Adjacent Edge Graceful Graphs

Full text

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Vol. 6, No. 1, 2014, 25-35

ISSN: 2279-087X (P), 2279-0888(online) Published on 5 May 2014

www.researchmathsci.org

25

Annals of

Adjacent Edge Graceful Graphs

T. Tharmaraj1 and P.B.Sarasija2 1

Department of Mathematics, Udaya School of Engineering Vellamodi, Tamil Nadu, India. E-mail: trajtr@gmail.com

2

Department of Mathematics, Noorul Islam University Kumaracoil, Tamil Nadu, India. E-mail: sijavk@gmail.com

Received 6 April 2014; accepted 20 April 2014

Abstract. Let G(V,E)be a graph with pvertices and qedges. A (p,q) graph G(V,E)

is said to be an adjacent edge graceful graph if there exists a bijection f E G: ( ) {1, 2,3,..., }q

→ such that the induced mapping f * from V(G) by f* ( )u

( )i i

f e

=

over all edges ei incident to adjacent vertices of u is an injection. The

function f is called an adjacent edge graceful labeling of G.In this paper, we prove path

) 5 (n

Pn , cycle Cn(n≥5), fan fn(n≥3), Helm Hn(n≥3), triangle snake

) 3 (n

Tn and alternate triangle snake A(Tn)(n=4,6,8,...)are the adjacent edge graceful graphs and we also prove n-bistar Bn,n, complete bipartite graph Km,n, star graph K1,n and graph gn are not the adjacent edge graceful graphs.

Keywords: adjacent edge graceful graph, adjacent edge graceful labeling AMS Mathematical Subject Classification (2010): 05C78

1. Introduction

All graphs in this paper are finite, simple and undirected graphs. Let (p,q)be a graph with p=|V(G)|vertices and q=|E(G)| edges. Graph labeling, where the vertices and edges are assigned real values or subsets of a set are subject to certain conditions. A detailed survey of graph labeling can be found in [1].Terms not defined here are used in the sense of Harary in [3].The concept of edge graceful labeling was first introduced in [2] and the concept of strong edge graceful labeling was introduced in [4]. Some results on strong edge graceful labeling of graphs are discussed in [4]. In this paper, we introduced a new edge graceful labeling. We use the following definitions in the subsequent sections.

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26 from V(G) by *( )=

( )

i i e f u

f over all edges ei incident to adjacent vertices of u

is an injection. The function f is called an adjacent edge graceful labeling of G.

Definition 1.2.[1] The bistar graph Bn,n is the graph obtained from two copies of star n

K1, by joining the vertices of maximum degree by an edge.

Definition 1.3.[1] The helm Hn is the graph obtained from a wheel by attaching a pendent edge at each vertex of the n-cycle.

Definition 1.4.[1] The fan fn(n≥2) is obtained by joining all nodes of Pn to a further node called the center and contains n+1 nodes and 2n−1 edges.

Definition 1.5.[1] A graph gn(n≥2) be a graph with n+2 nodes and 3n−1 edges

obtained by joining all nodes of Pn to two additional nodes.

Definition 1.6.[1] A triangular snake Tn is obtained from a path u1u2...un by joining ui and ui+1 to a new vertex vi for i=1,2,….,n-1.

Definition 1.7.[5] An alternate triangular snake A(Tn) is obtained from a path

n u u

u1 2... by joining ui and ui+1 (alternatively) to new vertex vi.

2. Main results

Theorem 2.1. The triangle snake Tn(n≥3) admits an adjacent edge graceful labeling.

Proof: Let V(Tn)={ui :1≤in; vi :1≤in−1}

Let

    

− ≤ ≤ =

− ≤ ≤ =

− ≤ ≤ =

=

+ + −

+ −

+

1 1

1 1

1 1

) (

1 1

2 1

1

n i if v u e

n i if v u e

n i if u u e

T E

i i i n

i i i n

i i i

n

Define a bijection f :E(Tn)→{1,2,3,....,3n−3} by 1

1 )

(e =i ifin

f i ; f(en−1+i)=n+2(i−1) if 1≤in−1;

1 1

2 2 2 )

(e2 1+ = n− + i ifin

f n i .

Let f * be the induced vertex labeling of f . The induced vertex labels are as follows:

If n=3 , f *(u1)=6n+1 ; f *(u2)=10n ; f *(u3)=7n+2 ;

1 5 ) (

* v1 = n+

f ; f *(v2)=6n+2.

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27

1 9 ) (

* u4 = n+

f ; f *(v1)=4n+3 ; f *(v2)=8n+2 ; f *(v3)=85n . If n≥5, f *(u1)=4n+7 ; f *(u2)=7n+19 ;

40 24 ) (

* u 1 = n

f n ; f *(un)=14n−19 ; f *(v1)=3n+7 ;

16 12 ) (

* v −1 = n

f n ; f *(ui+2)=8n+18+20i if 1≤in−4;

3 1

12 6 4 ) (

* v+1 = n+ + i ifin

f i .

Example 2.2. An adjacent edge graceful labeling of a triangular snake T8 is shown in Fig.1.

Figure 1:

Theorem 2.3. The alternate triangular snake A(Tn)(n=4,6,8,...) is an adjacent edge

graceful graph.

Proof: Case (i) If the triangle starts from u1.

Let }

2 1 : ; 1

: { )) (

(AT u i n v i n

V n = i ≤ ≤ i ≤ ≤ .

Let

  

   

 

≤ ≤ =

≤ ≤ =

− ≤ ≤ =

=

+ −

− + −

+

2 1

2 1

1 1

)) ( (

2 2

2 2 3

1 2 1

1

n i if v u e

n i if v u e

n i if u

u e

T A E

i i i n

i i i n

i i i

n

Define a bijection f :E(A(Tn))→{1,2,3,...,2n−1} by

1 1

2 )

(e = i ifin

f i ;

2 1

3 4 )

(e 1 i if i n f n+i = − ≤ ≤ ;

2 1

1 4 ) (

2 2 2 3

n i if i e

f n +i = − ≤ ≤ .

Let f * be the induced vertex labeling of f .In this case the induced vertex label are as follows:

If n=4 , f *(u1)=3n+1 ; f *(u2)=5n+2 ; f *(u3)=8n+2 ;

3 6 ) (

* u4 = n+

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28

If n≥6 , f *(u1)=13 ; f *(u2)=22 ;       − ≤ ≤ + = + 2 4 1 10 32 ) ( * 2 1

n i if i

u

f i ;

22 14 ) (

* u 1 = n

f n ; f *(un)=10n−13; 

     − ≤ ≤ + = + 2 4 1 22 32 ) ( * 2 2

n i if i

u

f i ;

12 ) ( * v1 =

f ; 

     ≤ ≤ + = + 2 4 1 12 24 ) (

* v 1 i if i n

f i ; *( ) 10 12

2

− = n v

f n .

Case (ii) If the triangle starts from u2

Let }

2 1 : ; 1 : { )) (

(AT u i n v i n

V n = i ≤ ≤ i ≤ ≤ .

Let                 − ≤ ≤ =       ≤ ≤ = − ≤ ≤ = = + + − + − + 2 2 1 2 2 1 1 1 )) ( ( 1 2 2 2 2 3 2 1 1 n i if v u e n i if v u e n i if u u e T A E i i i n i i i n i i i n

Define a bijection f :E(A(Tn))→{1,2,3,...,2n−3} by

1 1

1 2 )

(e = iifin

f i ; 

     − ≤ ≤ − = + − 2 2 1 2 4 ) ( 1 n i if i e

f n i ;

      ≤ ≤ = + − 2 2 1 4 ) ( 2 2 4 3 n i if i e

f n i .

In this case the induced vertex label are as follows:

If n=4 , f *(u1)=n+2 ; f *(u2)=4n+3 ; f *(u3)=4n+1 ; f *(u4)=3n;

2 4 ) (

* v1 = n+

f .

If n≥5 , f *(u1)=6 ; f *(u2)=19 ;       ≤ ≤ − = 2 2 1 6 24 ) (

* v i if i n

f i ;

12 6 ) (

* u = n

f n ; 

     − ≤ ≤ − = + 2 4 1 2 32 ) ( * 2 1

n i if i u

f i ;

      − ≤ ≤ + = + 2 4 1 18 32 ) ( * 2 2

n i if i

u

f i ; f *(un−1)=12n−31.

Example 2.4. An adjacent edge graceful labeling of an alternate triangular snake A(T8)

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29

Figure 2:

Figure 3:

Theorem 2.5. The Path Pn(n≥5) is an adjacent edge graceful graph.

Proof: Let V(Pn)={vi :1≤in}.Let E(Pn)={ei =vivi+1 :1≤in−1}. Define f :E(Pn)→{1,2,3,...,n−1} by

Case (i) n is odd , f(ei)=i if 1≤in−1.

Case (ii) n is even,

2 1

1 2 )

(e i if i n

f i = − ≤ ≤ ;

2 2 1

2 )

(

2 2

− ≤ ≤ −

=

+

n i if i n e

f n i .

Let f * be the induced vertex labeling of f .The induced vertex labels are as follows: If n is odd , f *(v1)=3 ; f *(v2)=6; f *(vn1)=3(n−2);

3 2 ) (

* v = n

f n ; f *(vi+2)=4i+6 if 1≤in−4.

If n=6, f *(v1)=n−2 ; f *(v2)=n+3 ; f *(v3)=2n+1 ; n

v

f *( 4)=2 ; f *(v5)=2n−5 ; f *(v6)=6.

If n=8, f *(v1)=n−4 ; f *(v2)=n+1 ; f *(v3)=2n ; f *(v4)=2n+5 ; n

v

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30

2 6 1

) 1 ( 8 ) (

* v+2 = i+ ifin

f i ; *( ) 4 11

2

− = n v

f n ; *( ) 4 10

2

2 = −

+ n

v

f n ;

13 4 ) ( *

2

4 = −

+ n

v

f n ;

2 8 1

8 12 4 ) (

*

2 2 4

− ≤ ≤ −

− =

+ +

n i if i n

v

f n i .

Example 2.6. The adjacent edge graceful labelings of P9 and P10 are shown in the Fig. 4 and Fig. 5 respectively.

Figure 4:

Figure 5:

Theorem 2.7. The cycle is an adjacent edge graceful graph. Proof: Let V(Cn)={vi :1≤in}. Let

} ;

1 1

: {

)

(Cn ei vivi 1 i n en v1vn

E = = + ≤ ≤ − = .

Define f :E(Cn)→{1,2,3,...,n} by

Case (i) n is odd , f(ei)=i if 1≤in.

Case (ii) n is even ,

2 1

1 2 )

(e i if i n

f i = − ≤ ≤ ;

2 1

) 1 ( 2 )

(

2 2

n i if i n e

f n+ i = − − ≤ ≤ .

Let f * be the induced vertex labeling of f .The induced vertex labels are as follows: If n is odd ,f *(v1)=2n+2 ; f *(v2)=n+6 ;

2 3 ) (

* v = n

f n ; f *(vi+2)=4i+6 if 1≤in−3 ;

If n=6, f *(v1)=n+4 ; f *(v2)=n+5 ; f *(v3)=2n+3 ; n

v

f *( 4)=3 ; f *(v5)=2n+5 ; f *(v6)=2n+1. If n≥8 , f *(v1)=10 ; f *(v2)=11 ;

2 6 1

) 1 ( 8 ) (

* v+2 = i+ ifin

f i ;

9 4 ) ( *

2

− = n v

f n ; *( ) 4 6

2

2 = −

+ n

v

f n ; *( ) 4 7

2

4 = −

+ n

v

f n ;

2 6 1

) 3 2 ( 4 ) (

* v = i+ ifin

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31

Example 2.8. The adjacent edge graceful labelings of C14 and C11 are shown in the Fig. 5 and Fig. 6 respectively.

Figure 5: Figure 6:

Theorem 2.9. The graph gn is not an adjacent edge graceful graph.

Proof: Let V(gn)={vi :1≤in; vn+1 ; vn+2}.

Let

    

≤ ≤ =

≤ ≤ =

− ≤ ≤ =

=

+ +

+ +

− +

n i if v v e

n i if v v e

n i if v v e g E

n i i n

n i i n

i i i

n

1 1

1 1

) (

2 1

2

1 1

1

Let f be a bijection from E(gn) to {1,2,3,...,3n−1}.

Let l1,l2,....,ln,ln+1,....,l3n−1 be the label of edges e1,e2,....,en,en+1,....,e3n−1 by f respectively. And f induces that f*:V(gn)→{1,2,3,...} by

) ( )

(

* =

i i e f u

f over all edges ei incident to adjacent vertices of u.

The vertices vn+1 and vn+2 have the same adjacent vertices {v1,v2,....,vn}. Then f *(vn+1)=2(l1+l2+...+ln−1)+(ln +ln+1+...+l3n−1)

=

= −

=

+3 1

1 1

1 n

i i n

i

i l

l =

= −

=

+3 1

1 1

1 n

i n

i

i i

l =

2 ) 1 3 ( 3 −

+ n n

m , where

=

= 1

1 n

i i l m

Since vn+1 and vn+2 have the same adjacent vertices,

2 ) 1 3 ( 3 )

(

* v +2 =m+ n n

f n . That is f * is not injective. Hence gn is not an adjacent

edge graceful graph.

Theorem 2.10. The graph n-bistar Bn,n is not an adjacent edge graceful graph.

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32 Let      = ≤ ≤ = ≤ ≤ = = + + + + + + 1 1 1 2 1 1 , 1 1 ) ( n n n n i i n n i i n n v u e n i if v v e n i if u u e B E

Let f be a bijection from E(Bn,n) to {1,2,3,...,2n+1}.

And f induces that f*:V(Bn,n)→{1,2,3,...} by *( )=

( )

i i e f u f

over all edges ei incident to adjacent vertices of u.

Step 1:

The adjacent vertices of un+1 are u1,u2,...,un,vn+1.

) ( ) ( ) ( * 1 2 1 1

1

+ + = = + = + n n i i n i i

n f e f e

u

f =

+ = 1 2 1 ) ( n i i e f

=1+2+3+...+(2n+1) =(n+1)(2n+1).

Step 2:

The adjacent vertices of vn+1 are v1,v2,...,vn,un+1.

) ( ) ( ) ( ) (

* 2 1

2 1 1 1 + + = =

+ =

+

+ n

n n i i n i i

n f e f e f e

v

f =

+ = 1 2 1 ) ( n i i e f

=1+2+3+...+(2n+1)=(n+1)(2n+1).

From step(1) and step(2), the vertex labels of un+1 and vn+1 are same. That is f * is not injective.Hence Bn,n is not an adjacent edge graceful graph.

Theorem 2.11. The Helm Hn(n≥3) is an adjacent edge graceful graph.

Proof: Let V(Hn)={ui :1≤in+1; vi :1≤in}

Let        ≤ ≤ = ≤ ≤ = = − ≤ ≤ = = + + + + n i if u u e n i if v v e u u e n i if u u e H E n i i n i i i n n n i i i n 1 1 1 1 ) ( 1 2 1 1

Define a bijection f :E(Hn)→{1,2,3,...,3n} by f(ei)=2i if 1≤in ; n

i if i e

f( n+i)=2 −1 1≤ ≤ ; f(e2n+i)=2n+i if 1≤in. Let f * be the induced vertex labeling of f .

The induced vertex label are as follows:

If n=3 , f *(u1)=22n ; f *(u2)=22n+1 ; f *(u3)=20n+2 ; n

u

f *( 4)=19 ; f *(v1)=5n+1 ; f *(v2)=5n+2 ; f *(v3)=8n. If n≥4 ,

2 18 23 5 ) ( * 2 1 + + = n n

u

f ;

2 50 13 5 ) ( * 2 2 + + = n n

u

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33

2 14 31 5

) ( *

2 + −

= n n

u

f n ;

2 5 11 ) ( *

2 1

n n u

f n

+ =

+

; 1 3

2

32 50 9 5 ) ( *

2

2 ≤ ≤ −

+ + + =

+ if i n

i n

n u

f i ;

4 4 ) (

* v1 = n+

f ; f *(vi+1)=2n+4+7i if 1≤in−1.

Theorem 2.12. Complete bipartite graph Km,n is not an adjacent edge graceful graph.

Proof: Let V(Km,n)={ui : 1≤im ;vi :1≤in}.

Let E(Km,n) {ei uivj :1 i m,1 i n}

j= ≤ ≤ ≤ ≤

= .

Define a bijection f from E(Km,n) to {1,2,3,...,n}.

And f induces that f*:V(Km,n)→{1,2,3,...} by *( )=

( )

i i e f u

f over all

edges ei incident to adjacent vertices of u..

Case (i) m<n

) ( )

( *

1 1

∑∑

= =

= m

i n

j i

i f ej

v

f =

2 ) 1 ( ...

3 2

1+ + + +mn=mn mn+ for 1≤im .

The label of each vertex vi is

2 ) 1 (mn+ mn

for 1≤im .

Case (ii) m=n

) ( * ui

f = *( ) ( )

1 1

∑∑

= =

= n

i n

j i

i f ej

v

f =

2 ) 1 ( ...

2 1

2 2

2= +

+ +

+ n n n for 1≤in .

The label of each vertex ui and vi are

2 ) 1 ( 2

2 +

n n

for 1≤in.

Case (iii) m>n

) ( )

( *

1 1

∑∑

= =

= m

i n

j i

i f ej

u

f =

2 ) 1 ( ...

3 2

1+ + + +mn=mn mn+ for 1≤in .

The label of each vertex ui is

2 ) 1 (mn+ mn

for 1≤in .

In the above three cases , f * is not injective.

Hence the complete bipartite graph Km,n is not an adjacent edge graceful graph.

Corollary 2.13. The Star graph K1,n is not an adjacent edge graceful graph.

Proof: By putting m=1 in the result of the above theorem 2.12 , we can get the result of this corollary.

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34

Proof: Let V(fn)={vi :1≤in+1}.

Let E(fn)={ei =vivi+1 :1≤in−1;en1+i =vivn+1 :1≤in}. Define a bijection f :E(fn)→{1,2,3,...,2n−1} by

Case (i) n is odd , f(ei)=i if 1≤in−1 and

n i if i n e

f( n1+i)= −1+ 1≤ ≤ .

Case (ii) n is even , f(ei)=2i−1 if 1≤in−1 ;

1 1

2 )

(e −1+ = i ifin

f n i ;f(e2n−1)=2n−1 . Let f * be the induced vertex labeling of f .

The induced vertex labels are as follows:

If n is odd ,

2 8 3

) ( *

2 1

+ + = n n

v

f ;

2 20 13 3 ) ( *

2 1

− + =

n n v

f n ;

2 10 7 3 ) ( *

2 +

= n n

v

f n ;

2 ) 3 5 ( ) (

* v +1 =n n

f n ;

3 1

) 1 ( 6 2

16 3 3 ) ( *

2

1 + − ≤ ≤ −

+ + =

+ i if i n

n n v

f i .

If n=4 , f *(v1)=6n+3 ; f *(v2)=9n ; f *(v3)=9n+3 ;

1 8 ) (

* v4 = n+

f ; f *(v5)=9n+1 .

If n≥6 , *( ) 2 7

1 =n +n+ v

f ; *( ) 2 16

2 =n +n+ v

f ;

21 11 )

(

* v −1 =n2 + n

f n ; *( ) 7 11

2 + −

=n n v

f n ;

1 3 3 ) (

* 2

1 = − +

+ n n

v

f n ; *( ) 2 15 12 1 4

2 = + + + ≤ ≤ −

+ n n i if i n

v

f i .

REFERENCES

1. J.A.Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 5(2002) # DS6, 1-144.

2. B.Gayathri and M.Subbiah, Strong edge graceful labelings of some graphs, Bull. Pure Appl. Sci., 27E (2008) 89-98.

3. H.Harary, Graph Theory, Addison-Wesley, Reading, Massachusetts, 1972. 4. S.Lo, On edge-graceful labelings of graphs, Congr. Numer., 50 (1985) 231-241. 5. S.S.Sandhya, S.Somasundaram and R.Ponraj, Harmonic mean labeling of some cycle

related graphs, Int. Journal of Math. Analysis, 6 (40) (2012) 1997-2005.

6. Ali Ahmad, E.T.Baskoro and M.Imran, Total vertex irregularity strength of disjoint union of Helm graphs, Discussiones Mathematicae Graph Theory, 32 (2012) 427– 434.

7. S. K. Vaidya and N. H. Shah , Some star and bistar related divisor cordial graphs, Annals of Pure and Applied Mathematics, 3(1) (2013) 67-77.

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35

9. R. L. Graham and N. J. A. Sloane, On additive bases and harmonious graphs, SIAM J. Alg. Disc. Meth., 1 (4) (1980) 382-404.

10. Z. Liang and Z. Bai, On the odd harmonious graphs with applications, J. Appl. Math. Comput., 29 (2009) 105-116.

11. V. Yegnanaryanan and P. Vaidhyanathan, Some interesting applications of graph labellings, J. Math. Comput. Sci., 2 (5) (2012) 1522-1531.

12. I. Cahit, Cordial Graphs, A weaker version of graceful and harmonious Graphs, Ars Combinatoria, 23 (1987) 201-207.

13. M. Andar, S. Boxwala and N. B. Limaye, Cordial labelings of some wheel related graphs, J. Combin. Math. Combin. Comput., 41 (2002) 203-208.

14. M. Andar, S. Boxwala and N. B. Limaye, A note on cordial labeling of multiple shells, Trends Math., (2002) 77-80.

Figure

 Figure 2:
Figure 2: p.5
 Figure 5: Theorem  2.7. The cycle  is an adjacent edge graceful graph.
Figure 5: Theorem 2.7. The cycle is an adjacent edge graceful graph. p.6
Fig. 5  and  Fig. 6  respectively.Example 2.8.
Fig. 5 and Fig. 6 respectively.Example 2.8. p.7

References

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