View of Decomposition of Product Path Graphs Into Graceful Graphs

(1)

Decomposition of Product Path Graphs Into Graceful Graphs

P.M.Sudhaa* _{And P.Senthilkumar}b

a _{Research Scholar, PG and Research Department of Mathematics, Government Arts and Science College, Kangeyam,}

Tiruppur – 638108, Tamil Nadu, India.

b_{Assistant Professor, PG and Research Department of Mathematics, Government Arts and Science College, Kangeyam,}

Tiruppur – 638108, Tamil Nadu, India.

* Corresponding author E-Mail: sudhasathees@gmail.com

Article History: Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published

online: 28 April 2021

Abstract: A decomposition of

G

is a collection



_g

=



H

₁

,

H

₂

,...

H

_r



such that

H

_iare edge disjoint and every edge in

H

_ibelongs to

G

. If each

H

_i is a graceful graph, then



_gis called a graceful decomposition of

G

. The minimum cardinality of a graceful decomposition of

G

is called the graceful decomposition number of

G

and it is denoted by



_g

(G

).

In this paper, we define graceful decomposition and graceful decomposition number

)

(G

g



of a graphs. Also, some bounds of



_g

(G

)

in product graphs like Cartesian product, composition etc. are investigated.

Keyword: Decomposition, Graceful graphs, Graceful decomposition and Graceful decomposition number. 1. Introduction

A graph is a well-ordered pair

G =

(

V

,

E

)

, where

V

is a non-empty finite set, called the set of vertices or nodes of G, and

E

is a set of unordered pairs (2-element subsets) of

V

, called the edges of

G

. If

xy 

E

, x and y are called adjacent and they are incident with the edge

xy

.

The complete graph on n vertices, denoted by

K

_n, is a graph on n vertices such that every pair of vertices is connected by an edge. The empty graph on n vertices, denoted by

E

_n, is a graph on n vertices with no edges. A graph

G



=

(

V



,

E



)

is a sub graph of

G =

(

V

,

E

)

if and only if

V 



V

and

E 



E

.The order of a graph

G =

(

V

,

E

)

is

V

, the number of its vertices. The size of G is

E

, the quantity of its edges. The degree of a node

x 

V

, represented by

d

( x

)

, is the quantity of edges incident with it.

A subgraph H of G is a graph such that

V

(

H

)



V

(

G

)

and

E

(

H

)



E

(

G

)

. For a graph

G

(

V

,

E

)

and a subset

W 

V

, the subgraph of G induced by W, denoted as

G

 

W

, is the graph

H

(

W

,

F

)

such that, for all

u

,

v



W

, if

uv 

E

, then

uv 

F

. We say H is an induced subgraph of G.

A graph

G

(

V

,

E

)

is said to be connected if every pair of vertices is connected by a path. If there is exactly one path connecting each pair of vertices, we say G is a tree. Equivalently, a tree is a connected graph with n − 1 edges. A pathgraph

P

_nis a connected graph on n vertices such that each vertex has degree at most 2. A cycle graph

C

_n is a connected graph on n vertices such that every vertex has degree 2.

A complete graph

P

_n is a graph with n vertices such that every vertex is adjacent to all the others. On the other hand, an independent set is a set of vertices of a graph in which no two vertices are adjacent. We denote In for an independent set with n vertices.

A bipartite graph

G

(

V

,

E

)

is a graph such that there exists a partition

P

(

A

,

B

)

of V such that every edge of G connects a vertex in A to one in B. Equivalently, G is said to be bipartite if A and B are independent sets. The bipartite graph is also denoted as

G

(

A

,

B

,

E

)

.

A graceful labelling of a graph G is a vertex labelling

f

:

V

→

 

0 ,

1

such that f is injective and the edge labelling

f

*

:

E

→

 

1 ,

m

defined by

f

*

(

uv

)

=

f

(

u

)

−

f

(

v

)

is also injective. If a graph G admits a graceful labelling, we say G is a graceful graph.

(2)

In this paper we define graceful decomposition and graceful decomposition number



_g

(G

)

of a graph

G

. Also investigate some bounds of



_g

(G

)

in product graphs like Cartesian product, composition etc.

2. Graceful Decomposition

In this section we define graceful decomposition of a graph

G V E

( , )

some and investigate some bounds of graceful decomposition number in

G V E

( , )

.

Definition 2.1:Let



_g

=



H

₁

,

H

₂

,...

H

_r



be a decomposition of a graph

G

. If each

H

_i is a graceful graph, then



_gis called a graceful decomposition of

G

. The minimum cardinality of a graceful decomposition of

G

is called the graceful decomposition number of

G

and it is denoted by



_g

(G

).

Definition 2.2: Let

G =

₁

(

V

₁

,

E

₁

)

and

G =

₂

(

V

₂

,

E

₂

)

be two simple graphs. The join

G +

₁

G

₂ of

2 1

and

G

with disjoint vertex set

V

₁

&V

₂ and the edge set E of

G +

₁

G

₂ is defined by the two vertices

)

,

(

u

_i

v

_j if one of the following conditions are satisfied i)

u

_i

v

_j



E

₁.

ii)

u

_i

v

_j



E

₂.

iii)

u

_i



V

₁

&

v

_j



V

₂ ,

u

i

v

j



E

Theorem 2.1: A graph

P +

_n

P

_mis a join of two path graceful graphs with (m>n) can be decomposed in to at least ‘m’ number of

P

_m,graceful graphs. Then the graceful decomposition number



_g

(

P

_n

+

P

_m

)



3 .

Proof:Let

P

_n and

P

_m be two path graceful graphs of order m and n (m>n)respectively and

P +

_n

P

_mis a join of

P

_n and

P

_mwith edge set E. Therefore

E

=

E

₁



E

₂



S

(

K

_m_,_n

)

, here

S

(

K

m,n

)

is a size of a bipartite

complete graph

K

_m_,_n. Note that

P

_n and

P

_m be two graceful graphs and complete bipartite graphs

K

_m_,_nalso graceful graph. The complete bipartite graphs

K

_m_,_ncan be decomposed in to m number of

P

_m. This implies















= mi m i g

P

1



and















= mi m i g

P

1



. Therefore we get



_g

(

P

_n

+

P

_m

)



m

.

Note that

P

_n and

P

_mare graceful graph also decomposed in to

P

_n and

P

_m paths, hence we get



_g

(

P

_n

+

P

_m

)



m

.

Illustration 2.1:The Join of two graceful graphs

P

₂

& P

₃is given in figure.2.1

3 2

P

P +

The graph

P +

₂

P

₃ is decomposed in to isomorphic graphs of

P

₂ ,

P

₃and

K

₃_,_,₂. Therefore the set



P

1

,

P

2

,

K

3,2



g

=

(3)

Figure.2.1:Graceful decomposition of

P +

₂

P

₃

G =

₁

(

V

₁

,

E

₁

)

and

G =

₂

(

V

₂

,

E

₂

)

be two simple graphs. The Cartesian product

2 1

G

G 

of

G

₁

and

G

₂, is a graph with vertex set

V

=

V

₁



V

₂ and the edge set of

G 

₁

G

(

u

i

,

v

j

)

&

(

u

k

,

v

l

)

if one of the following conditions are satisfied

i)

u =

₁

v

₁and

u

₂

, v

₂are adjacent vertices in

G =

₂

(

V

₂

,

E

₂

)

. ii)

u =

₂

v

₂and

u

₁

, v

₁are adjacent vertices in

G =

₁

(

V

₁

,

E

₁

)

.

P 

_m

P

_nis a Cartesian product of two graceful graphs

P &

_m

P

_n with order m and n can be decomposed in to at least

(

m +

n

)

graceful graphs (i.e.



g

(

G

1



G

2

)



(

m

+

n

)

).

Proof:Let

P

_m and

P

_n be two path graceful graphs of order m and n (m > n) respectively and

P 

_n

P

_m

and _is a Cartesian product of

P &

_n

P

_mwith edge set E the one of the following conditions are satisfied i)

u =

₁

v

₁and

u

₂

, v

G =

₂

(

V

₂

,

E

₂

)

.

ii)

u =

₂

v

₂and

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

.

Case (i):If

u =

₁

v

₁and

u

₂

, v

G =

₂

(

V

₂

,

E

₂

)

If

u =

₁

v

₁and

u

₂

, v

G =

₂

(

V

₂

,

E

₂

)

. Let the sub graph

H

_iis isomorphic to the graph

G =

₂

(

V

₂

,

E

₂

)

. The graph

G =

₂

(

V

₂

,

E

₂

)

be a graceful graph this implies

H

_iis also a graceful graph. This implies

H

_i





Case (ii):If

u =

₂

v

₂

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

If

u =

₂

v

₂

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

. Let the sub graph

H

_jis isomorphic to the graph

)

,

(

₁ ₁ 1

V

E

G =

. The graph

G =

₁

(

V

₁

,

E

₁

)

is a graceful graph this implies

H

_jis also a graceful graph. This implies





j

H

.

From case (i) and (ii), we get























 















=

= = j n j i m i 1

H

1

H



this implies

H

m

n

n j j m i i

+

=

+

=



= =1 1



. Hence we get



g

(

G

1



G

2

)

=

(

m

+

n

)

.

(4)

Figure.2.2:

P 

₂

P

₃

The graph

P 

₂

P

₃is decomposed in to isomorphic graphs of

P

₂and

P

₃, the set



contains n times

P

₂ and m times

P

₃as follows.

Isomorphic graphs of

P

₂ Isomorphic graphs of

P

₃

The graph

P 

₂

P

₃is decomposed in to

O

(

G

₂

)

number of

G

₁ graphs,

O

(

G

₁

)

number of

G

₂ Graphs.

G =

₁

(

V

₁

,

E

₁

)

and

G =

₂

(

V

₂

,

E

₂

)

be two simple graphs. The Composition

G 

₁

G

₂ of

G

₁

and

G

₂, is a graph with vertex set

V

=

V

₁



V

₂ and the edges in

G 

₁

G

)

,

(

&

)

,

(

u

₁

u

₂

v

₁

v

₂ if one of the following conditions are satisfied

i)

u =

₁

v

₁and

u

₂

, v

G =

₂

(

V

₂

,

E

₂

)

. ii)

u =

₂

v

₂and

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

. iii)

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

.

G 

₁

G

₂is a Composition of two graceful graphs

G

₁

& G

₂ with order m and n, can be decomposed in to at least

(

mn

+

m

+

n

)



g

(

G

1



G

2

)



(

mn

+

m

+

n

)

).

Proof:Let

G =

₁

(

V

₁

,

E

₁

)

and

G =

₂

(

V

₂

,

E

₂

)

be two graceful graphs of order m and n respectively and

2 1

G

G 

is a Composition of

G

₁

and

G

₂with edge set E the one of the following conditions are satisfied i)

u =

₁

v

₁and

u

₂

, v

G =

₂

(

V

₂

,

E

₂

)

.

ii)

u =

₂

v

₂and

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

. iii)

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

.

Case (i):If

u =

₁

v

₁and

u

₂

, v

G =

₂

(

V

₂

,

E

₂

)

If

u =

₁

v

₁and

u

₂

, v

G =

₂

(

V

₂

,

E

₂

)

. Let the sub graph

H

_iis isomorphic to the graph

G =

₂

(

V

₂

,

E

₂

)

. The graph

G =

₂

(

V

₂

,

E

₂

)

is a graceful graph this implies

H

_i





Case (ii):If

u =

₂

v

₂

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

If

u =

₂

v

₂

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

. Let the sub graph

H

_jis isomorphic to the graph

)

,

(

₁ ₁ 1

V

E

G =

. The graph

G =

₁

(

V

₁

,

E

₁

)

be a graceful graph this implies

H

_j





.

Case (iii):If

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

.

If

u

₁

, v

G =

₁

(

V

₁

,

E

₁

)

. The graph

G =

₁

(

V

₁

,

E

₁

)

be a graceful graph therefore we get mn number graceful graph isomorphic to

G =

₁

(

V

₁

,

E

₁

).

Hence we get mn times of

G =

₁

(

V

₁

,

E

₁

).

(5)

(

)























 













 















=

= = = j j mj n j j n j i m i 1

H

1

H

1

H

1

,

H

2

,...

H



this implies

H

m

n

mn

n j m i ij n j j m i i

+

=

+

=





= = = =1 1 1 1



. Hence we get

)

(

)

(

G

₁

G

₂

m

n

mn

g





+



.

Illustration 2.3: The Cartesian product of two graceful graphs

P

₂

& P

₃is given in Figure.2.3

2 1

G

G 

Decomposition of

G 

₁

G

₂ Isomorphic to

G

₁ Isomorphic to

G

₂ Isomorphic to ‘mn’ times of

G

₁ Figure.2.3

Definition 2.5:FortwosimplegraphsGandHtheirtensor product is denoted by

G 

H

, has vertex set

2 1

V

(6)

P

_mis a tensor product of two graceful graphs with order

(

m 

n

)

, can be decomposed in to

(m

)

number of

P

_mgraceful graphs (i.e.



g

(

P

m



P

n

)

=

(

m

)

).

Proof: A graph

P 

_m

P

_nis a tensor product of two graceful graphs with

(

m 

n

)

. Let the vertex

(

u

₁

,

v

₁

)

and

(

u

₂

,

v

₂

)

are adjacent whenever

u

₁

u

₂is an edge in

P

_mand

v

₁

v

₂is an edge in

P

_n. By the definition we identify ‘m’ number of

P

_min tensor product

P

_m.Hence we get



g

(

P

m



P

n

)

=

(

m

).

Illustration 2.4: The tensor product of two graceful graphs

P

₂

& P

₃is given in Figure.2.4

4 3

P

P 

4

P

Decomposition of

P 

₃

P

₄ Figure.2.4

Definition 2.6:The Strong product

G 

H

of graphs G and H has the vertex set

)

(

)

(

)

(

G

H

V

G

V

H

V



=



and

(

a

,

x

)(

b

,

y

)

is an edge of

G 

H

ere satisfied one of the following condition.

i)

a =

b

and

xy 

E

(H

)

. ii)

ab 

E

(G

)

and

x =

y

. iii)

ab 

E

(G

)

and

xy 

E

(H

)

.

P 

_m

P

_nis a Strong productof two graceful graphs with m > n, can be decomposed in to at least

(

2 m +

n

)



_g

(

P

_m



P

_n

)



(

2 m

+

n

)

).

(7)

Proof:Let

P

_m

=

(

V

₁

,

E

₁

)

and

P

_m

=

(

V

₂

,

E

₂

)

be two graceful graphs of order m and n respectively and n

m

P

P 

is a Strong productof

P

_m

and

P

_n with edges

(

a

,

x

)(

b

,

y

)



E

and the set is satisfied the one of the following conditions.

i)

a =

b

and

xy 

P

_m. ii)

ab

P

_n and

x =

y

. iii)

ab

P

_nand

xy 

P

_m.

Case (i): If

a =

b

and

xy 

P

_mare adjacent vertices in

P

_m.

If

a =

b

and

xy 

P

_m. Let the sub graph formed by these set of edges is

H

_i

isomorphic to the graph

P

_m. The graph

P

_mis a graceful graph this implies

H





i

H

Case (ii): If

ab

P

_nare adjacent vertices in

P

_nand

x =

y

.

If

ab

P

_n and

x =

y

.Let the sub graph formed by these set of edges is

H

_j isomorphic to the graph

P

_n. The graph

P

_nis a graceful graph this implies

H





j

H

.

Case (iii): If

ab

P

_n.and

xy 

P

_m.

If

ab

P

_n.and

xy 

P

_m. The graph

P

_mis a graceful graph therefore we get m number graceful graph isomorphic to

P

_m Hence we get m times of

P

_m.





































 















=

= = = mi m i j m n j i n m i 1

P

1

P

1

P



this implies

n

m

n

m

P

m i i m n j j m m i i n

+

=

+

=

+

=



= = =

2

1 1 1



.

Paths

P &

_m

P

_nare also decomposed in to graceful graphs .Hence we get



_g

(

P

_m



P

_n

)



(

2 m

+

n

)

.

Illustration 2.5: The strong product of two graceful graphs

P

₂

& P

₃and its possible decomposition are given in Figure.2.5

4 3

P

P 

(8)

Isomorphic to

P

₃ Isomorphic to

P

₃

Other

P

₄Decomposition of

P 

₃

P

₄

Figure.2.5 Conclusion:

In this paper, we define graceful decomposition and graceful decomposition number



g

(G

)

of a graph

G

. Also,

some bounds of



g

(G

)

in product graphs like Cartesian product, composition etc. are discussed. In future, we will

define different types of decomposition on labelling.v

References

1. Rosa, (1967), On certain valuation of graph, Theory of Graphs (Rome, July 1966), Goden and

Breach, N. Y. and Paris, , 349−355.

2. Barrientos, (2005), The gracefulness of unions of cycles and complete bipartite graphs, J. Combin. Math. Combin. Comput. 52, 69 − 78.

3. F. Harary, (1972), Graph theory Addition Wesley, Massachusetts.

4. J. A. Bondy and U. S. R. Murty, (1976), Graph Theory with Applications, Macmillan Press, London. 5. G. Chartrand, L. Lesniak, (1996), Graph and Digraphs, 3rd edition, Chapman & Hall, London,.