P9 factorization of Symmetric Complete Bipartite Digraph

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_{–Factorization of Symmetric Complete}

Bipartite Digraph

U S Ra jput

Department of mathematics and astronomy, Lucknow University,

Bal Go vind Shukla

Department of mathematics and astronomy, Lucknow University

,

ABSTRACT

In path factorization, Ushio [1] gave the necessary and sufficient conditions for -design when k is odd.

-factorization of a complete bipartite graph for an integer, was studied by Wang [2]. Further, Beiling [3] extended the work of Wang [2], and studied -factorization of complete bipartite multigraphs. For even value of in -factorization the spectrum problem is completely solved [1, 2, 3]. However, for odd value of k i.e. and , the path

factorization have been studied by a number of researchers [4, 5, 6, 7, 8]. The necessary and sufficient conditions for the existence of -factorization of symmetric complete bipartite digraph were given by Du B [9]. Earlier we have discussed the necessary and sufficient conditions for the existence of and -factorization of symmetric complete bipartite digraph [10, 11]. Now, in the present paper, we give the necessary and sufficient conditions for the existence of factorization of symmetric complete bipartite digraph, .

Mathemati cal

Clas sificati on

- 6 8 R 1 0 , 0 5 C 7 0 , 0 5 C 3 8 .

Key words

– Complete bipartite Graph, Factorization of Graph, Spanning Graph.

Introdu cti on:

Let be a complete bipartite symmetric digraph with two partite sets having and vertices. A spanning sub graph of is called a path factor if each component of is a path of order at least two. In particular, a spanning sub graph of is called a factor if each component of is isomorphic to . If is expressed as an arc disjoint sum of factors, then this sum is called

factorization of . Here, we take path of order 9. A is the symmetric directed path on vertices.

Mathemati cal Analysis:

The necessary and sufficient conditions for the existence of factorization of complete bipartite symmetric digraph are given below in theorem 1.

Theorem 1:

Let be the positive integers. Then has a factorization iff:

1. 2.

3.

4. is an integer

Proof of n ecessi ty

Proof: Let be the number of factors in the

factorization, and be the number of copies of in a factor. Since any –factor spans , hence the

number of copies of in any factor,

. … (1 )

Since every –factor has symmetric

edges in a factorization, therefore total number of edges in symmetric complete bipartite digraph will be,

Also, since total number of edges in a symmetric complete bipartite digraph are therefore,

H e n c e ,

. … ( 2 )

Obviously, and will be integers. Thus conditions 3 and 4 in theorem 1 are necessary. In a particular

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its end points in , and with end points in , then by simple arithmetic we can obtain,

and

From this, we can compute and , which are as follows:

… (3) ... (4). Since, by definition and are positive integers, therefore equations (3) and (4) imply,

and

This implies

and

Therefore, conditions 1 and 2 in theorem 1 are necessary.

This proves the necessity of theorem 1.

Proof of sufficien cy

Further, we need the following number theoretic result (lemma 1) to prove the sufficiency of theorem 1. The proof of lemma 1 can be found in any good text related to elementary number theory, one such is given in [12].

Lem m a 1:

If gcd then gcd

where and are positive integers. We prove the following result (lemma 2), which will be used further.

Lem m a 2:

If has -factorization, then

has a -factorization for every positive

integer .

Proof: Let is 1- factorable [13] and { }

be a 1- factorization of it. For each with replace every edge of by a to get a spanning subgraph of , such that the graphs

are pair wise edge disjoint, and there union is . Since has a -factorization, it

is clear that is also , and hence

is -

Now to prove the sufficiency of theorem 1, there are three cases to consider:

Case I: . In this case from lemma 2, has

-factorization. For instance (trivial case) has

-factorization. These factorizations are illustrated in figure from 1 to 5.

F i g ( 1 )

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F i g ( 2 ) y1x2y2x3y3x4y4x5y5 , y5x5y4x4y3x3y2x2y1; y6x6y7x7y8x8y9x1y1 0, y1 0x1y9x8y8x7y7x6y6:

.

F i g ( 3 ) y1x3y2x4y3x5y4x6y5 , y5x6y4x5y3x4y2x3y1; y6x7y7x8y8x1y9x2y1 0, y1 0x2y9x1y8x8y7x7y6:

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F i g ( 5 ) y1x4y2x5y3x6y4x7y5 , y5x7y4x6y3x5y2x4y1; y6x8y7x1y8x2y9x3y1 0, y1 0x3y9x 2 y8x1y7x8y6: .

C a s e I I : . O b v i o u s l y , h a s

-f a c t o r i z a t i o n .

C a s e I I I : a n d . I n t h i s c a s e , l e t

,

a n d

T h e n f r o m c o n d i t i o n s 1 - 4 o f t h e o r e m ( 1 ) ,

a r e i n t e g e r s a n d , . A s o b t a i n e d p r e v i o u s l y

a n d

. H e n c e ,

S i n c e i s a p o s i t i v e i n t e g e r , t h e r e f o r e ,

( s a y )

mu s t b e p o s i t i v e i n t e g e r .

A g a i n l e t t h e n a n d

w h e r e a r e i n t e g e r s a n d g c d . T h e n b e c o m e s

T h e s e e q u a l i t i e s i mp l y t h e f o l l o w i n g e q u a l i t i e s

T o c o mp u t e a l l p o s s i b l e v a l u e s o f a n d f o r c a s e ( I I I ) , w e e s t a b l i s h t h e f o l l o w i n g l e m m a , l e m m a 3 .

Lem m a 3:

C a s e ( 1 ) : I f g c d a n d g c d t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r s u c h t h a t

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a n d

C a s e ( 2 ) : I f g c d a n d g c d a n d l e t , t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r s u c h t h a t

a n d

C a s e ( 3 ) : I f g c d a n d g c d

a n d l e t , t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r , s u c h t h a t

a n d

C a s e ( 4 ) : I f g c d a n d g c d a n d l e t t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r s u c h t h a t

a n d

.

C a s e ( 5 ) : I f g c d a n d g c d (q, 2 5 ) = 5 , a n d l e t t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r s u c h t h a t

a n d

C a s e ( 8 ) : I f g c d g c d a n d l e t t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r s u c h t h a t

a n d

C a s e ( 9 ) : I f g c d g c d a n d l e t , t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r , s u c h t h a t

a n d

C a s e ( 1 0 ) : I f g c d g c d a n d l e t , t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r , s u c h t h a t

a n d

C a s e ( 1 1 ) : I f g c d g c d a n d l e t t h e n t h e r e e x i s t s a

p o s i t i v e i n t e g e r , s u c h t h a t

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a n d

a n d C a s e ( 1 4 ) : I f g c d g c d

a n d l e t , t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r , s u c h t h a t

a n d

C a s e ( 1 5 ) : I f g c d g c d a n d l e t , t h e n t h e r e e x i s t s a p o s i t i v e i n t e g e r s u c h t h a t

a n d

P r o o f : L e t g c d g c d a n d g c d h o l d .

T h e n g c d = g c d .

S i n c e g c d a n d g c d ,

h e n c e

g c d a n d g c d .

S i n c e

i s a p o s i t i v e i n t e g e r . T h e r e f o r e b y l e m m a 2 . 2 , w e s e e t h a t

g c d = g c d .

H e n c e mu s t b e a n i n t e g e r , w h e r e :

.

T h e r e f o r e , w i l l b e a p o s i t i v e i n t e g e r , w h e r e :

S i mi l a r l y a l l v a l u e s o f a n d r a r e p o s i t i v e i n t e g e r s i n c a s e ( 1 ) o f l e m m a 3 .

T h e p r o o f s o f o t h e r e q u a l i t i e s i n l e m m a 3 a r e s i m i l a r t o c a s e ( 1 ) .

T h i s p r o v e s l e m m a 3 .

B e l o w i n l e m m a 4 , w e g i v e t h e c o n s t r u c t i o n o f

-f a c t o r o f c o m p l e t e b i p a r t i t e s y m m e t r i c d i g r a p h ( f o r c a s e I I I o n l y ) , b y t a k i n g t h e v a l u e s o f a n d f r o m c a s e 1 o f l e m m a 3 a t

.

Lem m a (4 ) : -

L e t , a n d

b e p o s i t i v e i n t e g e r s , t h e n

h a s -f a c t o r i z a t i o n .

P r o o f : - F o r g i v e n v a l u e s o f a n d w e h a v e

n u mb e r o f c o p i e s o f , w i t h i t s e n d p o i n t s i n Y i n a p a r t i c u l a r - f a c t o r , a n d

n u mb e r o f c o p i e s o f , w i t h i t s e n d p o i n t s i n X i n a p a r t i c u l a r - f a c t o r .

H e n c e t o t a l n u m b e r o f c o p i e s o f

a n d t h e n u mb e r o f - f a c t o r s

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w h e r e

a n d

H e r e X , Y a r e t w o p a r t i t e s e t s o f a s ,

a n d

w i t h

a n d

N o w f o r e a c h l e t b e t h e s e t o f g r a p h s w i t h e n d p o i n t s i n Y , s u c h t h a t

w h e r e

F o r e a c h l e t b e a s e t o f g r a p h s

w i t h e n d p o i n t s i n X , s u c h t h a t

= { x5p + 5 ( i - 1 ) + 2 ( u - 1 ) + v , j + 2 ( 5p + 4q) v + 3 ( 5 p + 4 q ) w + ( 5 p + 4 q ) t y2 5p + 1 6 ( i - 1 ) + 4 u + 2 v + w + 1 , j + 1 0p+ 8 ( i - 1 ) + 2 u + v + w} ,

w h e r e

1 ≤ j ≤ 2 ( 5p + 4q), 1 ≤ u ≤3 , 0 ≤ v, w, t ≤1 . L e t O b v i o u s l y c o n t a i n s t

n u mb e r o f v e r t e x d i s j o i n t a n d e d g e d i s j o i n t c o mp o n e n t s s u c h t h a t

t = a + b

= 2 ( 5p + 4q) 2 _{= ( 5p + 4}_{q) n}

0 ,

A l s o s p a n s T h e n t h e g r a p h i s a -

f a c t o r o f . F u r t h e r , i n t h e g r a p h , e a c h o f t h e s e c o n d s u b s c r i p t o f s m e e t s

e a c h o f t h e s e c o n d s u b s c r i p t s o f s o n c e a n d

o n l y o n c e . Al s o i n g r a p h , t h e

s e c o n d s u b s c r i p t o f s m e e t s e a c h o f t h e

s e c o n d s u b s c r i p t s o f s o n c e a n d o n l y o n c e ,

F u r t h e r , i n t h e g r a p h , w e s e e t h a t e a c h o f t h e s e c o n d s u b s c r i p t o f s

c o n n e c t e a c h o f t h e s e c o n d s u b s c r i p t s o f s

o n c e a n d o n l y o n c e . H e n c e w e d e f i n e a b i j e c t i o n s u c h t h a t

:

onto

XUY







wi t h

a n d

f o r e a c h a n d e a c h

L e t

Fξ , η = { ξ( x) η( y ) ; x



X , y



Y , x y



F } b e t h e s e t o f g r a p h s , t h e n i t i s s h o w n t h a t t h e g r a p h s Fξ , η { 1 ≤ ξ ≤ r1, 1 ≤ ≤ r2} , a r e e d g e d i s j o i n t -f a c t o r s o f a n d t h e r e u n i o n i s

.

T h u s { Fξ , η: 1 ≤ ξ ≤r1, 1 ≤ η ≤r2} i s a -f a c t o r i z a t i o n o -f .T h i s p r o v e s l e m m a ( 4 ) .

P roof of t heorem ( 1) :

C o mb i n i n g l e m m a s 2 t o 4 , i t c a n b e s e e n t h a t w h e n t h e p a r a m e t e r s a n d s a t i s f y c o n d i t i o n s ( 1 ) - ( 4 )

i n t h e o r e m 1 , t h e g r a p h h a s -f a c t o r i z a t i o n .

Con clusion :

I n t h i s p a p e r , w e h a v e o b t a i n e d t h e c o n d i t i o n f o r p a t h f a c t o r i z a t i o n o f c o mp l e t e b i p a r t i t e s y m m e t r i c d i g r a p h w i t h 9 n u mb e r o f v e r t i c e s h a v i n g s y m m e t r i c d i s j o i n t e d g e s .

R e f e r e n c e s

[ 1 ] U s h i o K : G - d e s i g n s a n d r e l a t e d d e s i g n s , D i s c r e t e M a t h . , 1 1 6 ( 1 9 9 3 ) , 2 9 9 - 3 1 1 . [ 2 ] W a n g H : - f a c t o r i z a t i o n o f a c o mp l e t e

b i p a r t i t e g r a p h , d i s c r e t e m a t h . 1 2 0 ( 1 9 9 3 ) 3 0 7 - 3 0 8 .

[ 3 ] B e i l i n g D u : - f a c t o r i z a t i o n o f c o m p l e t e

b i p a r t i t e m u l t i g r a p h . A u s t r a l a s i a n J o u r n a l o f C o m b i n a t o r i c s 2 1 ( 2 0 0 0 ) , 1 9 7 - 1 9 9 .

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[ 5 ] W a n g J a n d D u B :

P

₅ - f a c t o r i z a t i o n o f c o mp l e t e b i p a r t i t e g r a p h s . D i s c r e t e m a t h . 3 0 8 ( 2 0 0 8 ) 1 6 6 5 – 1 6 7 3 .

[ 6 ] W a n g J :

P

₇ - f a c t o r i z a t i o n o f c o mp l e t e b i p a r t i t e g r a p h s . Au s t r a l a s i a n J o u r n a l o f C o mb i n a t o r i c s , v o l u m e 3 3 ( 2 0 0 5 ) , 1 2 9 -1 3 7 .

[ 7 ] U . S . R a j p u t a n d B a l Go v i n d S h u k l a :

9

P



f a c t o r i z a t i o n o f c o mp l e t e b i p a r t i t e g r a p h s . A p p l i e d M a t h e m a t i c a l S c i e n c e s , v o l u m e 5 ( 2 0 1 1 ) , 9 2 1 - 9 2 8 .

[ 8 ] D u B a n d W a n g J : - f a c t o r i z a t i o n o f

c o mp l e t e b i p a r t i t e g r a p h s . S c i e n c e i n C h i n a S e r . A M a t h e m a t i c s 4 8 ( 2 0 0 5 ) 5 3 9 – 5 4 7 .

[ 9 ] D u B : - f a c t o r i z a t i o n o f c o mp l e t e b i p a r t i t e s y m m e t r i c d i g r a p h s .

Au s t r a l a s i a n J o u r n a l o f C o mb i n a t o r i c s , v o l u m e 1 9 ( 1 9 9 9 ) , 2 7 5 - 2 7 8 .

[ 1 0 ] U . S . R a j p u t a n d B a l G o v i n d S h u k l a :

f a c t o r i z a t i o n o f c o mp l e t e b i p a r t i t e s y m m e t r i c d i g r a p h . N a t i o n a l S e m i n a r o n “Cu rr en t Tr en d s in Mat h e mati cs wi th Sp eci al Fo cu s o n O.R. an d Co mp u t er s” , D . R . M . L . A . U . F a i z a b a d , I n d i a , ( 2 0 1 0 ) .

[ 1 1 ] U . S . R a j p u t a n d B a l G o v i n d S h u k l a :

f a c t o r i z a t i o n o f c o mp l e t e b i p a r t i t e s y m m e t r i c d i g r a p h . I n t e r n a t i o n a l M a t h e m a t i c a l F o r u m, v o l . 6 ( 2 0 1 1 ) , 1 9 4 9 -1 9 5 4 .

[ 1 2 ] D a v i d M . B u r t o n : E l e m e n t a r y N u mb e r T h e o r y . U B S P u b l i s h e r s N e w D e l h i , 2 0 0 4 .