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# Edge even graceful labeling of some graphs

*Journal of the Egyptian Mathematical Society*
**volume 27**, Article number: 20 (2019)

## Abstract

Edge even graceful labeling is a new type of labeling since it was introduced in 2017 by Elsonbaty and Daoud (Ars Combinatoria 130:79–96, 2017). In this paper, we obtained an edge even graceful labeling for some path-related graphs like Y- tree, the double star *B*_{n,m}, the graph 〈*K*_{1,2n}:*K*_{1,2m}〉, the graph \( ~P_{2n-1}\odot \overline { K_{2m}}~ \), and double fan graph *F*_{2,n}. Also, we showed that some cycle-related graphs like the prism graph \( ~\prod _{n}~ \), the graph \( ~C_{n}\left (\frac {n}{2}\right)~ \), the flag FL_{n}, the graph *K*_{2}⊙*C*_{n}, the flower graph FL(*n*), and the double cycle {*C*_{n,n}} are edge even graphs.

## Introduction

A graph labeling is an assignment of integers to the edges or vertices, or both, subject to certain condition. The idea of graph labelings was introduced by Rosa in [1]. Following this paper, other studies on different types of labelings (Odd graceful, Chordal graceful, Harmonious, edge odd graceful) introduced by many others [2–4]. A new type of labeling of a graph called an edge even graceful labeling has been introduced by Elsonbaty and Daoud [5]. They introduced some path- and cycle-related graphs which are edge even graceful.

Graph labelings give us useful models for a wide range of applications such as coding theory, X-ray, astronomy, radar, and communication network addressing.

###
**Definition 1**

[5] An edge even graceful labeling of a graph *G*(*V*(*G*), *E*(*G*)) with *p*=|*V*(*G*)| vertices and *q*=|*E*(*G*)| edges is a bijective mapping *f* of the edge set *E*(*G*) into the set {2,4,6, ⋯,2*q* } such that the induced mapping *f*^{∗}:*V*(*G*) →{0,2,4,⋯,2*q* }, given by: \( f^{\ast }(x) = \left ({\sum \nolimits }_{xy \in E(G)} f(xy)~ \right)~mod~(2k) \), is an injective function, where *k*=*m**a**x*(*p*,*q*). The graph that admits an edge even graceful labeling is called an edge even graceful graph.

In Fig. 1, we present an edge even graceful labeling of the Peterson graph and the complete graph *K*_{5}.

## Edge even graceful for some path related graphs

A Y- tree is a graph obtained from a path by appending an edge to a vertex of a path adjacent to an end point, and it is denoted by *Y*_{n} where *n* is the number of vertices in the tree.

###
**Lemma 1**

The Y-tree *Y*_{n} is an edge even graceful graph when *n* is odd.

###
*Proof*

The number of vertices of Y-tree *Y*_{n} is *n* and the number of edges is *n*−1. Let the vertices and the edges of *Y*_{n} be given as in Fig. 2.

We define the mapping *f*:*E*(*Y*_{n}) →{2,4,⋯,2*n*−2} as follows:

Then, the induced vertex labels are

Similarly, \( ~~~f^{\ast }\left (v_{\frac {n+3}{2}}\right)= 6~,~~~~~~~~~~~~~ f^{\ast }\left (v_{\frac {n+5}{2}}\right)= 8~, f^{\ast }\left (v_{\frac {n+7}{2}}\right)= 10 ~\),

Clearly, all the vertex labels are even and distinct. Thus, Y-tree *Y*_{n} is an edge even graceful graph when *n* is odd. □

**Illustration:** The edge even labeling of the graph *Y*_{13} is shown in Fig. 3.

Double star is the graph obtained by joining the center of the two stars *K*_{1,n} and *K*_{1,m} with an edge, denoted by *B*_{n,m}. The graph *B*_{n,m} has *p*=*n*+*m*+2 and *q*=*n*+*m*+1.

###
**Lemma 2**

The double star *B*_{n,m} is edge even graceful graph when one (*m* *o**r* *n*) is an odd number and the other is an even number.

###
*Proof*

Without loss of generality, assume that *n* is odd and *m* is even. Let the vertex and edge symbols be given as in Fig. 4.

Define the mapping *f*:*E*(*B*_{m,n}) →{2,4,⋯,2*q*} as follows:

We realize the following:

So, the vertex labels will be

Also, each pendant vertex takes the labels of its incident edge which are different from the labels of the vertices *U* and *v*. □

The graph 〈*K*_{1,n}:*K*_{1,m}〉 is obtained by joining the center *v*_{1} of the star *K*_{1,n} and the center *v*_{2} of the another star *K*_{1,m} to a new vertex *u*, so the number of vertices is *p*=*n*+*m*+3 and the number of edges is *q*=*n*+*m*+2.

###
**Lemma 3**

The graph 〈*K*_{1,2n}:*K*_{1,2m}〉 is an edge even graceful graph.

###
*Proof*

Let the vertex and edge symbols be given as in Fig. 5.

Define the mapping *f*:*E*(*G*) →{2,4,⋯,2*p*} as follows:

We can see the following:

So, the vertex labels will be

which are even and distinct from each pendant vertices. Thus, the graph 〈*K*_{1,2n}:*K*_{1,2m}〉 is an edge even graceful graph. □

###
**Lemma 4**

The corona *[*6*]*\( ~P_{3}\odot \overline { K_{2m}}~ \) is an edge even graceful graph.

###
*Proof*

In this graph *p*=6*m*+3 and *q*=6*m*+2. Let the vertex and edge symbols be given as in Fig. 6.

We can define the mapping \(f: E(P_{3}\odot \overline { K_{2m}}~)~\rightarrow \{2,4,\cdots, 2q \}\) as follows:

It is clear that *f*(*A*_{ij})+*f*(*B*_{ij})≡0 mod (2*p*) for *j*=1,2,⋯,*m*

So the vertex labels will be,

Therefore, all the vertices are even and distinct which complete the proof. □

**Illustration:** The edge even labeling of the graph \( ~P_{3}\odot \overline { K_{6}}~ \) is shown in Fig. 7.

The generalization of the previous result is presented in the following theory

###
**Theorem 1**

The graph \( ~P_{2n-1}\odot \overline { K_{2m}}~ \) is an edge even graceful graph.

###
*Proof*

In this graph, *p*=4*n**m*+2(*n*−*m*)−1 and *q*=4*n**m*+2(*n*−*m*)−2. The middle vertex in the path *P*_{2n−1} will be *v*_{n}, and we start the labeling from this vertex. Let the vertex and edge symbols be given as in Fig. 8.

Define the mapping *f*:*E*(*G*) →{2,4,⋯,2*q*} by the following arrangement

*f*(*e*_{i})=2^{i} for *i*=1,2,⋯,*n*−1

*f*(*b*_{i})=2*p*− 2^{i} for *i*=1,2,⋯,*n*−1

*f*(*A*_{ij}) will take any number from the reminder set of the labeling not contains 2^{i} nor 2*p*− 2^{i} for *j*=1,2,⋯,*m* and

*f*(*D*_{ij})=2*p*−[*f*(*A*_{ij})] for *j*=1,2,⋯,*m*.

It is clear that [*f*(*A*_{ij})+*f*(*D*_{ij})] mod (2*p*)≡0 mod (2*p*) for *j*=1,2,⋯,*m*.

So, the vertex labels will be

*f*^{∗}(*v*_{n})=[*f*(*e*_{1})+*f*(*b*_{1})] mod (2*p*) ≡0 mod (2*p*),

For any vertex *v*_{k} when *k*<*n*, let *k*=*n*−*i* and *i*=1,2,⋯,*n*−2

When *k*>*n*, let *k*=*n*+*i* and *i*=1,2,⋯,*n*−2

The pendant vertices *v*_{1} *a**n**d* *v*_{2n−1} of the path *P*_{2n−1} will take the labels of its pendant edges of *P*_{2n−1},i.e.,

If *n* is even, then *f*^{∗}(*v*_{1})=*f*(*e*_{n−1}), and *f*^{∗}(*v*_{2n−1})=*f*(*b*_{n−1})

If *n* is odd, then *f*^{∗}(*v*_{1})=*f*(*b*_{n−1}), and *f*^{∗}(*v*_{2n−1})=*f*(*e*_{n−1})

Then, the labels of the vertices of the path *P*_{2n−1} takes the labels of the edges of the path *P*_{2n−1}, and each pendant vertex takes the labels of its incident edge. Then, there are no repeated vertex labels, which complete the proof. □

**Illustration:** The graph \( ~P_{11}\odot \overline { K_{4}}~ \) labeled according to Theorem 1 is presented in Fig. 9.

A double fan graph *F*_{2,n} is defined as the graph join \(\overline {K_{2}} + P_{n}~\) where \(\overline {K_{2}} \) is the empty graph on two vertices and *P*_{n} be a path of length *n*.

###
**Theorem 2**

The double fan graph *F*_{2,n} is an edge even graceful labeling when *n* is even.

###
*Proof*

In the graph *F*_{2,n} we have *p*=*n*+2 *a**n**d* *q*=3*n*−1. Let the graph *F*_{2,n} be given as indicated in Fig. 10.

Define the edge labeling function *f*:*E*(*F*_{2,n})→ {2,4,⋯, 6*n*−2} as follows:

*f*(*a*_{i})=2*i*; *i*=1,2,⋯*n*

*f*(*b*_{i})=2*q*−2*i*=6*n*−2(*i*+1); *i*=1,2,⋯*n*

Hence, the induced vertex labels are

\(~ f^{\ast }(u) = \left (\sum _{i=1}^{n}(f(a_{i}))\right)~\text {mod}(6n-2)\,=\, \left (\sum _{i=1}^{n}(2i)\right)~\text {mod}(6n-2) =(n^{2}+n)~\text {mod}(6n-2)~~\)

\(~ f^{\ast }(v) = \left (\sum _{i=1}^{n}(f(b_{i})) \right)~\text {mod}(6n-2)= \left (\sum _{i=1}^{n}(2q-2i)\right)~\text {mod}(6n-2) =\left (- n^{2}-n\right)~\text {mod}(6n-2) \)

\( f^{\ast }(v_{i}) \,=\, \left [\sum _{i=1}^{n}(f(a_{i})\,+\,f(b_{i})\,+\,f (e_{i}) \,+\,f(e_{i-1}))\right ]~\text {mod}(6n\,-\,2)~ \,=\,(4n\,+\,4i\,-\,2)~\text {mod}(6n\,-\,2)~,~~2 \leq i \leq \frac {n}{2}- 1 \)

\(f^{\ast }(v_{i}) = \left [\sum _{i=1}^{n}(f(a_{i})+f(b_{i})+f (e_{i}) +f(e_{i-1}))\right ]~\text {mod}(6n-2) =(4n+4i-6)~\text {mod}(6n-2),~~\frac {n}{2}+2 \leq i \leq n- 1 \)

Since [*f*(*a*_{i})+*f*(*b*_{i})] mod(6*n*−2)=0, we see that

*f*^{∗}(*v*_{1})= *f*(*e*_{1})=2*n*+2,

*f*^{∗}(*v*_{n})= *f*(*e*_{n−1})=4*n*−4,

\(~~~~~ f^{\ast }(v_{\frac {n}{2}})=~f (e_{\frac {n}{2}-1})= 3n-2 ~\) and

\( ~~~~f^{\ast }(v_{\frac {n}{2}+1})=~f (e_{\frac {n}{2}})= 3n ~\)

Thus, the set of vertices \( v_{1}, v_{2},v_{3},\cdots ~,v_{\frac {n}{2}-1},v_{\frac {n}{2} },v_{\frac {n}{2}+1},v_{\frac {n}{2}+2},v_{\frac {n}{2}+3},\cdots ~,v_{n-2},v_{n-1},v_{n} \) are labeled by 2*n*+2, 4*n*+6, 4*n*+10,⋯,6*n*−6, 3*n*−2, 3*n*, 4,8, ⋯,2*n*−12, 2*n*−8, 4*n*−4 respectively.

Clearly, *f*^{∗}(*u*) and *f*^{∗}(*v*) are different from all the labels of the vertices. Hence *F*_{2,n} is an edge even graceful when *n* is even. □

**Illustration:** The double fan *F*_{2,10} labeled according to Theorem 2 is presented in Fig. 11.

## Edge even graceful for some cycle related graphs

###
**Definition 2**

For *n*≥4, a cycle (of order n) with one chord is a simple graph obtained from an *n*-cycle by adding a chord. Let the *n*-cycle be *v*_{1}*v*_{2}⋯*v*_{n}*v*_{1}. Without loss of generality, we assume that the chord joins *v*_{1} with any one *v*_{i}, where 3≤*i*≤*n*−1. This graph is denoted by *C*_{n}(*i*).

###
**Lemma 5**

The graph \( ~C_{n}\left (\frac {n}{2}\right)~ \) is an edge even graceful graph if *n* is even.

###
*Proof*

Let \( \{ v_{1}, v_{2},\cdots, v_{\frac {n}{2}-1},v_{\frac {n}{2}},v_{\frac {n}{2}+1}, \cdots,v_{n} \}\) be the vertices of the graph \(C_{n}\left (\frac {n}{2}\right)\), and the edges are *e*_{i}=(*v*_{i}*v*_{i+1}) for *i*≤*i*≤*n*−1 and the chord \(e_{0}=(v_{1}v_{\frac {n}{2}}) \) connecting the vertex *v*_{1} with \(v_{\frac {n}{2}}\) as in Fig. 12.

Here, *p*=*n* and *q*=*n*+1, so 2*k*=2*q*=2*n*+2; first, we label the edges as follows:

Then, the induced vertex labels are as follows:

for any other vertex \(~~~v_{i}, ~~~i\neq 1,\frac {n}{2}\)

Hence, the labels of the vertices \(v_{0}, v_{1},v_{2}, \cdots,v_{\frac {n}{2}-1},v_{\frac {n}{2}},v_{\frac {n}{2}+1}, \cdots v_{n} \) are 6, 10, 14,⋯, 2*n*−2, 2, 4,⋯, 2*n* respectively, which are even and distinct. So, the graph \( ~C_{n}\left (\frac {n}{2}\right)~ \) is an edge even graceful graph if *n* is even. □

###
**Definition 3**

Let *C*_{n} denote the cycle of length *n*. The flag *F**L*_{n} is obtained by joining one vertex of *C*_{n} to an extra vertex called the root, in this graph *p*=*q*=*n*+1.

###
**Lemma 6**

The flag graph *F**L*_{n} is edge even graceful graph when *n* is even.

###
*Proof*

Let {*v*_{1},*v*_{2},⋯,*v*_{n}} be the vertices of the cycle *C*_{n} and the edges are *e*_{i}=(*v*_{i}*v*_{i+1}) *f**o**r* 1≤*i*≤*n* and the edge *e*=(*v*_{1}*v*_{0}) connecting the vertex *v*_{1} with *v*_{0} as in Fig. 13.

First, we label the edges as follows:

Then, the induced vertex labels are as follows *f*^{∗}(*v*_{0})=*f*(*e*_{0})=2,

Hence, the labels of the vertices \(v_{0}, v_{1},v_{2}, \cdots,v_{\frac {n}{2}-1},v_{\frac {n}{2}},v_{\frac {n}{2}+1}, \cdots v_{n} \) will be 2, 6, 10, 14,⋯, 2*n*−2, 0, 4,⋯, 2*n* respectively. □

###
**Lemma 7**

The graph *K*_{2}⊙*C*_{n} is edge even graceful graph when *n* is odd.

###
*Proof*

Let \( \left \{ v_{1}, v_{2},\cdots, v_{n},v^{'}_{1},v^{'}_{2}, \cdots,v^{'}_{n} \right \}\) be the vertices of the graph *K*_{2}⊙*C*_{n} and the edges are \( \left \{e_{1}, e_{2},\cdots, e_{n},~e^{'}_{1},e^{'}_{2}, \cdots,e^{'}_{n} \right \}\) as shown in Fig. 14. Here, *p*=2*n* and *q*=2*n*+1, so 2*k*=4*n*+2.

First, we label the edges as follows:

We can see that [*f*(*e*_{n})+*f*(*e*_{n+1})]mod(2*q*)=(4*n*+2) mod(4*n*+2)≡0

Then, the induced vertex labels are as follows

\( f^{\ast }(v^{\prime }_{i})= \left [ f(e^{\prime }_{i}) +f(e^{\prime }_{i+1})\right ]~\text {mod}(4n+2)=[8+4~(i-1)]~\text {mod}(4n+2)~,~~1 \leq i < n \)

*f*^{∗}(*v*_{n})= [*f*(*e*_{1})+*f*(*e*_{n})+*f*(*e*_{n+1}) ] mod(4*n*+2)= *f*(*e*_{1}) mod(4*n*+2) ≡2

\( ~~~f^{\ast }(v^{\prime }_{n})=~ \left [ f(e^{\prime }_{1}) +f(e_{n+1})~+~f\left (e^{\prime }_{n}\right)\right ]~\text {mod}(4n+2) ~\equiv ~4\)

Clearly, the vertex labels are all even and distinct. Hence, the graph *K*_{2}⊙*C*_{n} is edge even graceful for odd *n*. □

Let *C*_{n} denote the cycle of length *n*. Then, the corona of all vertices of *C*_{n} except one vertex {*v*_{1}} with the complement graph \(\overline {K_{2m-1}}\) is denoted by \( ~ \{C_{n}-\{v_{1} \} \} \odot \overline {K_{2m-1}}~ \), in this graph *p*=*q*=2*m*(*n*−1)+1.

###
**Lemma 8**

The graph \( \{C_{n}-\{v_{1} \} \} \odot \overline {K_{2m-1}}~ \) is an edge even graceful graph.

###
*Proof*

Let the vertex and edge symbols be given as in Fig. 15.

Define the mapping *f*:*E*(*G*) →{2,4,⋯,2*q*} as follows:

We realize the following:

[*f*(*A*_{ij})+*f*(*B*_{ij})] mod (2*q*)≡0 mod (2*q*) for *j*=1,2,⋯,*m*−1

Also, [*f*(*E*_{i−1})+*f*(*e*_{i})] mod (2*q*)≡0 mod (2*q*) *f**o**r* *i*=2,3,⋯,*n*

So, verifying the vertex labels, we get that,

*f*^{∗}(*v*_{1})=[*f*(*E*_{1})+*f*(*E*_{n})] mod (2*q*)=(2+2*q*) mod (2*q*)=2,

Hence, the labels of the vertices *v*_{1},*v*_{2},⋯,*v*_{n} takes 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. □

**Illustration:** In Fig. 16, we present an edge even graceful labeling of the graph \( ~ \{C_{6}-\{v_{1} \} \} \odot \overline {K_{3}}~ \).

###
**Lemma 9**

The double cycle graph {*C*_{n,n}} is an edge even graceful graph when *n* is odd.

###
*Proof*

Here, *p*=*n* and *q*=2*n*. Let the vertex and edge symbols be given as in Fig. 17.

Define the mapping *f*:*E*(*G*) →{2,4,⋯,4*n*} by

*f*(*e*_{i})=2*i* for *i*=1,2,⋯,*n*. So, the vertex labels will be

*f*^{∗}(*v*_{1})=[*f*(*e*_{1})+*f*(*e*_{n}) +*f*(*e*_{n+1}) +*f*(*e*_{2n})]mod (4*n*)=4

Hence, the labels of the vertices \( v_{1},v_{2},v_{3}, \cdots, v_{\frac {n+1}{2}}, v_{\frac {n+3}{2}}, \cdots, v_{n} \) will be 4,12,20,⋯,0,8,⋯,4*n*−4 □

The prism graph \( \prod _{n}\) is the cartesian product *C*_{n}□*K*_{2} of a cycle *C*_{n} by an edge *K*_{2}, and an *n*-prism graph has *p*=2*n* vertices and *q*=3*n* edges.

###
**Theorem 3**

The prism graph \( ~\prod _{n}~ \) is edge even graceful graph.

###
*Proof*

In the prism graph \( \prod _{n}\) we have two copies of the cycle *C*_{n}, let the vertices in one copy be *v*_{1},*v*_{2},⋯,*v*_{n} and the vertices on the other copy be \(~v^{\prime }_{1}, v^{\prime }_{2}, \cdots, v^{\prime }_{n}~\). In \( ~\prod _{n}~ \), the edges will be

\(v_{i}v_{i+1}, ~~ ~~~v^{\prime }_{i}v^{\prime }_{i+1},~~~ \text {and} ~~~v_{i}v^{\prime }_{i} \). Let the vertex and edge symbols be given as in Fig. 18.

Define the mapping \(f: E\left (\prod _{n}\right)~\rightarrow \{2,4,\cdots, 6n \}\) by

So, the vertex labels will be

Hence, the labels of the vertices *v*_{1},*v*_{2},⋯,*v*_{n} will be 4*n*+4, 4*n*+6,⋯,0, 2 respectively.

Also, \( f^{\ast }(v^{\prime }_{1}) = \left [~ f(e^{\prime }_{1}) +f(e^{\prime }_{n})~+ f(E_{1})\right ] \text {mod}~(6n)= 12n+ 4~ \text {mod}~(6n)= 4\)

Hence, the labels of the vertices \( v^{\prime }_{1},v^{\prime }_{2}, \cdots, v^{\prime }_{n} \) are 4, 6,8,⋯,2*n*, 2*n*+2 respectively. Overall, the vertices are even and different. Thus, the prism graph \( ~\prod _{n}~ \) is an edge even graceful graph. □

**Illustration:** In Fig. 19, we present an edge even graceful labeling of of prism graphs \( ~\prod _{5}~ \) and \( ~\prod _{6} \).

The flower graph FL(*n*) (*n*≥3) is the graph obtained from a helm *H*_{n} by joining each pendant vertex to the center of the helm.

###
**Theorem 4**

The flower graph FL(n) (*n*≥4) is an edge even graceful graph.

###
*Proof*

In the flower graph FL(*n*) (*n*≥4), we have *p*=2*n*+1 and *q*=4*n*. Let \(\{~v_{0},v_{1}, v_{2}, \cdots, v_{n}~,v^{\prime }_{1}, v^{\prime }_{2}, \cdots, v^{\prime }_{n}\}\) be the vertices of FL(*n*) and

{ *e*_{1},*e*_{2},*e*_{3},⋯,*e*_{3n},*E*_{1},*E*_{2},*E*_{3},⋯,*E*_{n} } be the edges of FL(*n*) as in Fig. 20.

First, define the mapping *f*:*E*(*F**l*(*n*))→{2,4,⋯,8*n* } as the following:

Then, the induced vertex labels are

Overall, all the vertex labels are even and distinct which complete the proof. □

**Illustration:** In Fig. 21, we present an edge even graceful labeling of of the flower graphs FL(6)and FL(7).

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## Acknowledgments

I am so grateful to the reviewers for their valuable suggestions and comments that significantly improved the paper.

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### Cite this article

Zeen El Deen, M.R. Edge even graceful labeling of some graphs.
*J Egypt Math Soc* **27, **20 (2019). https://doi.org/10.1186/s42787-019-0025-x

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### Keywords

- Edge-even graceful labeling
- Flag graph FL
_{n} - Double fan graph
*F*_{2,n} - Prism graph
- The flower graph FL(
*n*)

### Mathematics subject classification

- 05 C 78
- 05 C 76
- 05 C 90
- 05 C 99