3-2n+8mn-(1/2)r2m-(15/2)m-(15/2)nm2.(iv)
Forn≡r(modt),r∈[1,t-1], and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=-st+2nt-(9/2)m2t+8ntm2+(1/2)trm-8ntm+(3/2)smt-3t+8mt-(3/2)sm-(15/2)nm2+8mn+(9/2)m2+s-2n+3-(3/2)s2m+s2-(1/2)mr2-(15/2)m.
Therefore,Dm(Pn) is Dm(Pt)-supermagic for each m≥2, n≥4, and t∈[3,n-1].For an illustration, we give an example ofH-supermagic labeling of Dm(Pn) in Figure 1.Figure 1
D 3 ( P 3 )-supermagic labeling of D3(P7) with magic constant c = 697.


## 2.2.Dt(Cn)-Supermagic Labeling of Dm(Cn)

In this part, we focus on theH-supermagic labeling of m-shadow of cycles. Let Dm(Cn) be the m-shadow of cycles with vertex set V(Dm(Cn))={vij∣i∈[1,n],j∈[1,m]} and edge set E(Dm(Cn))={vijvi+1j∣i∈[1,n-1], j∈[1,m]}∪{vnjv1j∣j∈[1,m]}∪{vijvi+1j+1∣i∈[1,n-1], j∈[1,m-1]}∪{vnjv1j+1∣j∈[1,m-1]}∪{vi+1jvij+1∣i∈[1,n-1], j∈[1,m-1]}∪{v1jvnj+1∣j∈[1,m-1]}. Next, the H-supermagic labeling of Dm(Cn) will be shown in the following theorem.Theorem 2.
D m ( C n ) is Dt(Cn)-supermagic for any integer m≥4, n≥3, and t∈[3,m-1].Proof (letG≅Dm(Cn)).
Define a total labelingf:V(G)∪E(G)→{1,2,…,VG+EG} in the following way: (i)
Label the sets{vij∣i∈[1,n],j∈[1,m]}∪{vijvi+1j∣i∈[1,n-1],j∈[1,m]}∪{vnjv1j∣j∈[1,m]} as follows: (7)fvij=γ1j+i-1m,fori∈1,nfvijvi+1j=γ2j+mn+i-1m,fori∈1,n-1fvnjv1j=γ2j+mn+mn-1with(8)γ1j=j-kt+k-1mt+1form≡0modt;j=k,t+k,2t+k,…,m-t+k;k∈1,tj-kt+k-1m-rt+1+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r+k;k∈1,rj-kt+k-1m-rt+r+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r-t+k;k∈r+1,tγ2j=-j-kt+t-k+1mtform≡0modt;j=k,t+k,2t+k,…,m-t+k;k∈1,t-j-kt+r-k+1m-rt+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r+k;k∈1,r-j-kt+t-k+1m-rt+rm-rt+1form≡rmodt,r∈1,t-1;j=k,t+k,2t+k,…,m-r-t+k;k∈r+1,t(ii)
Label the sets{vijvi+1j+1∣i∈[1,n-1],j∈[1,m-1]}∪{vnjv1j+1∣j∈[1,m-1]}∪{vi+1jvij+1∣i∈[1,n-1],j∈[1,m-1]}∪{v1jvnj+1∣j∈[1,m-1]}. To label these sets, first, let q=m-1. Next, label the sets in the following way: (9)fvijvi+1j+1=γ3j+2mn+i-1m-1,fori∈1,n-1fvnjv1j+1=γ3j+2mn+m-1n-1fvi+1jvij+1=γ4j+2mn+m-1n+i-1m-1,fori∈1,n-1fv1jvnj+1=γ4j+2mn+m-1n+m-1n-1with(10)γ3j=j-kt-1+k-1m-1t-1+1forq≡0modt-1;j=k,t+k-1,2t-1+k,…,m-t+k;k∈1,t-1j-kt-1+k-1m-s-1t-1+1+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-1-s+k;k∈1,sj-kt-1+k-1m-s-1t-1+s+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-s-t+k;k∈s+1,t-1γ4j=-j-kt-1+t-km-1t-1forq≡0modt-1;j=k,t+k-1,2t-1+k,…,m-t+k;k∈1,t-1-j-kt-1+s-k+1m-s-1t-1+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-1-s+k;k∈1,s-j-kt-1+t-km-s-1t-1+sm-s-1t-1+1forq≡smodt-1,s∈1,t-2;j=k,t+k-1,2t-1+k,…,m-s-t+k;k∈s+1,t-1It can be seen that f(V(Dm(Cn)))∈[1,mn]. Let Dt(Cn(1)),Dt(Cn(2)),…,Dt(Cn(m-t+1)) be sub-m-shadow of cycles with V(Dt(Cn(z)))={vij∣i∈[1,n], j∈[z,z+t-1] and E(Dt(Cn(z)))={vijvi+1j∣i∈[1,n-1], j∈[z,z+t-1]∪{vnjv1j∣j∈[z,z+t-1]∪{vijvi+1j+1∣i∈[1,n-1], j∈[z,z+t-2]∪{vnjv1j+1∣j∈[z,z+t-2]∪{vi+1jvij+1∣i∈[1,n-1], j∈[z,z+t-2]∪{v1jvnj+1∣j∈[z,z+t-2]. It can be checked that (11)fvij=fvij+t-1,fori∈1,nandj∈1,m-tfvijvi+1j=fvij+tvi+1j+t+1,fori∈1,n-1andj∈1,m-tfvnjv1j=fvnj+tv1j+t+1,forj∈1,m-tfvijvi+1j+1=fvij+t-1vi+1j+t-1,fori∈1,n-1andj∈1,m-tfvnjv1j+1=fvnj+t-1v1j+t-1,forj∈1,m-tfvi+1jvij+1=fvi+1j+t-1vij+t+1,fori∈1,n-1andj∈1,m-tfv1jvnj+1=fv1j+t-1vnj+t+1,forj∈1,m-tFurthermore, for z∈[1,m-t+1], it can be shown that V(Dt(Cn(z)))∩V(Dt(Cn(z+1)))={vij∣i∈[1,n], j∈[z+1,z+t-1]} and E(Dt(Cn(z)))∩E(Dt(Cn(z+1)))={vijvi+1j∣i∈[1,n-1], j∈[z+1,z+t-1]}∪{vnjv1j∣j∈[z+1,z+t-1]}∪{vijvi+1j+1∣i∈[1,n-1], j∈[z+1,z+t-2]}∪{vnjv1j+1∣j∈[z+1,z+t-2]}∪{vi+1jvij+1∣i∈[1,n-1], j∈[z+1,z+t-2]}∪{v1jvnj+1∣j∈[z+1,z+t-2]}. By combining these pieces of information, we have (12)wtDtCnz=∑i=1n∑j=zz+t-1fvij+∑i=1n-1∑j=zz+t-1fvijvi+1j+∑j=zz+t-1fvnjv1j+∑i=1n-1∑j=zz+t-2fvijvi+1j+1+∑j=zz+t-2fvnjv1j+1+∑i=1n-1∑j=zz+t-2fvi+1jvij+1+∑j=zz+t-2fv1jvnj+1=∑i=1n∑j=z+1z+tfvij-n+∑i=1n-1∑j=z+1z+tfvijvi+1j+n-1+∑j=z+1z+tfvnjv1j+1+∑i=1n-1∑j=z+1z+t-1fvijvi+1j+1-n-1+∑j=z+1z+t-1fvnjv1j+1-1+∑i=1n-1∑j=z+1z+t-1fvi+1jvij+1+n-1+∑j=z+1z+t-1fv1jvnj+1+1=wtDtCnz+1Let wt(Dt(Cn(1)))=c be the magic constant. For the value of c, consider the following cases: (i)
Form≡0(modt) and q≡0(mod(t-1)), it can be verified that c=8mn2t-2tn2+2nt-n-6mn2+2n2.(ii)
Form≡0(modt) and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=8mn2t-2tn2+nts+2nt-n-ns-6mn2-ns2+2n2.(iii)
Form≡r(modt),r∈[1,t-1], and q≡0(mod(t-1)), it can be verified that c=8mn2t+2nt+nrt-2tn2-n-6mn2+2n2-nr2.(iv)
Form≡r(modt),r∈[1,t-1], and q≡s(mod(t-1)),s∈[1,t-2], it can be verified that c=8mtn2-2tn2+nts+2nt+nrt-ns2+2n2-nr2-n-6mn2-ns.
Hence,Dm(Cn) is Dt(Cn)-supermagic for each m≥4, n≥3, and t∈[3,m-1].For an illustration, an example ofH-supermagic labeling of Dm(Cn) can be seen in Figure 2.Figure 2
D 3 ( C 3 )-supermagic labeling of D6(C3) with magic constant c = 954.


## 3. Conclusion

We have studied theH-supermagic labeling of one type of graph operation, namely, m-shadow of graphs. In this paper, we have only shown the existence of H-supermagic labeling of m-shadow of paths and cycles. Meanwhile, the H-supermagic labeling of m-shadow of other classes of graphs is still widely open. Therefore, we propose open problems as follows.Open Problem 1.Determine H-supermagic labelings of m-shadow of other classes of graphs.Open Problem 2.Determine H-supermagic labelings of m-shadow of any connected graph.



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