# Extending of Edge Even Graceful Labeling of Graphs to Strongr-Edge Even Graceful Labeling

Authors: Mohamed R. Zeen El Deen; Nora A. Omar
Journal: Journal of Mathematics
 (2021)

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## Abstract

Edge even graceful labeling of a graphG with p vertices and q edges is a bijective f from the set of edge EG to the set of positive integers 2,4,…,2q such that all the vertex labels f∗VG, given by f∗u=∑uv∈EGfuvmod2k, where k=maxp,q, are pairwise distinct. There are many graphs that do not have edge even graceful labeling, so in this paper, we have extended the definition of edge even graceful labeling to r-edge even graceful labeling and strong r-edge even graceful labeling. We have obtained the necessary conditions for more path-related graphs and cycle-related graphs to be an r-edge even graceful graph. Furthermore, the minimum number r for which the graphs: tortoise graph, double star graph, ladder and diagonal ladder graphs, helm graph, crown graph, sunflower graph, and sunflower planar graph, have an r-edge even graceful labeling was found. Finally, we proved that the even cycle C2n has a strong 2-edge even graceful labeling when n is even.


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## Body

## 1. Introduction and Preliminaries

The field of graph theory plays vital role in various mathematical fields which are used in structural models. This structural arrangement of various objects or technologies lead to new inventions and modifications in the existing environment for improvement in these fields. Nowadays, graph labeling has become an important and rich branch of different research studies in graph theory because of its huge applications in many disciplines, e.g., coding theory, X-ray, crystallography, radar, communication networks, astronomy, communication network addressing, circuit design, and graph decomposition problems. For more interesting applications of graph labeling, see [1–3].Throughout this paper, a graphG=VG,EG always means a finite and a simple connected graph (without loops and parallel edges), where q=EG is the cardinality of EG and p=VG is that of VG.There are two main types of graceful labeling in graph theory. The oldest one is the vertex graceful labeling and the recent one is the edge graceful labeling. Here we are dealing with the recent one.A functionf is called an edge graceful labeling of a graph G if f:EG⟶1,2,…,q is bijective and the induced function f∗:VG⟶0,1,2,…,p−1, defined as f∗u=∑xy∈EGfxymodp, is bijective. This type of graph labeling was first introduced by Lo in 1985 [4].A functionf is called an edge odd graceful labeling of a graph if f:EG⟶1,3,…,2q−1 is bijective and the induced function f∗:VG⟶0,1,2,3,…,2q−1, defined as f∗u=∑xy∈EGfxymod2q, is bijective. This type of graph labeling was first introduced by Solairaju and Chithra in 2009 [5] (see also [6, 7]).A functionf is called an edge even graceful labeling of a graph G if f:EG⟶2,4,…,2q is bijective and the induced function f∗:VG⟶0,2,4,…,2q, defined as f∗x=∑xy∈EGfxymod2k, is an injective function, where k=maxp,q. This type of graph labeling was first introduced by Elsonbaty and Daoud in 2017 [8, 9]. They introduced some path- and cycle-related graphs which are edge even graceful. Following this, Daoud [10] investigated the edge even graceful labeling of polar grid graphs and Zeen El Deen et al. [11–13] studied more graphs having an edge even graceful labeling. The graph that admits an edge even graceful labeling is called an edge even graceful graph.A summary of the results on these three types of graceful labels as well as all known labels so far can be found in [14].Because there are many graphs which do not have edge even graceful labeling, we introduce the extension of the definition of edge even graceful labeling tor-edge even graceful labeling.Definition 1.
A functionf is called r-edge even graceful labeling of the graph G if f:EG⟶2,4,…,2q+2r−2 is an injective function such that the induced mapping f∗:VG⟶0,2,4,…,2q+2r−2, given by: f∗x=∑xy∈EGfxymod2k+2r−2, is an injective function, where k=maxp,q. We will call the graph that admits r-edge even graceful labeling an r-edge even graceful graph.Note that the standard definition of edge even graceful labeling corresponds to a1-edge even graceful labeling.Definition 2.
A functionf is called strong r-edge even graceful labeling of the graph G if f:EG⟶2,4,…,2q+2r−2 is an injective function such that the induced mapping f∗:VG⟶2r−2,2r,2r+2,…,2q+2r−2, given by f∗x=∑xy∈EGfxymod2k+2r−2, is an injective function, where k=maxp,q.Illustration. A 2-edge even and a strong 2-edge even labeling of the graph K4 is shown in Figure 1.Figure 1
(a) A 2-edge even graceful labeling forK4. (b) A strong 2-edge even graceful labeling for K4.
(a)(b)


## 2. Ther-Edge Even Graceful Labeling for Some Path-Related Graphs

### 2.1. Ther-Edge Even Graceful Labeling of the Path Graph Pn

The path graphPn has edge even graceful labeling only when n is an odd number [8].Theorem 1.
Whennis an even number, the path graph Pn has 3-edge even graceful labeling.Proof.
The path graphPn has p=n vertices and q=n−1 edges. Let ei denote the edge vivi+1 in Pn (see Figure 2).
Letr=3; then, 2p+2r−2=2n+4, and there are two cases.(i)
Case (1). If n≡0mod4, we define the labeling function f:EPn⟶2,4,…,2n+2 as follows:(1)fei=i+1,ifi=1,3,5,…,n−1,n+i,ifi=2,4,…,n2,n+i+4,ifi=n2+2,n2+4,…,n−2.Now, the labeling of the vertices is(2)f∗v1=fe1=2,f∗vn=fen−1mod2n+4=n,f∗vi=fei+fei−1mod2n+4=n+2imod2n+4,i=2,3,…,n2+1,f∗vi=fei+fei−1mod2n+4=n+2i+4mod2n+4,i=n2+2,n2+3,…,n−1.So, the labels of the verticesv2,v3,…,vn/2,vn/2+1 will be n+4,n+6,…,2n,2n+2, respectively, and the labels of the vertices vn/2+2,vn/2+3,…,vn−2,vn−1 will be 4,6,…,n−4,n−2, respectively.(ii)
Case (2). If n≡2mod4, we define the labeling f as follows:(3)fei=i+1,ifi=1,3,5,…,n−1,n+i,ifi=2,4,…,n2+1,n+i+4,ifi=n2+3,n2+5,…,n−2.Now, the labeling of the vertices is(4)f∗v1=fe1=2,f∗vn=fen−1mod2n+4=n,f∗vi=fei+fei−1mod2n+4=n+2imod2n+4,i=2,3,…,n2+2,f∗vi=fei+fei−1mod2n+4=n+2i+4mod2n+4,i=n2+3,n2+4,…,n−1.
So, the labels of the verticesv2,v3,…,vn/2,vn/2+1,vn/2+2 are n+4,n+6,…,2n,2n+2,0, respectively, and the labels of the vertices vn/2+2,vn/2+3,…,vn−2,vn−1 are 6,8,…,n−4,n−2 respectively.
Therefore, all the vertex labels are even and distinct numbers. Thus, the path graphPn has 3-edge even graceful labeling when n is even.Figure 2
The path graphPn with ordinary labeling when n is even.Illustration. A 3-edge even graceful labeling of the graphs P12 and P14 is shown in Figure 3.Figure 3
A3-edge even graceful labeling of the path graphs P12 and P14.


### 2.2. Ther-Edge Even Graceful Labeling of the Star Graph K1,n

It should be noted that the star graphK1,n has edge even graceful labeling only when n is an even number [8].Theorem 2.
The star graphK1,n has 3-edge even graceful labeling when n is odd.Proof.
The graphK1,n has p=n+1 vertices and q=n edges obtained by connecting one vertex v0 with each of the vi, where 1≤i≤n. Let r=3; then, 2p+2r−2=2n+6, and we define the labeling f:EK1,n⟶2,4,…,2n+4 as follows:(5)fv0vi=2i,ifi=1,2,3,…,n+32,2i+2,ifi=n+52,…,n.
Note that we do not take the labelsn+5 and 2n+4 in the label of edges. Since the vertices vi,i=1,2,3,…,n are pendant, then f∗vi take their edge labels which are even and distinct and(6)f∗v0=∑i=1nfv0vimod2n+6=∑i=1n+3/22i+∑i=n+5/2n2i+2mod2n+6=0.
Hence, the labels of the vertices are all even and distinct numbers.Illustration. A 3-edge even labeling of the graph K1,11 is shown in Figure 4.Figure 4
A3-edge even labeling of the graph K1,11.


### 2.3. Ther-Edge Even Graceful Labeling of the Comb Graph Pn⊙K1

Theorem 3.
The comb graph [15] Pn⊙K1 has 3-edge even graceful labeling for any positive integer n.Proof.
The graphPn⊙K1 has p=2n vertices and q=2n−1 edges. Let the comb graph Pn⊙K1 be given as indicated in Figure 5.
Letr=3; then, 2p+2r−2=4n+4, and we define the labeling f:EPn⊙K1⟶2,4,…,4n+2 as follows:(7)fai=4n−2i+2,ifi=1,2,…,n,fbi=2i+2,ifi=1,2,…,n−1.
Then, the induced vertex labels are(8)f∗v1=fa1+fb1mod4n+4=0,f∗vn=fan+fbn−1mod4n+4=4n+2,f∗vi=fbi+fbi−1+faimod4n+4=2i,fori=2,…,n−1.
So, the labels of the verticesv2,v3,v4,…,vn−1 are 4,6,8,…,2n−2, respectively. Again, each pendant vertex will take the labels of its edges; then, f∗ui=fai=4n−2i+2,∀i=1,2,3,…,n which are even and distinct from all the labels of the vertices vi. Hence, the comb graph Pn⊙K1 has 3-edge even graceful labeling for any number n.Figure 5
The comb graphPn⊙K1 with ordinary labeling.Illustration. A 3-edge even labeling of the comb graph P10⊙K1 is shown in Figure 6.Figure 6
A3-edge even labeling of the comb graph P10⊙K1.


### 2.4. Ther-Edge Even Graceful Labeling of the Graph Tn,n−2

The graphTn,n−2 obtained from a path Pn by attaching exactly one pendant edge to each internal vertex of the path Pn, i.e., Tn,n−2, has vertices vi,uj,i=1,2,…,n,j=1,2,…,n−2 and edges vivi+1,ujvj+1,i=1,2,…,n−1,j=1,2,…,n−2.Theorem 4.
The graphTn,n−2 has 3-edge even graceful labeling for all positive integer n≥4.Proof.
The graphTn,n−2 has p=2n−2 vertices and q=2n−3 edges (see Figure 7).
Letr=3; then, 2p+2r−2=4n. The labeling f:ETn,n−2⟶2,4,…,4n−2 can be defined as follows:(9)fuivi+1=fai=4n−2i,ifi=1,2,…,n−3,2n+2,ifi=n−2,fvivi+1=fbi=2i,ifi=1,2,…,n−2,2n,ifi=n−1.
Note that we do not take the labels2n−2 and 2n+4 in the label of edges.
Then, the induced vertex labels are(10)f∗v1=fb1=2,f∗vn=fbn−1=2n−2,f∗vi=fbi+fbi−1+fai−1mod4n=2i,fori=2,…,n−2.
So, the labels of the verticesv2,v3,v4,…,vn−2 are 4,6,8,…,2n−4, respectively, which are all even and distinct numbers. Once again, since the vertices ui are pendant, then f∗ui,i=1,2,…,n−2 take their edge labels. Therefore, there is no repetition of the labels, and Tn,n−2 has 3-edge even graceful labeling for all n.Figure 7
The graphTn,n−2 with ordinary labeling.Illustration. A 3-edge even labeling of the graph T9,7 is shown in Figure 8.Figure 8
A3-edge even labeling of the graph T9,7.


### 2.5. Ther-Edge Even Graceful Labeling of the Double Star Bn,m

The double star is the graph obtained by joining the center of two starsK1,n and K1,m with an edge, denoted by Bn,m. The number of vertices is p=n+m+2, and the number of edges is q=n+m+1. The double star Bn,m has edge even graceful labeling when one of (morn) is an odd number and the other is an even number [11].Theorem 5.
(1)
The double starBn,m is 2-edge even graceful graph when both m and n are odd numbers.(2)
The double starBn,m is 3-edge even graceful graph when both m and n are even numbers.Proof.
Let the vertex and edge symbols be given as in Figure9.Case (1). Ifmandnare odd numbers. Let r=2; then, 2q+2r−2=2m+2n+4.We define the labeling functionf:EG⟶2,4,…,2m+2n+4 as follows:(11)fuv=2,fei=2i+2,fori=1,2,…,m−12,fEi=2m+2n+6−2i,fori=1,2,…,m+12,fai=m+1+2i,fori=1,2,…,n+12,fbi=m+2n+5−2i,fori=1,2,…,n−12.In this labeling, we can note that(i)
fei+fEi+1mod2m+2n+6≡0mod2m+2n+6,fori=1,2,…,m−1/2.(ii)
fai+fbimod2m+2n+6≡0mod2m+2n+6,fori=1,2,…,n−1/2.(iii)
fuv+fE1mod2m+2n+6≡0mod2m+2n+6.So, the vertex labels will be(12)f∗u=∑i=1m−1/2fei+∑i=1m+1/2fEi+fuvmod2m+2n+6=0,f∗v=∑i=1n−1/2fbi+∑i=1n−1/2fai+fan+1/2+fuvmod2m+2n+6=m+n+4.Each pendant vertex takes the labels of its incident edge which are different fromf∗u and f∗v.Case (2). If m and n are even numbers. Let r=3; then, 2q+2r−2=2m+2n+6. Consider the vertex and edge symbols given in Figure 9(b). We define the mapping f:G⟶2,4,…,2m+2n+6 as follows:fuv=2,fei=2i+2 for i=1,2,…,m/2,fEi=2p−2i−1 for i=1,2,…,m/2(13)fai=m+2+2i,ifi=1,2,…,n2+1,m+4+2i,ifi=n2+2,…,n.In this labeling, we can see that(i)
fei+fEimod2m+2n+8≡0mod2m+2n+8,fori=1,2,…,m/2.(ii)
fai+fan−i+1mod2m+2n+8≡0mod2m+2n+8,fori=1,2,…,n/2−1.
So, the vertex labels will be(14)f∗u=∑i=1m/2fei+∑i=1m/2fEi+fuvmod2m+2n+8=fuv=2,f∗v=∑i=1n/2−1fan−i+1+∑i=1n/2−1fai+fan/2+fan/2+1+fuvmod2m+2n+8=0.
Also, each pendant vertex takes the labels of its incident edge. Thus, we get a3-edge even graceful graph.Figure 9
(a)Bm,n when mandn are odd numbers. (b) Bm,n when mandn are even numbers.
(a)(b)Illustration. A 2-edge even labeling of the star graph B9,7 and 3-edge even labeling of the star graph B10,8 are shown in Figure 10.Figure 10
(a) A2-edge even labeling of B9,7. (b) A 3-edge even labeling of B10,8.
(a)(b)


### 2.6. Ther-Edge Even Graceful Labeling of the Ladder Graph Ln=P2×Pn

Theorem 6.
The ladder graphLn=P2×Pn has 2-edge even graceful labeling.Proof.
The graphLn has p=2n vertices and q=3n−2 edges. Let r=2; then, 2q+2r−2=6n−2.
Consider the ladder graphLn=P2×Pn shown in Figure 11.
We define the labeling functionf:ELn⟶2,4,…,6n−2 as follows:(15)fai=2i,i=1,2,…,n,fci=6n−2i−2,i=1,2,…,n−1,fbi=4n−2i−2,ifi=1,2,…,n−2,6n−2,ifi=n−1.
Then, the induced vertex labels are(16)f∗v1=fa1+fc1mod6n−2=0,f∗vn=fan+fcn−1mod6n−2=2,f∗vi=fci+fci−1+faimod6n−2=6n−2i,i=2,…,n−1,f∗u1=fa1+fb1mod6n−2=4n−2,f∗un−1=fan−1+fbn−1+fbn−2mod6n−2=4n,f∗un=fan+fbn−1mod6n−2=2n,f∗ui=fbi+fbi−1+faimod6n−2=2n−2i,i=2,…,n−2.
So, the labels of the verticesv2,v3,…,vn−2,vn−1 will be 6n−4,6n−6,…,4n+4,4n+2, respectively, and the labels of the vertices u2,u3,…,un−3,un−2 will be 2n−4,2n−6,…,6,4, respectively. Clearly, all the labels of the vertices are even and there is no repetition on the labels. Hence, the ladder graph Ln has 2-edge even graceful labeling.Figure 11
The ladder graphLn=P2×Pn with ordinary labeling.Illustration. A 2-edge even labeling of the ladder graph L10 is shown in Figure 12.Figure 12
A 2-edge even labeling of the ladder graphL10=P2×P10.


### 2.7. Ther-Edge Even Graceful Labeling of the Diagonal Ladder DLn

Diagonal ladder graph is a ladder with additional edgesviui+1 and uivi+1,i=1,2,…,n−1, denoted by DLn, where n is half its vertices and the number of vertices VDLn=2n and edges EDLn=5n−4.Theorem 7.
The diagonal ladderDLn has 2-edge even graceful labeling.Proof.
Consider the diagonal ladderDLn shown in Figure 13. Let r=2; then, 2q+2r−2=10n−6.
The labelingf:EDLn⟶2,4,…,10n−6 can be defined as follows:(17)fviui+1=2i,i=1,2,…,n−1,fvi+1ui=2n+2i,i=1,2,…,n−1,fvivi+1=10n−6−2i,i=1,2,…,n−1,fuiui+1=8n−4−2i,i=1,2,…,n−1,fviui=10n−6,ifi=1,6n−2i,ifi=2,3,…,n.
Then, the induced vertex labels are(18)f∗v1=fv1v2+fv1u1+fv1u2mod10n−6=0,f∗u1=fu1u2+fv1u1+fu1v2mod10n−6=2.
Fori=2,3,…,n−1,(19)f∗vi=fvivi−1+fvivi+1+fviui+fviui+1+fviui−1mod10n−6=8n−2i,f∗ui=fuiui−1+fuiui+1+fviui+fuivi+1+fuivi−1mod10n−6=4n+4−2i.
So, the labels of the verticesv2,v3,…,vn−2,vn−1 are 8n−4,8n−6,…,6n+4,6n+2, respectively, and the labels of the vertices u2,u3,…,un−2,un−1 are 4n,4n−2,…,2n+8,2n+6, respectively, and(20)f∗vn=fvnvn−1+fvnun+fvnun−1mod10n−6=6n,f∗un=funun−1+fvnun+funvn−1mod10n−6=2n+2.
Clearly, all the labels of the vertices are even and distinct, and thus we get the result.Figure 13
The diagonal ladderDLn with ordinary labeling.Illustration. A 2-edge even labeling of the diagonal ladder DL8 is shown in Figure 14.Figure 14
A2-edge even graceful for the diagonal ladder DL8.


### 2.8. Ther-Edge Even Graceful Labeling of the Tortoise Graph Tn

The tortoise graphTn is the graph obtained from a path Pn, where n is odd, by attaching an edge between vi and vn−i+1 for 1≤i≤n/2, so VTn=VPn and ETn=EPn∪ei=vivn−i+1,i=n,n−1,…,n+3/2.Theorem 8.
The tortoise graphTn has 2-edge even graceful labeling.Proof.
The graphTn has n vertices and q=3n−1/2 edges. Consider the tortoise graph Tn shown in Figure 15.
Letr=2; then, 2q+2r−2=3n−1, and we define the labeling f:ETn⟶2,4,…,3n−1 as follows:(21)fei=2i,ifi=1,2,3,…,n+12,3n−2i+1,ifi=n+32,…,n−1,3n−1,ifi=n,5n−2i−1,ifi=n+1,n+2,…,3n−12.
In view of the above labeling pattern, we have(22)f∗v1=fe1+fenmod3n−1=2,f∗vn=fen+fen−1mod3n−1=n+3,f∗vi=fei+fei−1+fei+n−1mod3n−1=2i,i=2,3,…,n−12,f∗vn+1/2=fen+1/2+fen−1/2mod3n−1=2n,f∗vn+3/2=fen+3/2+fen+1/2fe3n−1/2mod3n−1=2n+2,f∗vi=fei+fei−1+fe2n−imod3n−1=n−2i+5,i=n+52,…,n−1.
Clearly all the label of the vertices are even and distinct, changed inside the set0,2,4,…,2q+2r−2=0,2,…,3n−1. Hence the tortoise Tn has a 2- edge even labeling.Figure 15
The tortoise graphTn with ordinary labeling.Illustration. A 2-edge even labeling of the tortoise graph T11 is shown in Figure 16.Figure 16
A2-edge even graceful labeling for the tortoise graph T11.


## 2.1. Ther-Edge Even Graceful Labeling of the Path Graph Pn

The path graphPn has edge even graceful labeling only when n is an odd number [8].Theorem 1.
Whennis an even number, the path graph Pn has 3-edge even graceful labeling.Proof.
The path graphPn has p=n vertices and q=n−1 edges. Let ei denote the edge vivi+1 in Pn (see Figure 2).
Letr=3; then, 2p+2r−2=2n+4, and there are two cases.(i)
Case (1). If n≡0mod4, we define the labeling function f:EPn⟶2,4,…,2n+2 as follows:(1)fei=i+1,ifi=1,3,5,…,n−1,n+i,ifi=2,4,…,n2,n+i+4,ifi=n2+2,n2+4,…,n−2.Now, the labeling of the vertices is(2)f∗v1=fe1=2,f∗vn=fen−1mod2n+4=n,f∗vi=fei+fei−1mod2n+4=n+2imod2n+4,i=2,3,…,n2+1,f∗vi=fei+fei−1mod2n+4=n+2i+4mod2n+4,i=n2+2,n2+3,…,n−1.So, the labels of the verticesv2,v3,…,vn/2,vn/2+1 will be n+4,n+6,…,2n,2n+2, respectively, and the labels of the vertices vn/2+2,vn/2+3,…,vn−2,vn−1 will be 4,6,…,n−4,n−2, respectively.(ii)
Case (2). If n≡2mod4, we define the labeling f as follows:(3)fei=i+1,ifi=1,3,5,…,n−1,n+i,ifi=2,4,…,n2+1,n+i+4,ifi=n2+3,n2+5,…,n−2.Now, the labeling of the vertices is(4)f∗v1=fe1=2,f∗vn=fen−1mod2n+4=n,f∗vi=fei+fei−1mod2n+4=n+2imod2n+4,i=2,3,…,n2+2,f∗vi=fei+fei−1mod2n+4=n+2i+4mod2n+4,i=n2+3,n2+4,…,n−1.
So, the labels of the verticesv2,v3,…,vn/2,vn/2+1,vn/2+2 are n+4,n+6,…,2n,2n+2,0, respectively, and the labels of the vertices vn/2+2,vn/2+3,…,vn−2,vn−1 are 6,8,…,n−4,n−2 respectively.
Therefore, all the vertex labels are even and distinct numbers. Thus, the path graphPn has 3-edge even graceful labeling when n is even.Figure 2
The path graphPn with ordinary labeling when n is even.Illustration. A 3-edge even graceful labeling of the graphs P12 and P14 is shown in Figure 3.Figure 3
A3-edge even graceful labeling of the path graphs P12 and P14.


## 2.2. Ther-Edge Even Graceful Labeling of the Star Graph K1,n

It should be noted that the star graphK1,n has edge even graceful labeling only when n is an even number [8].Theorem 2.
The star graphK1,n has 3-edge even graceful labeling when n is odd.Proof.
The graphK1,n has p=n+1 vertices and q=n edges obtained by connecting one vertex v0 with each of the vi, where 1≤i≤n. Let r=3; then, 2p+2r−2=2n+6, and we define the labeling f:EK1,n⟶2,4,…,2n+4 as follows:(5)fv0vi=2i,ifi=1,2,3,…,n+32,2i+2,ifi=n+52,…,n.
Note that we do not take the labelsn+5 and 2n+4 in the label of edges. Since the vertices vi,i=1,2,3,…,n are pendant, then f∗vi take their edge labels which are even and distinct and(6)f∗v0=∑i=1nfv0vimod2n+6=∑i=1n+3/22i+∑i=n+5/2n2i+2mod2n+6=0.
Hence, the labels of the vertices are all even and distinct numbers.Illustration. A 3-edge even labeling of the graph K1,11 is shown in Figure 4.Figure 4
A3-edge even labeling of the graph K1,11.


## 2.3. Ther-Edge Even Graceful Labeling of the Comb Graph Pn⊙K1

Theorem 3.
The comb graph [15] Pn⊙K1 has 3-edge even graceful labeling for any positive integer n.Proof.
The graphPn⊙K1 has p=2n vertices and q=2n−1 edges. Let the comb graph Pn⊙K1 be given as indicated in Figure 5.
Letr=3; then, 2p+2r−2=4n+4, and we define the labeling f:EPn⊙K1⟶2,4,…,4n+2 as follows:(7)fai=4n−2i+2,ifi=1,2,…,n,fbi=2i+2,ifi=1,2,…,n−1.
Then, the induced vertex labels are(8)f∗v1=fa1+fb1mod4n+4=0,f∗vn=fan+fbn−1mod4n+4=4n+2,f∗vi=fbi+fbi−1+faimod4n+4=2i,fori=2,…,n−1.
So, the labels of the verticesv2,v3,v4,…,vn−1 are 4,6,8,…,2n−2, respectively. Again, each pendant vertex will take the labels of its edges; then, f∗ui=fai=4n−2i+2,∀i=1,2,3,…,n which are even and distinct from all the labels of the vertices vi. Hence, the comb graph Pn⊙K1 has 3-edge even graceful labeling for any number n.Figure 5
The comb graphPn⊙K1 with ordinary labeling.Illustration. A 3-edge even labeling of the comb graph P10⊙K1 is shown in Figure 6.Figure 6
A3-edge even labeling of the comb graph P10⊙K1.


## 2.4. Ther-Edge Even Graceful Labeling of the Graph Tn,n−2

The graphTn,n−2 obtained from a path Pn by attaching exactly one pendant edge to each internal vertex of the path Pn, i.e., Tn,n−2, has vertices vi,uj,i=1,2,…,n,j=1,2,…,n−2 and edges vivi+1,ujvj+1,i=1,2,…,n−1,j=1,2,…,n−2.Theorem 4.
The graphTn,n−2 has 3-edge even graceful labeling for all positive integer n≥4.Proof.
The graphTn,n−2 has p=2n−2 vertices and q=2n−3 edges (see Figure 7).
Letr=3; then, 2p+2r−2=4n. The labeling f:ETn,n−2⟶2,4,…,4n−2 can be defined as follows:(9)fuivi+1=fai=4n−2i,ifi=1,2,…,n−3,2n+2,ifi=n−2,fvivi+1=fbi=2i,ifi=1,2,…,n−2,2n,ifi=n−1.
Note that we do not take the labels2n−2 and 2n+4 in the label of edges.
Then, the induced vertex labels are(10)f∗v1=fb1=2,f∗vn=fbn−1=2n−2,f∗vi=fbi+fbi−1+fai−1mod4n=2i,fori=2,…,n−2.
So, the labels of the verticesv2,v3,v4,…,vn−2 are 4,6,8,…,2n−4, respectively, which are all even and distinct numbers. Once again, since the vertices ui are pendant, then f∗ui,i=1,2,…,n−2 take their edge labels. Therefore, there is no repetition of the labels, and Tn,n−2 has 3-edge even graceful labeling for all n.Figure 7
The graphTn,n−2 with ordinary labeling.Illustration. A 3-edge even labeling of the graph T9,7 is shown in Figure 8.Figure 8
A3-edge even labeling of the graph T9,7.


## 2.5. Ther-Edge Even Graceful Labeling of the Double Star Bn,m

The double star is the graph obtained by joining the center of two starsK1,n and K1,m with an edge, denoted by Bn,m. The number of vertices is p=n+m+2, and the number of edges is q=n+m+1. The double star Bn,m has edge even graceful labeling when one of (morn) is an odd number and the other is an even number [11].Theorem 5.
(1)
The double starBn,m is 2-edge even graceful graph when both m and n are odd numbers.(2)
The double starBn,m is 3-edge even graceful graph when both m and n are even numbers.Proof.
Let the vertex and edge symbols be given as in Figure9.Case (1). Ifmandnare odd numbers. Let r=2; then, 2q+2r−2=2m+2n+4.We define the labeling functionf:EG⟶2,4,…,2m+2n+4 as follows:(11)fuv=2,fei=2i+2,fori=1,2,…,m−12,fEi=2m+2n+6−2i,fori=1,2,…,m+12,fai=m+1+2i,fori=1,2,…,n+12,fbi=m+2n+5−2i,fori=1,2,…,n−12.In this labeling, we can note that(i)
fei+fEi+1mod2m+2n+6≡0mod2m+2n+6,fori=1,2,…,m−1/2.(ii)
fai+fbimod2m+2n+6≡0mod2m+2n+6,fori=1,2,…,n−1/2.(iii)
fuv+fE1mod2m+2n+6≡0mod2m+2n+6.So, the vertex labels will be(12)f∗u=∑i=1m−1/2fei+∑i=1m+1/2fEi+fuvmod2m+2n+6=0,f∗v=∑i=1n−1/2fbi+∑i=1n−1/2fai+fan+1/2+fuvmod2m+2n+6=m+n+4.Each pendant vertex takes the labels of its incident edge which are different fromf∗u and f∗v.Case (2). If m and n are even numbers. Let r=3; then, 2q+2r−2=2m+2n+6. Consider the vertex and edge symbols given in Figure 9(b). We define the mapping f:G⟶2,4,…,2m+2n+6 as follows:fuv=2,fei=2i+2 for i=1,2,…,m/2,fEi=2p−2i−1 for i=1,2,…,m/2(13)fai=m+2+2i,ifi=1,2,…,n2+1,m+4+2i,ifi=n2+2,…,n.In this labeling, we can see that(i)
fei+fEimod2m+2n+8≡0mod2m+2n+8,fori=1,2,…,m/2.(ii)
fai+fan−i+1mod2m+2n+8≡0mod2m+2n+8,fori=1,2,…,n/2−1.
So, the vertex labels will be(14)f∗u=∑i=1m/2fei+∑i=1m/2fEi+fuvmod2m+2n+8=fuv=2,f∗v=∑i=1n/2−1fan−i+1+∑i=1n/2−1fai+fan/2+fan/2+1+fuvmod2m+2n+8=0.
Also, each pendant vertex takes the labels of its incident edge. Thus, we get a3-edge even graceful graph.Figure 9
(a)Bm,n when mandn are odd numbers. (b) Bm,n when mandn are even numbers.
(a)(b)Illustration. A 2-edge even labeling of the star graph B9,7 and 3-edge even labeling of the star graph B10,8 are shown in Figure 10.Figure 10
(a) A2-edge even labeling of B9,7. (b) A 3-edge even labeling of B10,8.
(a)(b)


## 2.6. Ther-Edge Even Graceful Labeling of the Ladder Graph Ln=P2×Pn

Theorem 6.
The ladder graphLn=P2×Pn has 2-edge even graceful labeling.Proof.
The graphLn has p=2n vertices and q=3n−2 edges. Let r=2; then, 2q+2r−2=6n−2.
Consider the ladder graphLn=P2×Pn shown in Figure 11.
We define the labeling functionf:ELn⟶2,4,…,6n−2 as follows:(15)fai=2i,i=1,2,…,n,fci=6n−2i−2,i=1,2,…,n−1,fbi=4n−2i−2,ifi=1,2,…,n−2,6n−2,ifi=n−1.
Then, the induced vertex labels are(16)f∗v1=fa1+fc1mod6n−2=0,f∗vn=fan+fcn−1mod6n−2=2,f∗vi=fci+fci−1+faimod6n−2=6n−2i,i=2,…,n−1,f∗u1=fa1+fb1mod6n−2=4n−2,f∗un−1=fan−1+fbn−1+fbn−2mod6n−2=4n,f∗un=fan+fbn−1mod6n−2=2n,f∗ui=fbi+fbi−1+faimod6n−2=2n−2i,i=2,…,n−2.
So, the labels of the verticesv2,v3,…,vn−2,vn−1 will be 6n−4,6n−6,…,4n+4,4n+2, respectively, and the labels of the vertices u2,u3,…,un−3,un−2 will be 2n−4,2n−6,…,6,4, respectively. Clearly, all the labels of the vertices are even and there is no repetition on the labels. Hence, the ladder graph Ln has 2-edge even graceful labeling.Figure 11
The ladder graphLn=P2×Pn with ordinary labeling.Illustration. A 2-edge even labeling of the ladder graph L10 is shown in Figure 12.Figure 12
A 2-edge even labeling of the ladder graphL10=P2×P10.


## 2.7. Ther-Edge Even Graceful Labeling of the Diagonal Ladder DLn

Diagonal ladder graph is a ladder with additional edgesviui+1 and uivi+1,i=1,2,…,n−1, denoted by DLn, where n is half its vertices and the number of vertices VDLn=2n and edges EDLn=5n−4.Theorem 7.
The diagonal ladderDLn has 2-edge even graceful labeling.Proof.
Consider the diagonal ladderDLn shown in Figure 13. Let r=2; then, 2q+2r−2=10n−6.
The labelingf:EDLn⟶2,4,…,10n−6 can be defined as follows:(17)fviui+1=2i,i=1,2,…,n−1,fvi+1ui=2n+2i,i=1,2,…,n−1,fvivi+1=10n−6−2i,i=1,2,…,n−1,fuiui+1=8n−4−2i,i=1,2,…,n−1,fviui=10n−6,ifi=1,6n−2i,ifi=2,3,…,n.
Then, the induced vertex labels are(18)f∗v1=fv1v2+fv1u1+fv1u2mod10n−6=0,f∗u1=fu1u2+fv1u1+fu1v2mod10n−6=2.
Fori=2,3,…,n−1,(19)f∗vi=fvivi−1+fvivi+1+fviui+fviui+1+fviui−1mod10n−6=8n−2i,f∗ui=fuiui−1+fuiui+1+fviui+fuivi+1+fuivi−1mod10n−6=4n+4−2i.
So, the labels of the verticesv2,v3,…,vn−2,vn−1 are 8n−4,8n−6,…,6n+4,6n+2, respectively, and the labels of the vertices u2,u3,…,un−2,un−1 are 4n,4n−2,…,2n+8,2n+6, respectively, and(20)f∗vn=fvnvn−1+fvnun+fvnun−1mod10n−6=6n,f∗un=funun−1+fvnun+funvn−1mod10n−6=2n+2.
Clearly, all the labels of the vertices are even and distinct, and thus we get the result.Figure 13
The diagonal ladderDLn with ordinary labeling.Illustration. A 2-edge even labeling of the diagonal ladder DL8 is shown in Figure 14.Figure 14
A2-edge even graceful for the diagonal ladder DL8.


## 2.8. Ther-Edge Even Graceful Labeling of the Tortoise Graph Tn

The tortoise graphTn is the graph obtained from a path Pn, where n is odd, by attaching an edge between vi and vn−i+1 for 1≤i≤n/2, so VTn=VPn and ETn=EPn∪ei=vivn−i+1,i=n,n−1,…,n+3/2.Theorem 8.
The tortoise graphTn has 2-edge even graceful labeling.Proof.
The graphTn has n vertices and q=3n−1/2 edges. Consider the tortoise graph Tn shown in Figure 15.
Letr=2; then, 2q+2r−2=3n−1, and we define the labeling f:ETn⟶2,4,…,3n−1 as follows:(21)fei=2i,ifi=1,2,3,…,n+12,3n−2i+1,ifi=n+32,…,n−1,3n−1,ifi=n,5n−2i−1,ifi=n+1,n+2,…,3n−12.
In view of the above labeling pattern, we have(22)f∗v1=fe1+fenmod3n−1=2,f∗vn=fen+fen−1mod3n−1=n+3,f∗vi=fei+fei−1+fei+n−1mod3n−1=2i,i=2,3,…,n−12,f∗vn+1/2=fen+1/2+fen−1/2mod3n−1=2n,f∗vn+3/2=fen+3/2+fen+1/2fe3n−1/2mod3n−1=2n+2,f∗vi=fei+fei−1+fe2n−imod3n−1=n−2i+5,i=n+52,…,n−1.
Clearly all the label of the vertices are even and distinct, changed inside the set0,2,4,…,2q+2r−2=0,2,…,3n−1. Hence the tortoise Tn has a 2- edge even labeling.Figure 15
The tortoise graphTn with ordinary labeling.Illustration. A 2-edge even labeling of the tortoise graph T11 is shown in Figure 16.Figure 16
A2-edge even graceful labeling for the tortoise graph T11.


## 3. Ther-Edge Even Graceful Labeling of Some Cycle-Related Graphs

### 3.1. Ther-Edge Even Graceful Labeling of the Even Cycle Graph Cm

The cycle graphCm has an edge even graceful labeling only when m is an odd number [8], so we will study the case when m is an even number, i.e., m=2n.Theorem 9.
The cycle graphC2n has 2-edge even graceful labeling. If n is even, the label is strong.Proof.
In the cycle graphC2n, p=q=2n. If we assume that r=2, then 2q+2r−2=4n+2. We will try to define an injective mapping f:EG⟶2,4,…,4n+2 such that f∗v=∑u∈VGfuvmod4n+2 is an injective. Now there are two cases.Case (1). When n is odd. Let the cyclic graph C2n be given as shown in Figure 17(a). The cyclic edges EC2n=e1,e2,…,en,en+1,en+2,…,e2n−1,e2n will be labeled as e1,e2,…,en,a1,a2,…,an−1,an, where ai=en+i,i=1,…,n such that ai is put opposite to ei.We define the labeling functionf:EG⟶2,4,…,4n+2 as follows:(23)fei=2i,i=1,2,…,n,fai=4n+4−2i,i=1,2,…,n,we see that(24)f∗v2=fe2+fe1mod4n+2=6,f∗vn=fen+fan−1mod4n+2=4n+6mod4n+2=4,f∗vi=fei+fei−2mod4n+2=4i−4,ifi=4,6,…,n−1,fei+fei+2mod4n+2=4i+4,ifi=1,3,…,n−2,f∗u1=fa3+fa1mod4n+2=fa3=4n−2,f∗u2=fa2+fa1mod4n+2=fa2=4n,f∗un=fen−1+fanmod4n+2=0,f∗ui=fai+fai−2mod4n+2=4n+10−4i,ifi=4,6,…,n−1,fai+fai+2mod4n+2=4n+2−4i,ifi=3,5,…,n−2.Hence, the labels of the verticesv1,v3,v5,…,vn−2 are 8,16,24,…,4n−4, respectively, and the labels of the vertices v4,v6,v8,…,vn−1 are 12,20,28,…,4n−8, respectively. The labels of the vertices u3,u5,…,un−2 are 4n−10,4n−18,…,10 and the labels of the vertices u4,u6,…,vn−1 will be 4n−6,4n−14,…,14, respectively. Therefore, the vertex labels are even and distinct. So, the graph has 2-edge even graceful labeling.Case (2). When n is even. Let C2n be given as shown in Figure 17(b), and we define the labeling function f:EG⟶2,4,…,4n+2 as follows:(25)fei=2i,i=1,2,…,n,fai=4n+2−fe2i=4n+2−4i,i=1,2,…,n2,fbi=4n+2−fe2i−1=4n+4−4i,i=1,2,…,n2.
Thus, the vertex labels are(26)f∗vi=fei+fei+1mod4n+2=4i+2;i=1,2,…,n−1,f∗vn=fen+fbn/2mod4n+2=4n+4mod4n+2=2,f∗v2n=fe1+fa1mod4n+2=4n,f∗v2n−i=fai+1/2+fbi+1/2mod4n+2=4n−4i,ifi=1,3,5,…,n+1,fai/2+1+fbi/2mod4n+2=4n−4i,ifi=2,4,6,…,n+2.
Hence, the labels of the verticesv1,v2,v3,v4,…,vn−1 will be 6,10,14,18,…,4n−2, respectively, and the labels of the vertices vn+1,vn+2,vn+3,vn+4,…,v2n−1 will be 4,8,12,16,…,4n−4. Therefore the vertex labels are even and distinct and changed inside the set 2r−2,2r,…,2q+2r−2=2,−4,…,4n+2. So, the graph C2.n has strong2-edge even graceful labeling when n is even.Figure 17
(a)C2n with ordinary labeling when n is odd. (b) C2n with ordinary labeling when n is even.
(a)(b)Illustration. The graphs C18 and C16 labeled according to Theorem 9 are presented in Figure 18.Figure 18
(a) A2-edge even graceful labeling of C18=C2.9. (b) A strong2-edge even graceful labeling of C16=C2.8.
(a)(b)


### 3.2. Ther-Edge Even Graceful Labeling of the Crown Rn and Cn⊙K2m−1¯

A crownRn is formed by adding n more pendent vertices u1,u2,u3,…,un and n more edges viui,i=1,2,…,n for n≥3 to the n vertices v1,v2,v3,…,vn of a cycle Cn.Proposition 1.
The crown graphRn has 2-edge even graceful labeling.Proof.
The crown graphRn has p=2n vertices and q=2n edges. Consider the crown graph Rn shown in Figure 19, where ai=viui and ei=vivi+1,1≤i≤n.
We can define the labelingf:ERn⟶2,4,…,4n+2 as follows:(27)fei=2i,i=1,2,…,n,fai=4n−2i+4,i=1,2,…,n.
Now, the induced labels of the vertices are(28)f∗v1=fe1+fen+fa1mod4n+2=6n+4mod4n+2=2n+2,f∗vi=fei+fei−1+faimod4n+2=4n+2+2imod4n+2=2i,i=2,3,…,n.
Also, the verticesui take the labels of the pendant edges which are 0,4n,4n−2,…,2n+4, so overall all the vertex labels are even and there is no repetition. Thus, the crown graph Rn has 2-edge even graceful labeling.Figure 19
The crown graphRn with ordinary labeling.The generalization of the previous result is presented in the following theory.Theorem 10.
The graphCn⊙K2m−1¯ has 2-edge even graceful labeling.Proof.
In the graphCn⊙K2m−1¯, the cardinality of the vertex set and the edge set is p=4mn and q=4mn, respectively. Let the vertex and edge symbols be given as in Figure 20.
Letr=2, and we define the labeling map f:ECn⊙K2m−1¯⟶2,4,…,4mn+2 as follows:(29)fEi=2i,fori=1,2,…,n,fei=4mn+21−i,fori=1,2,…,n,fAij=2n+4i−1+2j,forj=1,2,…,m−1,fBij=4mn−2n−4i−2j+6,forj=1,2,…,m−1.
We realize the following:(30)fAij+fBijmod4mn+2≡0mod4mn+2,forj=1,2,…,m−1,fEi+feimod4mn+2≡0mod4mn+2,fori=1,2,3,…,n.
Then, we have(31)f∗vi=∑j=1m−1fAij+∑j=1m−1fBij+fEi+fEi−1+feimod4mn+2,i=2,3,…,n=fEi−1mod4mn+2=2i−1,i=2,3,…,n.
Similarly,f∗v1=fEn=2n.
Hence, the labels of the verticesv1,v2,…,vn take the label of the edges of the cycles and each of the pendant vertices takes the label of its edge, so they are all even and different numbers.Figure 20
Cn⊙K2m−1¯ with ordinary labeling.
(a)(b)Illustration. In Figure 21, we present 2-edge even graceful labeling of the crown R9 and the graph C6⊙K5¯.Figure 21
(a)2-edge even graceful labeling of the crown R9. (b) 2-edge even graceful labeling of C6⊙K5¯.
(a)(b)


### 3.3. Ther-Edge Even Graceful Labeling of the Helm Graph Hn

The helmHn is formed from the wheel graph Wn=v0,v1,v2,v3,…,vn where v0 is the center vertex by adding n more pendent vertices u1,u2,u3,…,un and n more edges viui,i=1,2,…,n.Theorem 11.
Forn≥3, the helm graph Hn has 2-edge even graceful labeling.Proof.
The helm graphHn has p=2n+1 vertices and q=3n edges and K=maxp,q=3n. Let the helm graph Hn be given as indicated in Figure 22. We assume that r=2; then, 2q+2r−2=6n+2.
To find the labeling functionf:EHn⟶2,4,…,6n+2, we have two cases.Case (1). When n is even, we define the labeling function f:EHn⟶2,4,…,6n+2 as follows:(32)fei=3n+2i,ifi=1,2,…,n2,6n+2,ifi=n2+1,3n+2i−2,ifi=n2+2,…,n,fv0vi=2i,ifi=1,2,…,n2,4n+2i,ifi=n2+1,n2+2,…,n,fviui=n+2,ifi=1,3n+4−2i,ifi=2,3,…,n.In this labeling, we notice that(i)
fv0vi+fv0vn−i+1mod6n+2≡0mod6n+2,fori=1,2,…,n/2.(ii)
fviui+fei−1mod6n+2≡0mod6n+2,fori=2,…,n/2.(iii)
fviui+feimod6n+2≡0mod6n+2,fori=n/2+2,…,n.In view of the above labeling pattern, the induced vertex labels are(33)f∗v0=∑i=1nfv0vimod6n+2=0,f∗v1=fe1+fen+fv0v1+fv1u1mod6n+2=3n+2,f∗vn/2+1=fen/2+1+fen/2+fv0vn/2+1+fvn/2+1un/2+1mod6n+2=5n+2,f∗vn/2+2=fen/2+2+fen/2+1+fv0vn/2+2+fvn/2+2un/2+2mod6n+2=5n+4.Sincef∗vi=fei+fei−1+fv0vi+fviuimod6n+2,(34)f∗vi=3n+4imod6n+2,ifi=2,3,…,n2,n−6+4imod6n+2,ifi=n2+3,…,n.Hence, the labels of the verticesv2,v3,…,vn/2−1,vn/2 will be 3n+8,3n+12,…,5n−4,5n, respectively, and the labels of the vertices vn/2+3,vn/2+4,…,vn−1,vn will be 3n+6,3n+10,…,5n−10,5n−6, respectively, which are all even and distinct numbers.Case (2). When n is odd, by considering the helm graph Hn shown in Figure 22(b), we can define the labeling f:EHn⟶2,4,…,6n+2 as follows:(35)fei=3n+2i−1,ifi=1,2,…,n−12,3n+2i+1,ifi=n+12,…,n,fv0vi=2i,ifi=1,2,…,n−12,4n+2i+2,ifi=n+12,…,n,fviui=n+1,ifi=1,3n+3−2i,ifi=2,3,…,n.In this labeling, we notice that(i)
fv0vi+fv0vn−imod6n+2≡0mod6n+2,fori=1,2,…,n−1/2.(ii)
fviui+feimod6n+2≡0mod6n+2,fori=2,…,n−1/2.(iii)
fviui+fei−1mod6n+2≡0mod6n+2,fori=n+3/2,…,n.
Considering the vertex labels, the induced vertex labels are(36)f∗v0=∑i=1nfv0vimod6n+2=0,f∗v1=fe1+fen+fv0v1+fv1u1mod6n+2=3n+3,f∗vn+1/2=fen+1/2+fen−1/2+fv0vn+1/2+fvn+1/2un+1/2mod6n+2=3n+1.
Sincef∗vi=fei+fei−1+fv0vi+fviuimod6n+2,(37)f∗vi=3n+4i−3mod6n+2,ifi=2,3,…,n−12,n+1+4imod6n+2,ifi=n+32,…,n.
Hence, the labels of the verticesv2,v3,…,vn−3/2,vn−1/2 are 3n+5,3n+9,…,5n−9,5n−5, respectively, and the labels of the vertices vn+3/2,vn+5/2,…,vn−1,vn are 3n+7,3n+11,…,5n−3,5n+1, respectively, which are all even and distinct numbers. Thus, the helm graph Hn has 2-edge even graceful labeling.Figure 22
(a) The graphHn when n is even. (b) The graph Hn when n is odd.
(a)(b)Illustration. In Figure 23, we present a 2-edge even graceful labeling of H11 and H12.Figure 23
(a)2-edge even graceful labeling of H11. (b) 2-edge even graceful labeling of H12.
(a)(b)


### 3.4. Ther-Edge Even Graceful Labeling of the Sunflower Graph sfn

The sunflower graph,sfn, is defined as a graph obtained by starting with an n-cycle Cn with consecutive vertices v1,v2,…,vn and creating new vertices u1,u2,…,un, with ui connected to vi and vi+1, so(38)Esfn=ai=viui,bi=uivi+1,ci=vivi+1.Theorem 12.
The sunflower graph,sfn, has 2-edge even graceful labeling.Proof.
The graphsfn has p=2n vertices and q=3n edges and K=maxp,q=3n (see Figure 24). Assume that r=2; then, 2q+2r−2=6n+2, and we consider two cases.Case (1). When n is even. The labeling f:Esfn⟶2,4,…,6n+2 can be defined as follows:(39)fai=2i,fori=1,2,3,…,n,fbi=6n+4−4i,fori=1,2,3,…,n,fci=2n+2i,ifi=1,3,…,n−1,6n+2−2i,ifi=2,4,…,n.The induced vertex labels are(40)f∗ui=fai+fbimod6n+2=6n+4−2imod6n+2,fori=1,2,…,n,f∗v1=fa1+fbn+fcn+fc1mod6n+2=2n+8.Sincef∗vi=fai+fbi−1+fci+fci−1mod6n+2,(41)f∗vi=2n+8−2imod6n+2,ifi=3,5,…,n−1,2n+4−2imod6n+2,ifi=2,4,…,n.Hence, the labels of the verticesu1,u2,u3,…,un−1,un are 0,6n,6n−2,…,4n+6,4n+4, respectively, and the labels of the vertices v3,v5,…,vn−3,vn−1 will be 2n+2,2n−2,…,14,10, respectively. Also, the labels of the vertices v2,v4,…,vn−2,vn will be 2n,2n−4,…,8,4, respectively, which are all even and distinct numbers.Case (2). When n is odd, we define the labeling f:Esfn⟶2,4,…,6n+2 as follows:(42)fai=2i,fori=1,2,3,…,n,fbi=6n+4−4i,fori=1,2,3,…,n,fci=2n+2i,ifi=1,3,…,n,6n+2−2i,ifi=2,4,…,n−1.
The induced vertex labels are(43)f∗ui=fai+fbimod6n+2=6n+4−2imod6n+2,fori=1,2,…,n,f∗v1=fa1+fbn+fcn+fc1mod6n+2=2n+6.
Sincef∗vi=fai+fbi−1+fci+fci−1mod6n+2,(44)f∗vi=2n+8−2imod6n+2,ifi=3,5,…,n,2n+4−2imod6n+2,ifi=2,4,…,n−1.
Hence, the labels of the verticesu1,u2,u3,…,un−1,un will be 0,6n,6n−2,…,4n+6,4n+4, respectively, and the labels of the vertices v3,v5,…,vn−2,vn will be 2n+2,2n−2,…,12,8, respectively. Also, the labels of the vertices v2,v4,…,vn−3,vn−1 will be 2n,2n−4,…,10,6, respectively, which are all even and distinct numbers. Thus, the sunflower graph, sfn, has 2-edge even graceful labeling.Figure 24
The graphsfn with ordinary labeling.Illustration. In Figure 25, we present 2-edge even graceful labeling of sf9 and sf10.Figure 25
(a) A2-edge even graceful labeling of sf9. (b) A 2-edge even graceful labeling of sf10.
(a)(b)


### 3.5. Ther-Edge Even Graceful Labeling of the Sunflower Planar Graph SFn

The sunflower planar graph,SFn, is defined as a graph obtained by starting with an n-wheel Wn with a center vertex v0 and consecutive vertices v1,v2,…,vn and creating new vertices u1,u2,…,un, with ui connected to vi and vi+1, so Esfn=v0vi,ai=viui,bi=uivi+1,ci=vivi+1.Theorem 13.
The sunflower planar graph,SFn, has 2-edge even graceful labeling.Proof.
The graphSFn always has an odd number of vertices p=2n+1 and a number of edges q=4n and K=maxp,q=4n. Consider the graph SFn shown in Figure 26. Assume that r=2; then, 2q+2r−2=8n+2.
To find the label mapf:ESFn⟶2,4,…,8n+2, there are two cases.Case (1). When n is even. The labeling f can be defined as follows:(45)fai=n+2i,∀i=1,2,3,…,n,fbi=3n+2i,∀i=1,2,3,…,n,fvivi+1=7n+2−2i,∀i=1,2,3,…,n,fv0vi=2i,ifi=1,2,…,n2,6n+2i,ifi=n2+1,n2+2,…,n.In this labeling, we can check that(i)
fv0vi+fv0vn−i+1mod8n+2≡0mod8n+2,fori=1,2,…,n/2.(ii)
fvivi+1+faimod8n+2≡0mod8n+2,fori=1,2,…,n.In view of the above labeling pattern, the induced vertex labels are(46)f∗v0=∑i=1nfv0vimod8n+2=0,f∗ui=fai+fbimod8n+2=4n+4imod8n+2,fori=1,2,…,n.Since(47)f∗vi=fai+fbi−1+fv0vi+fvivi+1+fvivi−1mod8n+2,f∗vi=2n+2imod8n+2,ifi=1,2,3,…,n2,2imod8n+2,ifi=n2+1,n2+2,…,n.Hence, the labels of the verticesu1,u2,u3,…,un−1,un are 4n+4,4n+8,4n+12,…,8n−4,8n, respectively, and the labels of the vertices v1,v2,…,vn/2−1,vn/2 are 2n+2,2n+4,…,3n−2,3n, respectively. Also, the labels of the vertices vn/2+1,vn/2+2,…,vn−1,vn will be n,3n+2,…,2n−4,2n−2, respectively, which are all even and distinct numbers.Case (2). When n is odd. Define the labeling f:ESFn⟶2,4,…,8n+2 as follows:(48)fai=n−1+2i,∀i=1,2,3,…,n,fbi=3n−1+2i,∀i=1,2,3,…,n,fvivi+1=7n+3−2i,∀i=1,2,3,…,n,fv0vi=2i,ifi=1,2,…,n−12,6n+2i+2,ifi=n+12,n+32,…,n.We can check that(i)
fv0vi+fv0vn−imod8n+2≡0mod8n+2,fori=1,2,…,n/2.(ii)
fvivi+1+faimod8n+2≡0mod8n+2,fori=1,2,…,n.
In this labeling, the induced vertex labels are(49)f∗v0=∑i=1nfv0vimod8n+2=0,f∗ui=fai+fbimod8n+2=4n+4i−2mod8n+2,fori=1,2,…,n.
Since(50)f∗vi=fai+fbi−1+fv0vi+fvivi+1+fvivi−1mod8n+2,f∗vi=2n+2imod8n+2,ifi=1,2,3,…,n−12,2imod8n+2,ifi=n+12,n+32,…,n.
Hence, the labels of the verticesu1,u2,u3,…,un−1,un are 4n+2,4n+6,4n+10,…,8n−6,8n−2, respectively, and the labels of the vertices v1,v2,…,vn−3/2,vn−1/2 are 2n+2,2n+4,…,3n−3,3n−1, respectively. Also, the labels of the vertices vn+1/2,vn+3/2,…,vn−1,vn are n+1,n+3,…,2n−2,2n, respectively, and these are all even and distinct numbers. Thus, the sunflower planar graph, SFn, has 2-edge even graceful labeling.Figure 26
The graphSFn with ordinary labeling.Illustration. In Figure 27, we present 2-edge even graceful labeling of SF9 and SF10.Figure 27
(a)2-edge even graceful labeling of SF9. (b) 2-edge even graceful labeling of SF10.
(a)(b)


## 3.1. Ther-Edge Even Graceful Labeling of the Even Cycle Graph Cm

The cycle graphCm has an edge even graceful labeling only when m is an odd number [8], so we will study the case when m is an even number, i.e., m=2n.Theorem 9.
The cycle graphC2n has 2-edge even graceful labeling. If n is even, the label is strong.Proof.
In the cycle graphC2n, p=q=2n. If we assume that r=2, then 2q+2r−2=4n+2. We will try to define an injective mapping f:EG⟶2,4,…,4n+2 such that f∗v=∑u∈VGfuvmod4n+2 is an injective. Now there are two cases.Case (1). When n is odd. Let the cyclic graph C2n be given as shown in Figure 17(a). The cyclic edges EC2n=e1,e2,…,en,en+1,en+2,…,e2n−1,e2n will be labeled as e1,e2,…,en,a1,a2,…,an−1,an, where ai=en+i,i=1,…,n such that ai is put opposite to ei.We define the labeling functionf:EG⟶2,4,…,4n+2 as follows:(23)fei=2i,i=1,2,…,n,fai=4n+4−2i,i=1,2,…,n,we see that(24)f∗v2=fe2+fe1mod4n+2=6,f∗vn=fen+fan−1mod4n+2=4n+6mod4n+2=4,f∗vi=fei+fei−2mod4n+2=4i−4,ifi=4,6,…,n−1,fei+fei+2mod4n+2=4i+4,ifi=1,3,…,n−2,f∗u1=fa3+fa1mod4n+2=fa3=4n−2,f∗u2=fa2+fa1mod4n+2=fa2=4n,f∗un=fen−1+fanmod4n+2=0,f∗ui=fai+fai−2mod4n+2=4n+10−4i,ifi=4,6,…,n−1,fai+fai+2mod4n+2=4n+2−4i,ifi=3,5,…,n−2.Hence, the labels of the verticesv1,v3,v5,…,vn−2 are 8,16,24,…,4n−4, respectively, and the labels of the vertices v4,v6,v8,…,vn−1 are 12,20,28,…,4n−8, respectively. The labels of the vertices u3,u5,…,un−2 are 4n−10,4n−18,…,10 and the labels of the vertices u4,u6,…,vn−1 will be 4n−6,4n−14,…,14, respectively. Therefore, the vertex labels are even and distinct. So, the graph has 2-edge even graceful labeling.Case (2). When n is even. Let C2n be given as shown in Figure 17(b), and we define the labeling function f:EG⟶2,4,…,4n+2 as follows:(25)fei=2i,i=1,2,…,n,fai=4n+2−fe2i=4n+2−4i,i=1,2,…,n2,fbi=4n+2−fe2i−1=4n+4−4i,i=1,2,…,n2.
Thus, the vertex labels are(26)f∗vi=fei+fei+1mod4n+2=4i+2;i=1,2,…,n−1,f∗vn=fen+fbn/2mod4n+2=4n+4mod4n+2=2,f∗v2n=fe1+fa1mod4n+2=4n,f∗v2n−i=fai+1/2+fbi+1/2mod4n+2=4n−4i,ifi=1,3,5,…,n+1,fai/2+1+fbi/2mod4n+2=4n−4i,ifi=2,4,6,…,n+2.
Hence, the labels of the verticesv1,v2,v3,v4,…,vn−1 will be 6,10,14,18,…,4n−2, respectively, and the labels of the vertices vn+1,vn+2,vn+3,vn+4,…,v2n−1 will be 4,8,12,16,…,4n−4. Therefore the vertex labels are even and distinct and changed inside the set 2r−2,2r,…,2q+2r−2=2,−4,…,4n+2. So, the graph C2.n has strong2-edge even graceful labeling when n is even.Figure 17
(a)C2n with ordinary labeling when n is odd. (b) C2n with ordinary labeling when n is even.
(a)(b)Illustration. The graphs C18 and C16 labeled according to Theorem 9 are presented in Figure 18.Figure 18
(a) A2-edge even graceful labeling of C18=C2.9. (b) A strong2-edge even graceful labeling of C16=C2.8.
(a)(b)


## 3.2. Ther-Edge Even Graceful Labeling of the Crown Rn and Cn⊙K2m−1¯

A crownRn is formed by adding n more pendent vertices u1,u2,u3,…,un and n more edges viui,i=1,2,…,n for n≥3 to the n vertices v1,v2,v3,…,vn of a cycle Cn.Proposition 1.
The crown graphRn has 2-edge even graceful labeling.Proof.
The crown graphRn has p=2n vertices and q=2n edges. Consider the crown graph Rn shown in Figure 19, where ai=viui and ei=vivi+1,1≤i≤n.
We can define the labelingf:ERn⟶2,4,…,4n+2 as follows:(27)fei=2i,i=1,2,…,n,fai=4n−2i+4,i=1,2,…,n.
Now, the induced labels of the vertices are(28)f∗v1=fe1+fen+fa1mod4n+2=6n+4mod4n+2=2n+2,f∗vi=fei+fei−1+faimod4n+2=4n+2+2imod4n+2=2i,i=2,3,…,n.
Also, the verticesui take the labels of the pendant edges which are 0,4n,4n−2,…,2n+4, so overall all the vertex labels are even and there is no repetition. Thus, the crown graph Rn has 2-edge even graceful labeling.Figure 19
The crown graphRn with ordinary labeling.The generalization of the previous result is presented in the following theory.Theorem 10.
The graphCn⊙K2m−1¯ has 2-edge even graceful labeling.Proof.
In the graphCn⊙K2m−1¯, the cardinality of the vertex set and the edge set is p=4mn and q=4mn, respectively. Let the vertex and edge symbols be given as in Figure 20.
Letr=2, and we define the labeling map f:ECn⊙K2m−1¯⟶2,4,…,4mn+2 as follows:(29)fEi=2i,fori=1,2,…,n,fei=4mn+21−i,fori=1,2,…,n,fAij=2n+4i−1+2j,forj=1,2,…,m−1,fBij=4mn−2n−4i−2j+6,forj=1,2,…,m−1.
We realize the following:(30)fAij+fBijmod4mn+2≡0mod4mn+2,forj=1,2,…,m−1,fEi+feimod4mn+2≡0mod4mn+2,fori=1,2,3,…,n.
Then, we have(31)f∗vi=∑j=1m−1fAij+∑j=1m−1fBij+fEi+fEi−1+feimod4mn+2,i=2,3,…,n=fEi−1mod4mn+2=2i−1,i=2,3,…,n.
Similarly,f∗v1=fEn=2n.
Hence, the labels of the verticesv1,v2,…,vn take the label of the edges of the cycles and each of the pendant vertices takes the label of its edge, so they are all even and different numbers.Figure 20
Cn⊙K2m−1¯ with ordinary labeling.
(a)(b)Illustration. In Figure 21, we present 2-edge even graceful labeling of the crown R9 and the graph C6⊙K5¯.Figure 21
(a)2-edge even graceful labeling of the crown R9. (b) 2-edge even graceful labeling of C6⊙K5¯.
(a)(b)


## 3.3. Ther-Edge Even Graceful Labeling of the Helm Graph Hn

The helmHn is formed from the wheel graph Wn=v0,v1,v2,v3,…,vn where v0 is the center vertex by adding n more pendent vertices u1,u2,u3,…,un and n more edges viui,i=1,2,…,n.Theorem 11.
Forn≥3, the helm graph Hn has 2-edge even graceful labeling.Proof.
The helm graphHn has p=2n+1 vertices and q=3n edges and K=maxp,q=3n. Let the helm graph Hn be given as indicated in Figure 22. We assume that r=2; then, 2q+2r−2=6n+2.
To find the labeling functionf:EHn⟶2,4,…,6n+2, we have two cases.Case (1). When n is even, we define the labeling function f:EHn⟶2,4,…,6n+2 as follows:(32)fei=3n+2i,ifi=1,2,…,n2,6n+2,ifi=n2+1,3n+2i−2,ifi=n2+2,…,n,fv0vi=2i,ifi=1,2,…,n2,4n+2i,ifi=n2+1,n2+2,…,n,fviui=n+2,ifi=1,3n+4−2i,ifi=2,3,…,n.In this labeling, we notice that(i)
fv0vi+fv0vn−i+1mod6n+2≡0mod6n+2,fori=1,2,…,n/2.(ii)
fviui+fei−1mod6n+2≡0mod6n+2,fori=2,…,n/2.(iii)
fviui+feimod6n+2≡0mod6n+2,fori=n/2+2,…,n.In view of the above labeling pattern, the induced vertex labels are(33)f∗v0=∑i=1nfv0vimod6n+2=0,f∗v1=fe1+fen+fv0v1+fv1u1mod6n+2=3n+2,f∗vn/2+1=fen/2+1+fen/2+fv0vn/2+1+fvn/2+1un/2+1mod6n+2=5n+2,f∗vn/2+2=fen/2+2+fen/2+1+fv0vn/2+2+fvn/2+2un/2+2mod6n+2=5n+4.Sincef∗vi=fei+fei−1+fv0vi+fviuimod6n+2,(34)f∗vi=3n+4imod6n+2,ifi=2,3,…,n2,n−6+4imod6n+2,ifi=n2+3,…,n.Hence, the labels of the verticesv2,v3,…,vn/2−1,vn/2 will be 3n+8,3n+12,…,5n−4,5n, respectively, and the labels of the vertices vn/2+3,vn/2+4,…,vn−1,vn will be 3n+6,3n+10,…,5n−10,5n−6, respectively, which are all even and distinct numbers.Case (2). When n is odd, by considering the helm graph Hn shown in Figure 22(b), we can define the labeling f:EHn⟶2,4,…,6n+2 as follows:(35)fei=3n+2i−1,ifi=1,2,…,n−12,3n+2i+1,ifi=n+12,…,n,fv0vi=2i,ifi=1,2,…,n−12,4n+2i+2,ifi=n+12,…,n,fviui=n+1,ifi=1,3n+3−2i,ifi=2,3,…,n.In this labeling, we notice that(i)
fv0vi+fv0vn−imod6n+2≡0mod6n+2,fori=1,2,…,n−1/2.(ii)
fviui+feimod6n+2≡0mod6n+2,fori=2,…,n−1/2.(iii)
fviui+fei−1mod6n+2≡0mod6n+2,fori=n+3/2,…,n.
Considering the vertex labels, the induced vertex labels are(36)f∗v0=∑i=1nfv0vimod6n+2=0,f∗v1=fe1+fen+fv0v1+fv1u1mod6n+2=3n+3,f∗vn+1/2=fen+1/2+fen−1/2+fv0vn+1/2+fvn+1/2un+1/2mod6n+2=3n+1.
Sincef∗vi=fei+fei−1+fv0vi+fviuimod6n+2,(37)f∗vi=3n+4i−3mod6n+2,ifi=2,3,…,n−12,n+1+4imod6n+2,ifi=n+32,…,n.
Hence, the labels of the verticesv2,v3,…,vn−3/2,vn−1/2 are 3n+5,3n+9,…,5n−9,5n−5, respectively, and the labels of the vertices vn+3/2,vn+5/2,…,vn−1,vn are 3n+7,3n+11,…,5n−3,5n+1, respectively, which are all even and distinct numbers. Thus, the helm graph Hn has 2-edge even graceful labeling.Figure 22
(a) The graphHn when n is even. (b) The graph Hn when n is odd.
(a)(b)Illustration. In Figure 23, we present a 2-edge even graceful labeling of H11 and H12.Figure 23
(a)2-edge even graceful labeling of H11. (b) 2-edge even graceful labeling of H12.
(a)(b)


## 3.4. Ther-Edge Even Graceful Labeling of the Sunflower Graph sfn

The sunflower graph,sfn, is defined as a graph obtained by starting with an n-cycle Cn with consecutive vertices v1,v2,…,vn and creating new vertices u1,u2,…,un, with ui connected to vi and vi+1, so(38)Esfn=ai=viui,bi=uivi+1,ci=vivi+1.Theorem 12.
The sunflower graph,sfn, has 2-edge even graceful labeling.Proof.
The graphsfn has p=2n vertices and q=3n edges and K=maxp,q=3n (see Figure 24). Assume that r=2; then, 2q+2r−2=6n+2, and we consider two cases.Case (1). When n is even. The labeling f:Esfn⟶2,4,…,6n+2 can be defined as follows:(39)fai=2i,fori=1,2,3,…,n,fbi=6n+4−4i,fori=1,2,3,…,n,fci=2n+2i,ifi=1,3,…,n−1,6n+2−2i,ifi=2,4,…,n.The induced vertex labels are(40)f∗ui=fai+fbimod6n+2=6n+4−2imod6n+2,fori=1,2,…,n,f∗v1=fa1+fbn+fcn+fc1mod6n+2=2n+8.Sincef∗vi=fai+fbi−1+fci+fci−1mod6n+2,(41)f∗vi=2n+8−2imod6n+2,ifi=3,5,…,n−1,2n+4−2imod6n+2,ifi=2,4,…,n.Hence, the labels of the verticesu1,u2,u3,…,un−1,un are 0,6n,6n−2,…,4n+6,4n+4, respectively, and the labels of the vertices v3,v5,…,vn−3,vn−1 will be 2n+2,2n−2,…,14,10, respectively. Also, the labels of the vertices v2,v4,…,vn−2,vn will be 2n,2n−4,…,8,4, respectively, which are all even and distinct numbers.Case (2). When n is odd, we define the labeling f:Esfn⟶2,4,…,6n+2 as follows:(42)fai=2i,fori=1,2,3,…,n,fbi=6n+4−4i,fori=1,2,3,…,n,fci=2n+2i,ifi=1,3,…,n,6n+2−2i,ifi=2,4,…,n−1.
The induced vertex labels are(43)f∗ui=fai+fbimod6n+2=6n+4−2imod6n+2,fori=1,2,…,n,f∗v1=fa1+fbn+fcn+fc1mod6n+2=2n+6.
Sincef∗vi=fai+fbi−1+fci+fci−1mod6n+2,(44)f∗vi=2n+8−2imod6n+2,ifi=3,5,…,n,2n+4−2imod6n+2,ifi=2,4,…,n−1.
Hence, the labels of the verticesu1,u2,u3,…,un−1,un will be 0,6n,6n−2,…,4n+6,4n+4, respectively, and the labels of the vertices v3,v5,…,vn−2,vn will be 2n+2,2n−2,…,12,8, respectively. Also, the labels of the vertices v2,v4,…,vn−3,vn−1 will be 2n,2n−4,…,10,6, respectively, which are all even and distinct numbers. Thus, the sunflower graph, sfn, has 2-edge even graceful labeling.Figure 24
The graphsfn with ordinary labeling.Illustration. In Figure 25, we present 2-edge even graceful labeling of sf9 and sf10.Figure 25
(a) A2-edge even graceful labeling of sf9. (b) A 2-edge even graceful labeling of sf10.
(a)(b)


## 3.5. Ther-Edge Even Graceful Labeling of the Sunflower Planar Graph SFn

The sunflower planar graph,SFn, is defined as a graph obtained by starting with an n-wheel Wn with a center vertex v0 and consecutive vertices v1,v2,…,vn and creating new vertices u1,u2,…,un, with ui connected to vi and vi+1, so Esfn=v0vi,ai=viui,bi=uivi+1,ci=vivi+1.Theorem 13.
The sunflower planar graph,SFn, has 2-edge even graceful labeling.Proof.
The graphSFn always has an odd number of vertices p=2n+1 and a number of edges q=4n and K=maxp,q=4n. Consider the graph SFn shown in Figure 26. Assume that r=2; then, 2q+2r−2=8n+2.
To find the label mapf:ESFn⟶2,4,…,8n+2, there are two cases.Case (1). When n is even. The labeling f can be defined as follows:(45)fai=n+2i,∀i=1,2,3,…,n,fbi=3n+2i,∀i=1,2,3,…,n,fvivi+1=7n+2−2i,∀i=1,2,3,…,n,fv0vi=2i,ifi=1,2,…,n2,6n+2i,ifi=n2+1,n2+2,…,n.In this labeling, we can check that(i)
fv0vi+fv0vn−i+1mod8n+2≡0mod8n+2,fori=1,2,…,n/2.(ii)
fvivi+1+faimod8n+2≡0mod8n+2,fori=1,2,…,n.In view of the above labeling pattern, the induced vertex labels are(46)f∗v0=∑i=1nfv0vimod8n+2=0,f∗ui=fai+fbimod8n+2=4n+4imod8n+2,fori=1,2,…,n.Since(47)f∗vi=fai+fbi−1+fv0vi+fvivi+1+fvivi−1mod8n+2,f∗vi=2n+2imod8n+2,ifi=1,2,3,…,n2,2imod8n+2,ifi=n2+1,n2+2,…,n.Hence, the labels of the verticesu1,u2,u3,…,un−1,un are 4n+4,4n+8,4n+12,…,8n−4,8n, respectively, and the labels of the vertices v1,v2,…,vn/2−1,vn/2 are 2n+2,2n+4,…,3n−2,3n, respectively. Also, the labels of the vertices vn/2+1,vn/2+2,…,vn−1,vn will be n,3n+2,…,2n−4,2n−2, respectively, which are all even and distinct numbers.Case (2). When n is odd. Define the labeling f:ESFn⟶2,4,…,8n+2 as follows:(48)fai=n−1+2i,∀i=1,2,3,…,n,fbi=3n−1+2i,∀i=1,2,3,…,n,fvivi+1=7n+3−2i,∀i=1,2,3,…,n,fv0vi=2i,ifi=1,2,…,n−12,6n+2i+2,ifi=n+12,n+32,…,n.We can check that(i)
fv0vi+fv0vn−imod8n+2≡0mod8n+2,fori=1,2,…,n/2.(ii)
fvivi+1+faimod8n+2≡0mod8n+2,fori=1,2,…,n.
In this labeling, the induced vertex labels are(49)f∗v0=∑i=1nfv0vimod8n+2=0,f∗ui=fai+fbimod8n+2=4n+4i−2mod8n+2,fori=1,2,…,n.
Since(50)f∗vi=fai+fbi−1+fv0vi+fvivi+1+fvivi−1mod8n+2,f∗vi=2n+2imod8n+2,ifi=1,2,3,…,n−12,2imod8n+2,ifi=n+12,n+32,…,n.
Hence, the labels of the verticesu1,u2,u3,…,un−1,un are 4n+2,4n+6,4n+10,…,8n−6,8n−2, respectively, and the labels of the vertices v1,v2,…,vn−3/2,vn−1/2 are 2n+2,2n+4,…,3n−3,3n−1, respectively. Also, the labels of the vertices vn+1/2,vn+3/2,…,vn−1,vn are n+1,n+3,…,2n−2,2n, respectively, and these are all even and distinct numbers. Thus, the sunflower planar graph, SFn, has 2-edge even graceful labeling.Figure 26
The graphSFn with ordinary labeling.Illustration. In Figure 27, we present 2-edge even graceful labeling of SF9 and SF10.Figure 27
(a)2-edge even graceful labeling of SF9. (b) 2-edge even graceful labeling of SF10.
(a)(b)


## 4. Conclusions

Graph labeling has become a large and rich branch of various research studies in graph theory. It has enormous applications in many disciplines, like coding theory, X-rays, radar, communication networks, astronomy, and communication network. In this paper, we have extended the definition of edge even graceful labeling tor-edge even graceful labeling and strong r-edge even graceful labeling. We have obtained the necessary conditions for path-related graphs such as path graph Pn when n is odd, double star graph, ladder and diagonal ladder graphs, and tortoise graph and cycle-related graphs such as helm graph, crown graph, sunflower graph, and sunflower planar graph to be an r-edge even graceful graph. The minimum number r for which these graphs have an r-edge even graceful labeling was found.



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