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1st Int Conference on Computer Science from Algorithms to Applications (CSAA-2009), 8-10 2009, Cairo, Egypt.

ODD GRACEFUL LABELINGS OF CROWN GRAPHS

M. I. Moussa a , E. M. Badr b

a Faculty of Computers & Information, Benha University, Benha, Egypt,

moussa_6060@yahoo.com

b Mathematics and Computer Department, Faculty of Science, Benha University, Egypt,

badrgraph@gmail.com

ABSTRACT

A graph G of size q is odd-graceful, if there is an

injection f from V(G) to {0, 1, 2, …, 2q-1} such that,

when each edge xy is assigned the label or weight | f(x) -

f(y)|, the resulting edge labels are {1, 3, 5, …, 2q-1}. This

definition was introduced in 1991 by Gnanajothi [1] who

proved that the graphs obtained by joining a single

pendant edge to each vertex of

only if n is even. In this paper we generalize Gnanajothi's

result on cycles by showing that the graphs obtained by

joining m pendant edges to each vertex of Cn are odd

graceful if and only if n is even.

KEY WORDS

Vertex labeling, edge labeling, odd graceful.

1. Introduction.

The study of graceful graphs and graceful labeling

methods was introduced by Rosa [2]. Rosa defined a β-

valuation of a graph G with q edges an injection from the

vertices of G to the set {0, 1, 2, …, q}such that when each

edge xy is assigned the label | f(x) - f(y)|, the resulting

edges are distinct. β- valuations are functions that produce

graceful labelings. However, the term graceful labeling

was not used until Golomb studied such labelings several

years later [3].

A graph G of size q is odd-graceful, if there is an

injection f from V(G) to {0, 1, 2, …, 2q-1} such that,

when each edge xy is assigned the label or weight | f(x) -

f(y)|, the resulting edge labels are {1, 3, 5, …, 2q-1}. This

definition was introduced by Gnanajothi [1] in 1991.

Gnanajothi [1] proved that every cycle graph is odd

graceful if and only if n is even . We denote the crown

graphs (the graphs obtained by joining a single pendant

edge to each vertex of

n

C ) by

n

C are odd graceful, if and

1

n

CK

?

therefore the

crown graphs ( the graphs obtained by joining m pendant

edges to each vertex of Cn ) denoted

1

n

CmK

?

.

Gnanajothi [1] proved that

1

n

CK

?

are odd graceful if

and only if n is even. In our study we generalize

Gnanajothi's result on cycles by showing that the graphs

obtained by joining m pendant edges to each vertex of Cn

(

1

n

CmK

?

) are odd graceful if and only if n is even.

2. Main Results about the Development of

Algorithms with Respect to Labeling of

C mK

?

1

n

We are introducing a special method for labeling the

vertices of the cycle graph. By this method, we can get an

odd graceful cycle graph.

Consider a cycle Cn with n vertices and q edges ( n = q)

Algorithm ODDGL for Odd Graceful Labeling of a

cycle Cn.

1. Draw the cycle graph as the graph which consists of

two paths as Left (L) u u1 u2 u3 …

1

2

n

u

−and Right (R)

v1 v2 v3 … v =

2

n

v

. In order to get the cycle graph,

connect the vertex u with the vertex v1 and connect the

vertex v with the vertex

n

u

1

2

−.

2. Number the vertices in (L) u u1 u2 u3 …ui …

1

2

3. Number the vertices in (R) v1 v2 v3 …vi … v =

n

u

−consecutively as 0, q – i (i odd), q + i (i even).

2

n

v

consecutively as 2q – i (i odd), i (i even).

4. Compute the edge labels by taking the absolute value

of the difference of incident vertex labels.

5. The resulting labeling is odd graceful.

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1st Int Conference on Computer Science from Algorithms to Applications (CSAA-2009), 8-10 2009, Cairo, Egypt.

The Figure1(a) shows that how to construct the cycle

according with the algorithm ODDGL. The odd graceful

cycles (C8 and C10) obtained using Algorithm ODDGL

are shown in Figure1(b,c). We asummed that the number

of vertices u1 u2 u3 …

1

2

vertices v1 v2 v3 …

1

2

odd numbers in Figure 1(b) and are even numbers in

Figure 1(c).

n

u

−be x and the number of

n

v

−be y. It is clear that x and y are

(a) (b) (c)

Figure 1. C8 and C10 are odd graceful.

Theorem 2.1. The Algorithm ODDGL proves that the

cycle graphs

n

Algorithm ODDGL1 for Odd Graceful Labeling of

joining a single pendant edge to each vertex of the

cycle Cn.

1. Draw the cycle graph as the graph which consists of

two paths as Left (L) u u1 u2 u3 …

C is odd graceful, if and only if n is even .

1

2

n

u

−and Right (R)

v1 v2 v3 … v =

2

n

v

. In order to get the cycle graph,

connect the vertex u with the vertex v1 and connect the

vertex v with the vertex

n

u

1

2

−. Let the number of vertices

u1 u2 u3 …

1

2

n

u

−be x and the number of vertices v1 v2 v3

…

1

2

n

v

−be y.

2. Join one pendant edge for each vertex

iv on the right

n. Join one

path (

1

i

i

v v

) where i = 1, 2, 3, …,2

pendant edge for each vertex

iu on the left path (

n− . Finally, join one pendant edge

1

i

i

u u )

where i = 1, 2, 3,…,

1

2

for the vertex u (

3. If ( x and y are even number ) then

Begin

1

uu ).

3-1. Number the vertices in (L)

u u1 u2 …ui…

1

2

n

u

−consecutively as

0, q + 2( i-1) – 1 (i odd), 2q – 2( i + 1) (i even).

3-2. Number the vertices in (R) v1 v2 v3 …vi … v =

2

n

v

consecutively as 2q - 2( i - 1) - 1 (i odd) , 2i (i even).

3-3. Number the vertices in (L)

u1

1

2

2 q - 2( i+1) (i odd), q + 2( i - 1) - 1 (i even). Finally,

number the vertex u1 by 3

3-4. Number the vertices in (R)

v1

n

v

consecutively as

1 u2

1 u3

1 …ui

1 …

1

n

u

−consecutively as

1 v2

1 v3

1 …vi

1 …

1

2

2( i – 1) + 2 (i odd), 2q-2( i -1) - 1 (i even).

End

4- If (x and y are odd number ) then

Begin

4-1. Number the vertices in (L)

u u1 u2 u3 …ui …

u

1

2

n

−consecutively as

0, q + 2( i+1) – 1 (i odd), q – 2( i - 1) (i even).

4-2. Number the vertices in (R)

v1 v2 v3 …vi …

1

2

2q - 2( i - 1) - 1 (i odd) , 2i (i even).

number the vertex v by 10.

4-3. Number the vertices in (L)

u1

u

n

v

− consecutively as

1 u2

1 u3

1 …ui

1 …

1

n

1

2

−consecutively as

q - 2( i-1) (i odd), q + 2( i + 1) - 1 (i even).

number the vertex u1 by 1.

4-4. Number the vertices in (R)

v1

v

1 v2

1 v3

1 …vi

1 …

1

n

2

consecutively as

2( i – 1) + 2 (i odd), 2q-2( i -1) - 1 (i even).

End

5. Compute the edge labels by taking the absolute value of

the difference of incident vertex labels.

6. The resulting labeling is odd graceful.

CK

?

obtained using the Algorithm

The graphs

1

n

ODDGL1 are shown in Fig. 2. We asummed that the

number of vertices u1 u2 u3 …

1

2

n

u

−be x and the number

of vertices v1 v2 v3 …

1

2

n

v

−be y. It is clear that x and y are

odd numbers in Figure 2(a) and are even numbers in

Figure 2(b).

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1st Int Conference on Computer Science from Algorithms to Applications (CSAA-2009), 8-10 2009, Cairo, Egypt.

(a) (b)

Figure 2.

8

1

CK

?

and

10

1

CK

?

are odd-graceful.

Theorem 2.2. The Algorithm ODDGL1 proves that the

crown graphs

1

even .

Algorithm ODDGL2 for Odd Graceful Labeling of

joining two pendant edges to each vertex of the cycle

Cn.

1. Draw the cycle graph as the graph which consists of

two paths as Left (L) u u1 u2 u3 …

n

CK

?

is odd graceful, if and only if n is

1

2

n

u

−and Right (R)

v1 v2 v3 … v =

2

n

v

.In order to get the cycle graph,

connect the vertex u with the vertex v1 and connect the

vertex v with the vertex

1

2

u1 u2 u3 …

1

2

v

n

u

−. Let the number of vertices

n

u

−be x and the number of vertices v1 v2 v3 …

1

2

n

−be y.

2. Join two pendant edges for each vertex

iv on the right

path (

1

i

i

v v

and

2

i

i

v v

) where i = 1, 2, 3,…, 2

iu on the left path

n.

Join two pendant edges for each vertex

(

1

i

i

u u and

2

i

i

u u ) where i = 1, 2, 3, …,

1

2

n− .

Finally, join two pendant edges for the vertex u

(

uu and

uu

).

3. If ( x and y are even number ) then

Begin

3-1. Number the vertices in (L) u u1 u2 u3 …ui …

12

1

2

n

u

−

consecutively as

0, q + 3( i-1) – 1 (i odd), 2q – 3( i + 1) + 1 (i even).

3-2. Number the vertices in (R) v1 v2 v3 …vi … v =

2

n

v

consecutively as 2q - 3( i - 1) - 1 (i odd) , 3i (i even).

3-3. Number the vertices in (L) u1

1

2

(i odd), q + 3( i - 1) + 2k - 4 (i even).

Finally, number the vertex uk by 2k + 1where k = 1, 2.

3-4. Number the vertices in (R) v1

n

v

consecutively as

k u2

k u3

k …ui

k …

k

n

u

−consecutively as 2 q - 3( i+1) – 2k + 4

k v2

k v3

k …vi

k …

2

k

3( i – 1) + 2k (i odd), 2q-3( i -1) – 2k + 2 (i even).

End

4. If (x and y are odd number ) then

Begin

4-1. Number the vertices in (L) u u1 u2 u3 …ui …

1

2

0, q + 3( i+1) – 1 (i odd), q – 3( i - 1) - 1 (i even).

4-2. Number the vertices in (R) v1 v2 v3 …vi …

n

u

−consecutively as

1

2

n

v

−

consecutively as

2q - 3( i - 1) - 1 (i odd) , 3i (i even).

number the vertex v by 14.

4-3. Number the vertices in (L) ) u1

1

2

q + 3( i + 1) + 2k - 4 (i even).

Finally, number the vertex uk by 2k – 1 where k = 1, 2.

4-4. Number the vertices in (R) v1

n

v

consecutively as 3( i – 1) + 2k (i odd),

k u2

k u3

k …ui

k …

k

n

u

−consecutively as q - 3( i-1) – 2k + 2 (i odd),

k v2

k v3

k …vi

k …

2

k

2q-3( i -1) – 2k + 2 (i even).

End

5- Compute the edge labels by taking the absolute value

of the difference of incident vertex labels.

6- The resulting labeling is odd graceful.

The graphs

21

ODDGL2 are shown in Fig. 3. We asummed that the

number of vertices u1 u2 u3 …

n

CK

?

obtained using the Algorithm

1

2

n

u

−be x and the number

of vertices v1 v2 v3 …

1

2

n

v

−be y. It is clear that x and y are

odd numbers in Figure 3(a) and are even numbers in

Figure 3(b).

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1st Int Conference on Computer Science from Algorithms to Applications (CSAA-2009), 8-10 2009, Cairo, Egypt.

(a) (b)

21

Theorem 2.3. The Algorithm ODDGL2 proves that the

crown graphs

21

is even .

Algorithm ODDGL3 for Odd Graceful Labeling of

joining m pendant edges to each vertex of the cycle Cn.

1. Draw the cycle graph as the graph which consists of

two paths as Left (L) u u1 u2 u3 …

Figure 3.

8

CK

?

and

10

21

CK

?

are odd-graceful.

n

CK

?

is odd graceful, if and only if n

1

2

n

u

−and Right (R)

v1 v2 v3 … v =

2

n

v

.In order to get the cycle graph,

connect the vertex u with the vertex v1 and connect the

vertex v with the vertex

1

2

u1 u2 u3 …

1

2

v

n

u

−. Let the number of vertices

n

u

−be x and the number of vertices v1 v2 v3 …

1

2

n

−be y.

2. Join m pendant edges for each vertex

iv on the right

) where i =

path (

1, 2, 3, …, n/2. Join m pendant edges for each vertex

iu on the left path (

ii

u u ,

u u ,…, and

n− . Finally, join m pendant

1

i

i

v v

,

2

i

i

v v

, …, and

m

ii

v v

12

ii

m

ii

u u

)

where i =1, 2, 3,…,

1

2

edges for the vertex u (

3. If ( x and y are even number ) then

Begin

3-1. Number the vertices in (L) u u1 u2 u3 …ui …

1

2

2q – (m+1)( i + 1) + (m-1) (i even).

3-2. Number the vertices in (R) v1 v2 v3 …vi … v =

1

uu ,

2

uu

,…, and

m

uu

).

n

u

−consecutively as 0, q + (m+1)(i-1)-1 (i odd),

2

n

v

consecutively as 2q – (m+1)( i - 1) - 1 (i odd) ,

(m+1)i (i even).

3-3. Number the vertices in (L)

u1

n

u

k u2

k u3

k …ui

k …

1

2

k

−consecutively as

2 q - (m+1) ( i+1) - 2k + 2m (i odd),

q + (m+1) ( i - 1) + 2k – m-2 (i even).

Finally, number the vertex uk by 2k + 1

where k = 1, 2, 3,…, m

3-4. Number the vertices in (R) v1

k v2

k v3

k …vi

k …

2

k

n

v

Consecutively as (m+1) ( i – 1) + 2k (i odd),

2q-(m+1) ( i -1) – 2k + m (i even).

End

4- If (x and y are odd number ) then

Begin

4-1. Number the vertices in (L) u u1 u2 u3 …ui …

1

2

q – (m+1)( i - 1) - (m-1) (i even).

4-2. Number the vertices in (R) v1 v2 v3 …vi …

n

u

−consecutively as 0, q + (m+1)(i+1)-1 (i odd),

1

2

n

v

−

consecutively as 2q – (m+1)( i - 1) - 1 (i odd) ,

i (m+1) (i even).Number the vertex v by 4(m+1)+2.

4-3. Number the vertices in (L) ) u1

1

2

q + (m+1) ( i + 1) + 2k – m-2 (i even). Finally,

number the vertex uk by 2k – 1where k = 1, 2, 3,…, m

4-4. Number the vertices in (R) v1

k u2

k u3

k …ui

k …

k

n

u

−consecutively as q - (m+1) ( i-1) - 2k + 2 (i odd),

k v2

k v3

k …vi

k …

2

k

n

v

consecutively as (m+1) ( i – 1) + 2k (i odd),

2q-(m+1) ( i -1) – 2k + m (i even).

End

5. Compute the edge labels by taking the absolute value of

the difference of incident vertex labels.

6. The resulting labeling is odd graceful.

The graphs

1

n

CmK

?

obtained using the Algorithm

ODDGL3 are shown in Fig. 4. We asummed that the

number of vertices u1 u2 u3 …

1

2

n

u

−be x and the number

of vertices v1 v2 v3 …

1

2

n

v

−be y. It is clear that x and y are

odd numbers in Figure 4(a) and are even numbers in

Figure 4(b).

Theorem 2.4. The Algorithm ODDGL3 proves that the

crown graphs

1

is even .

n

CmK

?

is odd graceful, if and only if n

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1st Int Conference on Computer Science from Algorithms to Applications (CSAA-2009), 8-10 2009, Cairo, Egypt.

2m + 8

q+2m+5

q+2m+7

2m + 4

2m + 6

2q-4m-3

4m+10

2q-2m-9

2q-2m-7

2q-2m-5

6m+4

3

7 5

2(q-m-3)

2(q-m-4)

2(q-m-5)

q+4m+1

4m+8

q+2m+3

4m+6

4

6

2(q-2)

2(q-3)

2(q-4)

2(q-2m-2)

2

4m + 2

2q - 3

2(q-2m-3)

4(m+1)

q+2m+1

2(q-m)-3

0

2(q-2m)-5

2m

q-1

2q-1

2(q-m-1)

2m+1

2q - 7

2q - 5

2q-2m-1

q+1

q+3

q+5

q+2m-1

2(q-m-2)

2(m+1)

(a) (b)

CmK

?

and

Figure 4.

8

1

10

1

C mK

?

are odd-graceful.

3. Conclusion

Since labeled graphs serve as practically useful models for

wide ranging applications such as communications

network, circuit design, coding theory, radar, astronomy,

x-ray and crystallography. It is desired to have

generalized results or results for a whole class, if possible.

This work has presented the generalized result to obtain

the graphs (obtained by joining m pendant edges to each

vertex of

n

C ) which are odd graceful if and only if n is

even.

Acknowledgements

The authors are grateful to Prof. Dr. E. I. Alagamy for his

help and advice in the writing of this paper.

References:

[1] R.B. Gnanajothi (1991), Topics in Graph Theory, Ph.

D. Thesis, Madurai Kamaraj University, India.

[2] A. Rosa (1967), On certain valuations of the vertices

of a graph, Theory of Graphs (Internat. Symposium,

Rome, July 1966), Gordon and Breach, N. Y. and

Dunod Paris 349-355.

[3] S. W. Golomb (1972), How to number a graph: graph

theory and computing, R. C. Read, ed., Academic

press: 23-37.

[4] S. M. Hegde and S. Shetly(2003), On Graceful Tree,

Applied Mathematics E-Notes. 2, 192-197.

[5] K. Kathiresan, Odd graceful Graphs, preprint.

[6] A. Krishnaa (2004), A Study of the Major Graph

Labelings of Trees, Informatica 15, No. 4, 515-524.

[7] Michael Horton(2003), Graceful Trees: Statistic and

Algorithms, Bachelor Computing Thesis, University

of Tamania,

[8] L. Graham and N. J. A. Sloane (1980), On additive

bases and harmonious graphs, SIAM J. Alg. Discrete

Math., 1, 382-404.

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