[0, k i] 1^m -factorizations orthogonal to a subgraph

Applied Mathenmtie~ and Mech:mics (English Edition, Vol 22, No 5, May 2001)

Published by Shanghai University, Shanghai, China

Article ID: 0253-4827(2001)05-0593-04

[ 0 , ki ] ~n -FACTORIZATIONS ORTHOGONAL TO A SUBGRAPH * MA Run-nian ( Z ~ ) I ,

XU Jill (],~

~)1,

GAO Hang-shan ( ~ L L ] ) ~ -

(1 .Electronic Engineering Research Institute, Xidian University, Xi' an 710071, P R China; 2. Department of Engineering Mechanics, Northwestern Polyteclmical University, X_i'an 710072, P R China) (Communicated by ZHANG Ru-qing) AbsUmct : Let G be a graph, k 1 , " ' , k,, be positive integers. If the edges of graph G can be decomposed into some edge disjoint [ O, k I I-factor I t , " ' , [ 0, k~ ]-factor F,, , then we can say F = { F t ,..., F= } , is a [ O , ki ] ~-factorization of G . lf H is a subgraph with m edges in graphGand I E ( H ) f~ E(Fi) I = l f o r a l l l < i 1 E ( H )

I = m ( f o r example a tree, forest and cycle with m edges, denoted

respectively by m- tree, m- forest and m- c y c l e ) , then g r a p h G has a [ O, ki ] ~n_ factorization orthogonal to H . Theorem

Let kl , " ' , k .

be positive integers. If G is a [ 0 , k l + "'" +km - m + 1]-graph and

HIS a subgraph in G with m edges, then graph Ghas a [0, kj ]~'- factorizafion orthogonal to H. Proof

By induction on m , obviously, the theorem is true if m = 1. When m = 2 , by

Lemma 2, the theorem is also true. Now we assume m >I 3 and E ( H ) = { el , ' " , e,, } as follows. Obviously, we may assume each k~ ~> 2(1 ~< i ~< m ) . Otherwise, without loss of generality, we assume k., = 1, then, de(x)

t-1

S u b c a s e ~)

+1 V(H') I- d~,,(V(H')) >>.

2m-2+1 V(W) I-2(m1) = I V ( W ) 1>>.2>I e ( S , T ) . If at least one of the kl , " " , k,, is equal to 2, without loss of generality, we

may assume k m = 2 and

I T1 I~> 4, then

~e ( S , T) = t I T1 I+ k~ I S I - d c ( T l )

S u b c a s e (~)

+ de(T)

(t+

kin) I T1 I - k~, - d e ( T 1 )

(t+

kin) I T 1 I - d e ( T 1 )

- ee(S,T)

+ de(T)

=

- ee(S,T)

- k,, >~1 Tt I - k., >>. 2 >I r

If at least one of the k l , " " , km is equal to 2, without loss of generality, we

may assume km = 2 and I T1 I 3 4)) =- m+ land

(3 •

(m-

596

MA Run-nian, XU Jin and GAO Hang-shan ~e(S,T)

= t I T 1 I+ k., I S I - d e ( T 1 ) + d e ( T )

- ee(S,T)

t + ( t + k,~) I S I - d e ( T 1 ) + d e ( T )

- e~(S,T)

t + de( S) +1 S I - e e ( S , T )

=

+ d e ( T ) - de(T1) =

t-l+

d e ( S ) + d ~ ( S N V(H')) +

I Tll-

ee(S,T)

t-l+

dH,(S N V(H')) +1 T1 I+ d e ( T ) - de(T1) >.

t-l+

dH,(S N V(H')) +1 T1 I+ d e ( T 1 ) - de(T1) =

t-l+

dH,(S n V(H')) +1 T1 I - d~r(T1)

t-l+

dx,(S n V(H')) +1 T' I - dH,(T')

m-

+ d e ( T ) - de(T1) >.

1 + dn.(S n

V(H'))

- m + 1 >~

d,,,(S N V(H')). W h e n x ~ , y~ E S, t h e n d t e ( S N V ( H ' ) ~> 2 >~ e ( S , (S n

V ( l t , ) >I 1 ~ e ( S , T ) ;

As is proved above, V S , T

T ) ; whenx., ory,~ E S, then dte

w h e n x ~ , , y., ~ S , then d t r ( S n

V(W)

>~ 0 ~ e ( S , T ) .

C V ( G ' ) , S N T = 0 , we h a v e S e ( S , T )

>I e ( S , T ) .

Hence graph G' has a ( g , f ) - factor Fm containing given edge e.,, that is, graph G has a ( g , f ) factor F,, containing given edge e,,, but not containing edges e l , "'", e.,_l. Note that V x E V(G), f(x)

= k . , , s o F . , is a [ 0 , k . ] -

factor.

On the other hand, letG = G - E ( F . , ) , then0 ~< d e ( x ) - d e ( x ) da(x)-

d e ( x ) + kl + " " + k~,_l - ( m -