Fault-tolerant ring embedding in faulty arrangement graphs

Fault-Tolerant Ring Embedding in Faulty Arrangement Graphs

Sun-yuan Hsieh

Gen-Huey Chen

Dept. of Computer Science

&

Info. Engg.

National Taiwan University, Taiwan

Dept. of Computer Science

8~

Info.

Engg.

National Taiwan University, Taiwan

e-mail:

[email protected]~

e-mail: ghchenQcsie.ntu.edu.tw

Chin-Wen

Ho

Dept. of Computer Science

&

Info. Engg.

National Central University, Taiwan

e-mail: hocwQcsie.ncu.edu.tw

Abstract

The arrangement graph A n , k , which is a general-

ization of the star graph (n - k = l), presents more flexibility than the star graph in adjusting the major design parameters: number o f nodes, degree, and di- ameter. Previously the arrangement graph has proven hamiltonian. I n this p a p e r 'we further show that the arrangement graph remains hamiltonian even if it is faulty. Let IFe/ and IF,[ denote the numbers of edge faults and vertex faults, respectively.

We

show that

A n , k is hamiltonian when (1) (k = 2 and n - k

>_

4, o r k

2

3 and n - k

2

4 +

r;]),

and IF,I

5

k ( n - k ) - 2 , or (2) k

2

2 , n - k

2

2+[$1, and IF,/

5

k ( n - - k - 3 ) - 1 , or (3) k

2

2 , n - k

2

3, and IF,]

5

k .

1

Introduction

T h e star graph [1] has been recognized as an attrac- tive alternative t o the hypercube network. Unfortu- nately, there is a major drawback for the star graph: a

rigorous restriction t o its size (i.e., n ! ) . Recently, t h e arrangement graph [4], which is a generaliztion of the star graph, has been proposed as a possible solution to

the problem. T h e arrangement graph presents more flexibility t h a n the star graph in terms of choosing the major design parameters: number of vertices, de- gree, and diameter, while preserving most of the nice properties of the star graph. Previous work on the arrangement graph can be found in [3] [4] [5] [B].

Since node faults and link faults may happen when a network is p u t in use, it is practically meaningful

to consider faulty networks. Many related works have appeared in the literature, for example, [7] [8] [9] [lo].

In [8], Latifi, Zheng, and Bagherzadeh showed t h a t an n-dimensional hypercube with a t most n

-

2 link faults is hamiltonian. In [9], Rowley and Bose showed t h a t with slight modifications a base-d undirected de Bruijn graph with a t most d-1 edge faults is hamilto- nian. In [ 7 ] , Fernandes, Friesen, and Kanevsky showed t h a t a WK-recursive network od degree d with a t most d - 3 link faults is hamiltonian. In [lo], Tseng, Chang and Sheu showed t h a t an n-dimensional star graph with a t most n - 3 edge faults is hamiltonian.

Although Day and Tripathi [4] have shown that the arrangement graph is hamiltonian, they have assumed the arrangement graph is fault-free. In the remainding sections, we consider faulty arrangement graphs, and investigate their hamiltonicity.

2

Prelimiaries

First we briefly review the arrangement graph, de- noted by A n , k , where l

<

k

<

n.

Definition

1 The vertex set of A,,,, is denoted by

( p 1 p 2 . . . p k l , p i E { 1 , 2 , ..., n } for a l l 1

5

i

5

k and

pi

#

p j f o r a

#

j }. Vertex adjacency is defined as fol- lows: p l p 2 . . . p k is adjacent to q 1 q 2 . . . q k if and only

i f pi

#

qi f o r some I

5

i

5

k and p i = qj f o r a11 l < j < k a n d j # i .

Often k is referred to as the dimensions of An,kl and the position of pi is said t o be the i t h dimension.

An edge of A n , k is said to be of the ith dimension if its

two incident vertices differ between their correspond- ing arrangements in the ith dimension.

T h e structure of A n , k is recursive. We note t h a t A n , k contains embedded An-k+,., for all 1

5

T

5

k .

An embedded A n - k + r , r is conveniently denoted by ( s 1 s z . . . s k ) n 8 , , where si E {*, 1 , 2 , ..., n} for all I

5

i

<

IC (* represents a "don't care" symbol) and ex- actly r of SI, s 2 ,

...,

s k are

*.

Two basic operations on ( ~ 1 . ~ 2

...

s k ) , , , are defined as follows.

Definition 2 S u p p o s e ( s l ~ z . . . s k ) ~ , , is a n e m b e d d e d

An-k+r,r of A n , k . For each si =

*

(1

5

i

5

k ) , we define the i-partition on (s1sz ... s k ) , , , t o be

an operation that partitions the e m b e d d e d A n - k + r , r

into n - k

+

r embedded An-k+,.-1,,-l 's, denoted b y

( ~ 1...~i-1qsi+l...sk)~,,-~, ~ 2 where q is a missing ele-

ment of { 1 , 2 ,

...,

n } in s 1 s ~ . . . s k .

Definition 3

Suppose {SIS~...S~)~,, represents an e m b e d d e d A n - k + r , r o f A n , k . Let I = ( i 1 , i 2 ,

...,

im), where 1

5

i j

_<

k and si, =

*

f o r all 1

5

j

5

m.

We

define the I-parition on (s1sz

...

s k ) , , , t o be a sequence of operations that perform il-partition, iz-partition, ...,

i,-partztion, sequentially, on ( s ~ s z . . . ~ ~ ) ~ , , . A b e r the I-partition, the e m b e d d e d An-k+,,, is partitioned into ( n - IC + r ) ( n - k + r - 1) . . . ( n - k + r - m + 1 ) e m b e d d e d

Suppose (s1sz ... s k ) , , , and ( t l t 2 ... tk),,, represnts two embedded An-k+,,,’s of A n , k . They are said

to be adjacent if there exists some 1

5

j

<I

k so t h a t sj

#

*,

tj

#

t , and sj

#

t . , and si =

t ;

for all 1

5

i

5

k and

i

#

j . fiurther, we use d i f ( ( s 1 ~ 2 ... s k ) , , , , ( t l t 2

...

t k ) , , , )

(=

j ) to indicate t h e different position between S L S ~ . . . S ~ and t l t 2 ... t k .

Definition 4 Let

bedded

A,-k+,,,

’s of A n , k that result from executing

a ( i l l i z ,

...,

ik-,)-partition on A n , k , where i l i ~ . . . i k - ,

is

a

permutation of k - r elements from { 1 , 2 !

...,

k } .

If Ai is adjacent to A(i-l)mo,n(, -,,(,-z)...(,-,+r+,)

n ( n - I ) ( n - 2)...(n - k

+

r

+

1) - 1, then f o r m an r-ring (0

5

r

5

k ) , denoted b y

R,

=

If we regard each Ai as a supervertex called r- vertex, r

2

1, and define a superedge called r-edge

R, f o r m s a c y c l e o f n ( n - l ) ( n - 2 ) . . . ( n - I C $ - r + l ) r-vertices. Actually, each r-edge consists of ( n

-

k

+

r - l ) ( n - k + r - 2 ) - . ~ ( n - k ) e d g e s o f A n , ~ . Each T -

vertex can be partitioned into n - k + r (r-1)-vertices, and any two of them are connected with an ( r - 1)- edge. So, each r-vertex can be viewed as a complete graph of n - k

+

T ( r - 1)-vertices. For easy reference,

we use K ~ ~ ~ + , . to denote such a complete graph in subsequent discussion.

Definition 5 W e define an i-partition on

R, =

[Ao, A I ,

...,

A,(n-i)(,-2) ...( n - k + r + l ) - i ] t o be a se- quence of operations that perform an i-partition on Ao,

A I ,

...,

A,(,-

1)(,-2)...(~-k+,.+l)- 1 , respectively, where

l < i < k a n d r z l .

Lemma 2.1 For A,,k with n -

L

2

3

(resp. 11

-

k

=

a),

an

R,

of (maximal) length n ( n - l ) ( n - 2) ’ . . n -

k + r + 1) contains an

R,-1

o f (maximal) length n

i

n - l ) ( n - 2) . . . ( n - IC

+

r

+

l ) ( n - IC

+

r ) , where r

2

1

(resp. r

2

2).

Proof:

First a partition on

R,

is executed. Then, for 0

<

i

5

n ( n - 1)(n - 2 ) .

.

. ( n

-

L

+

r

+

1) - 1, two ( r - 1)-vertices, say

Xi

and

Y i ,

are determined in each

Ai

so t h a t ~ i - l ) m o d n ( n - I ) ( n - z ) . . . ( , - ~ + r + l ) and

X ( i + 1 ) m o d n ( n - 1 )(n - 2 ) . . (n - k +r

+

1 ) are connectec-1 to

Xi

and Y , through ( r - 1)-edges, respectively. T h e exis- tence of X i a n d Y , is assured by n - k + r

2

4 (because

A n - k + r - m , r - m ’5.

AO, A l ,

...,

A n ( n - l ) ( n - Z ) - - ( n - k + r + l ) - 1 represent em-

and A ( i + l ) m o d n ( n - l ) ( n - Z ) . . . ( n - k + r + l ) f o r all 0

L

i

I:

A o , A I , A,(,-1)(,-2)...(n-k+r+1)-1 are said t o

W O , All

”’,

A n ( n - l ) ( n - Z ) - - ( n - k + r + l ) - l l .

between

Ai

a n d A(i+I)modn(n-l)(n-z)---(n-li+r+l) 1 then

Suppose Rr = [A,, A I , ..., A,(n-l)(n-~)..-(n-lc+r+1)-1].

n -

k

2

3

and

r

2

I , or n -

k

=

2

and r

2

2). Since each Ai can be viewed as a complete graph

I i ‘ ~ ~ ~ + , . ,

there exists a p a t h , denoted by Pi, from

Xi

to that

passes all ( r

-

1)-vertices in

Ai

exactly once. Clearly, all t h e T - 1)-vertices encountered along the path: Po,

(Y,(,-i)(,-~)...(~-k+r+i)-i, X o ) , form a n Er-1 of the

desired length. Q.E.D.

Remark. In the rest of this paper, X i and as de- scribed above are referred to as the entry (r-l)-vertex and the exit (r-1)-vertex of

A ; ,

respectively.

Given a graph

G ,

we use

V ( G )

and E ( G ) to repre- sent the sets of vertices and edges of

G, respectively.

We say

G

is Connected if there exists a path in

G

be- tween any two of its vertices. T h e degree of U E V ( G ) ,

denoted by d e g U ) , is the number of edges incident

with it. A cycle

‘i

path) in G is called a hamiltonian cy- cle (path) if i t contains every vertex of G exactly once.

G is said to be hamiltonian if it contains a hamiitonian cycle. A hamiltonian graph

G

is said t o be k vertex ( e d g e ) fault-tolerant hamiltonian if it remains hamil- toriian after removing at most

IC

vertices (edges) [ll].

G is said

to be hamiltonian-connected if there exists a hamiltonian path between every two vertices of

G.

Definition 6 Suppose

G

is

a

hamiltonian graph. The vertex (resp. edge) fault-tolerant hamiltonicity of

G ,

denoted b y H,(G) (resp. H e ( G ) ) , is defined as max{lc

G

is k vertex (resp. edge) fault-tolerant hamil- tonian

!I

.

Lemma 2.2 [2]

If

IV(G)l

2

3 and deg(u)+deg(v)

2

IV(G)I for every two non-adjacent vertices U , U of G ,

then G is hamiltonian.

i s a connected graph and deg(u)

+

deg(w

i”’

2

If

IV(G)I +

1 for every t w o non- Lemma 2.3

adjacent vertices u , v of

GI

then

G

is hamiltonian- connected.

(Yo, XI

i

I PI, (Yl, XZ),..., ~n(n-l)(n-2)...(n-k+r+l)-l,

Lemma 2.4 Removing any d

-

3 edges ( x l , y l ) ,

(.a, y z ) , . , . , ( 2 d - 3 , y d - 3 ) f r o m Kd (the complete graph

of d vertices) results in a hamiltonian graph (denoted b y

IC;).

Further, if f o r each ( x i , y i ) , 1

5

i

5

d - 3, there exists anothei-(xj, y j ) , j

#

i , so that they are not incident t o the same vertex, then

l<i

is hamiltonian- connected.

Proof: We first prove t h a t I<$ is hamiltonian by show- ing d e g ( x ; )

+

deg(yi)

>

d for all 1

<

i

<

d

-

3.

Let p and q denote the number of removed edges, exclusive of ( x i , y i ) , t h a t are incident to xi and y i , respectively. Clearly, we have p

+

q

5

d

-

4 , and so

d e g ( z i ) + d e g ( y i ) =: ( d - 1 - ( p + 1)) + ( d - 1- ( q + 1))

=

2 d - 4 - ( p + q )

2

2 d - 4 - ( d - 4 ) = d. By Lemma2.2,

l<i

is hamiltonian. It can be proved similarly that

IC$

is hamiltonian-connected if t h e condition is satis- fied. We have p

+

q

<

d -

5 under the condition, and

thus d e g ( z i )

+

d e g ( y i )

2

d

+

1. By Lemma 2 . 3 , 1.2 is

hamiltonian-connected. Q.E.D.

Slight modifications of the proof above can lead t o the following lemma.

Lemma 2.5 Suppose ( 2 1 , yl), ( 2 2 , Y2),..,, Z d - 2 , Yd-.2) are any d - 2 edges of K d . I f for each

I

zi,y!), 1

5

i

5

d

-

2 , there exists m o t h e r ( z j , yj),

j

#

z, so that they are not incident to the same ver- tex, then removing d - 2 edges f r o m I<d results in a h amilt oni an graph.

T h e following two lemmas can be proved similarly.

Lemma 2.6 Removing any d - 4 edges f r o m Itrd re-

sults a h amilt onian- connect ed graph.

Lemma 2.7 Removing any d - 6 edges and any two vertices of l i d results an a hamiltonian-connected

graph.

3

Fault-Free Hamiltonian Cycles

in

Let Fe denote t h e set of faulty edges. According t o the dirnensions where faulty edges belong to, Fe can be p a h t i o n e d into k subsets, denoted by D i , i =

1 , 2 , ..., k , where each Di is the set of faulty edges t h a t belong t o the i t h dimension.

Lemma 3.1 Suppose IF,[ = k ( n - k ) - 2 , let S = a l a 2 ... ak be a permutation o f l , 2 , ..., k so that ID,,

I

2

IDa2[

2

. .

.

2

IDak[. Then, we have

lDukl

5

n - 3 if

k = 1, and IDak[

5

( n - k ) - 1 i f k

2

2.

Proof: We have / D u k /

5

5

(k(n-t)-2).

T h u s the maximal value of ID,,

1

is

[ ( k ( ( " - ~ ) ' " ) - 2 ) 1 ,

which is equal t o n - 3 as IC = 1, and ( n - k ) - 1 as k

2

2. Q.E.D. Lemma 3.2 Suppose U = ( u ~ u z . . . u ~ ) ~ , ~ , V = (wlva ... v k ) , , , , and

w

= (w1wz ... wk),,,, are arbitrary three consecutive r-vertices in R,, where r

2

1. Also let p

=

d i f ( U , V ) and q = d i f ( V , W ) . If

up

#

w4, then a f l e r executing a partition on R, each (r

-

1)-

vertex of V is adjacent to one (r - 1)-vertex of

U

o r one (r

-

1)-vertex of W .

Proof: Without loss of generality, we assume t h a t a j - partition is executed on

R,,

where 1

5

j

5

IC. Hence, we have u j = vj = wj =

*.

Since p

=

d i f ( U , V ) and

q = d i f ( V , W ) , we have up

#

v p , v q

#

w g , ui

=

vi for all 1

5

i

5

k and i

#

p , and vi = wi for all 1

5

i

5

k and i

# q .

Suppose u p

#

w q , and after t h e j-partition, there exists an ( r

-

l)-vertex, say I/' = ( V l W 2 ...llj- l v ' ' ~ j + 1 . . . v k ) ~ , , - ~ , of

v

t h a t is not adjacent t o any ( r - 1)-vertex of U and

W .

Thus, we have

w'

=

up for otherwise V' is adjacent t o some ( r - 1)-vertex of U . Similarly, we have 2)' = zug. This implies u q =

U'

=

w q r which contradicts our assumption. Q.E.D.

For 1

5

i

5

IC - 1, an i-edge of A,,! is faulty if it contains faulty edges. An R, of An,k is said t o be fault-free if it does not contain faulty r-edges. In the rest of this section, we use Ei,j t o denote t h e set of faulty i-edges t h a t belong t o the j t h dimension.

Lemma 3.3 Suppose IF,/ = k ( n

-

k ) - 2, and [et S

=

a l a 2 ... a k be a permutation of 1 , 2 , ..., k so that /Duk-21 = n

-

k , there is a fault-free R2 in A n , k .

A n , k

with Faulty Edges

] D a l l

2

/ D u 2 1

2

. . .

2

If

/Dall =

ID,*!

= . . .

=

Proof: First we construct fault-free Rk-1

,

R k - 2 ,

...,&

for the faulty A , , k . We only need to apply a ( a l , a 2 , ..., u k - 3 ) - partition to the faulty A n , k . Initially, an al-partition is executed on the faulty A n , k , and so a I<;-' is obtained. We note t h a t all edges of the I<:-' are ( k

-

1)-edges and l E k - l , a l /

( 5

] D a l l = n - k ) of them are faulty. By Lemma 2.4, the

I<;-'

with

I E I E - ~ , , ~

I

faulty ( k - 1)-edges removed is hamiltonian. Hence, a fault-free Rk-1 can be generated.

In general, when an aiTpartition is executed on a fault-free R k - ; + l for 2

5

z

5

k

-

3 , each ( k - i

+

1)- vertex of the fault-free Rk-i+l forms a Since

IEk-i,,,(

5

ID,,

I

= n - k , each contains n - k faulty ( k - i)-edges a t most. Now t h a t each

IC:Ij+l

contains n - i f 1

2

n-k+4 (k-+vertices, Lemma2.6 assures that it is hamiltonian-connected, even if all faulty ( k - i)-edges are removed. As a result, there is a fault-free hamiltonian path for each I i ~ ~ ~ + l t h a t goes from the entry ( k - i)-vertex t o the exit ( k -

i)-

vertex. All these hamiltonian paths interleaved with ( k - +edges form a fault-free R k - i .

Next we show how t o obtain a fault-free RZ from a

fault,-free

R3

=

[A,,

A I , ..., A n ( , - l ) ( , - 2 ) . . . ( n - k + 4 ) - l ] . An ak-2-partition is first executed on

RS

so that each Ai forms a I<:-,+,, where 0

5

i

₅

n ( n - l ) ( n -

2 ) - " ( n - k

+

4) - 1. There are totally IE2,a,-,l

(

I

= n

-

k ) faulty 2-edges contained in all Ai's. If IEa,ak-zl

<

n

-

k or IEz,,,-,l faulty 2-edges are distributed over two or more Ai's, then it is assured by Lemma 2.6 that each with faulty 2-edges removed is hamiltonian-connected. Consequently, as described in the proof of Lemma 2.1, a fault-free

R2

can be generated by properly selecting the entry 2- vertex and the exit 2-vertex for each izi.

On the other hand, if IE2,u,--2[

=

n - k and all faulty 2-edges are located in some A t , then it is as- sured by Lemma 2.4 t h a t the

I<:-,+,

resulting from At with n - IC faulty 2-edges removed is hamiltonian. T h a t is, the contains a fault-free hamilto- nian cycle, denoted by C =

(XO,

X I , ..., Xn--k+z, X o ) , where each Xj(0

5

j

5

n - k

+

2) is a 2-vertex of A t . Since there are n - k

+

2 2-edges between each Pair of

Ai

and A(i+l)modn(n-l)(n-2)..:(n-k+4), there are

n - k

+

2 %vertices in C t h a t are adjacent t o n - k

+

2 %vertices of A( t-l)modn(n-l)(n-2)...(,-k+4), and there

are n - k + 2 2-vertices in

C

t h a t are adjacent t o n-k+ 2 2-vertices of A ( t + ~ ) m o d n ( n - ~ ) ( n - a ) ( " - k + 4 ) . T h u s ,

there must exist a 2-edge, say ( X l , X ( l + 1 ) m o d n - k + 3 ) ,

of

C

so that

Xr

and X ( r + l ) m o d n - k f 3 are adja- cent t o 2-vertices of A(t- 1 - 1 )(,

-

2 ) . . .(, - k+4) and

il(l+l)modn(n

- l ) ( n

-

2 ) . . .(, - k +4), respectively. Since t h e

other K:-,+,'s that result from all A j ' s with j

#

t

are hamiltonian-connceted and their 2-edges are not faulty, a fault-free R2 can be generated by taking X I and X ( l + l ) m o d n - k f 3 as t h e entry and exit 2-vertices of

At,

respectively, and properly selecting the entry and exit 2-vertices for all A j ' s with j

#

t .

Q.E.D.

For ease of descrip- is said to be good if after a partition each ( r - 1)- vertex of

Ai

is adjacent to one ( r - 1)-vertex of A(i- l ) m o d n ( n - ~ ) ( ~ - ~ ) - . . ( ~ - i c + ~ + l ) or one ( r - I)-vertex Lemma 3.4 Suppose IF,I = k ( n - k ) - 2, and let S = a l a 2 ... u k be a permutation of 1 , 2 ,

.._,

k so that

l D , k - 2 /

= n

-

k

>

5

and

IDakwl

1

= n -

k

- 1, there is a fault-free good

R I

in An,k.

Proof: With the aid of Lemmas 2.4, 3.2 and Lemma 3.3, the result can be shown. Q.E.D. Lemma 3.5 Suppose /Pel = k ( n - I C )

-

2 , and let

S

= a l a 2 ... a k be a permutation of 1 , 2 , ..., k so that ID,,

1

2

ID,,/

2

. . .

2

ID,,

I.

When k = 1, there is a

faiil2-free hamiltonian cycle f o r An,k i f ID,,

I

= 12 - 3.

Wh,en k

=

2 , there is a fault-free hamiltonaan cycle f v r An,k i f n - IC

2

3 and lDall = IDu2i = n - 3 . W h e n

k

2

3, there is a fault-free ham,iltonian cycle f o r A,,k

z f n - IC

>

5, lDak-,l = n - k , and

IDak-,

=

lDak1

= n - l e - 1 .

Proof: When k = 1, A n , l is a complete graph of n vertices. By Lemma2.4, An,l with ID,,

1

= n-3 ifaulty edges removed hamiltonian.

When IC = 2, we first apply an al-partition t o A,,2 t o form a I<;. Since /El,,l

1

5

ID,,

1

= n - 3 , tlie I<: contains a t most n - 3 faulty Ledges. By t h e aid of Lemma 2.4, a fault-free

12

= [Ao, A I ,

...,

A,-1], can

be determined in An,2. We use X = (z~~~...zk),,~, Y = ( ~ 1 ~ 2 . . . y k ) , , ~ , and

2

= ( Z ~ Z ~ . . . Z L ) , , ~ t o denote

arbitrary three consecutive 1-vertices of the RI. Since

z d i f ( x , y )

#

z d i f ( Y , Z ) , it is assured by Lemma 3.2 t h a t t h e

RI

is good. We then apply an az-partition to the RI so t h a t each Ai (0

5

i

5

n - 1) forms a I<:-l.

Since there are totally JFo,,,J = JD,,I = n - 3 faulty edges contained in all Ai’s, each IC:-l contains n - 3 faulty edges a t most. Below we construct a fault-free hamiltonian cycle for An,2 according t o three situa- tions.

The first situation is t h a t all n - 3 faulty edges are located in the same At for some 0

5 t

5

n - 1, and t,here exists a faulty edge, say ( U , U ) , so t h a t the other

n - 4 faulty edges are incident t o U or w. We note t h a t n

-

2 vertices of At are connected t o A(t-l+,dn

,

and n

-

2 vertices of At are connected to A ( t + l ) m o d n . Moreover, since t h e

RI

is good, one vertex of

At

is

connected t o A(t-l)modn only, another is connected t o

A(t+l)modn only, and t h e others are each coninected t o both A(t-l)modn and A ( t + l ) m o d n . Thus, it can be assured t h a t one of U and v is connected t o A(t-l)modn

and the other is connceted to A ( t + l ) m o d n , and v is

adjacent t o a vertex, say

w’,

of A ( t + ~ ) ~ ~ d , . We then show t h a t there exists a fault-free hamil- tonian path between U and U in At. Since such a path

exists for the trivial case of n = 4, we assume n

2

5.

tion, an Rr

=

[Ao, A l , ‘.., An(n-l)(n-2).. (n-k+r+l)-ll

of A(i+l)mod,(n-1)(n-2)...(n--iE+r+l)-

laLll

2

P a , I

2

” ’

2

PQI.

IfIDa,I = P a 2 I =

Let K’ denote the rlesulting of

At

with n -

3

faulty edges removed, and H

=

I<’ - { U , U > be a com- plete graph of n - 3 vertices. Since n

2

5,

N

contains two or more vertices. Let 2 and y be two vertices of H t h a t are adjacent t o U and U , respectively. Clearly a

hamiltonian path between x and y in H combined with edges ( U , , z) and (y, ?I) constitutes a fault-free hamil- tonian path between U and v in At.

Since all the other Ais with i

#

t

do not contain faulty edges, they are hamiltonian-connected. A fault- free

&,

i.e., a fault-firee hamiltonian cycle, in

An,2

can be generated by taking U and U as the entry and exit vertices of A,, respectively, and properly selecting the entry and exit vertices for all Ai’s with i

#

t .

T h e second situation is t h a t all n-3 faulty edges are located in the A t , but the faulty edge ( U , w mentioned

of At with n -- 3 faulty edges removed contains a fault-free hamiltoiiian cycle. Let

f

and g be any two adjacent vertices in the hamiltonian cycle. Since all other Ai’s with

i

#

t

are hamiltonian-connected, a fault-free hamiltonian cycle for An,2 can be generated by taking f and g as the entry and exit vertices of A t , respectively, and properly selecting the entry and exit vertices for all Ai’s with i

#

t .

T h e third situation is t h a t n

-

3 faulty edges are distributed over two or more Ai’s. By Lemma 2.4, each resulting with faulty edges removed is hamiltonian-connected. As described in the proof of Lemma 2.1, a fault-free hamiltonian cycle for A,,2 can be generated by properly selecting t h e entry and exit vertices for each Ai.

When k

2

3 ,

w e have \Dall

= \Du,l =

IDa,-,l = n -.

k

because ( F e / = k ( n - k )

-

fault-free hamiltonian cycle for An,k is very similar t o t h e case of IC = 2, we only show t h e difference.

As

before an

RI

in AFd3k is first obtained by executing

a ( a l , u 2 , ..., ak-l)-partition. Like t h e case of k = 2, a good R I is necessary for constructing a fault-free hamiltonian cycle for A n , k . But, unlike the case of k = 2, a good

R I

is now assured by Lemma 3.4 be-

cause n - k

>

5. Q.E.D.

Similar t o the proof of the above, we have the fol- lowing.

Lemma 3.6 Suppose IF,/ = k ( n - IC) - 2 , and let

S

= a l a 2 ... a k be a permutation of 1 , 2 ,

...,

k

so that ID,,

1

2

lDu21

2

. . .

2

[Da,l. When k

=

2, there i s a fault-free hamiltonian cycle f o r A,,k if ID,,

I

=

n - k

2

3 arid ID,,

I

=

n - k

-

2. When k

2

3 , there is a fault- free hamiltonian cycle for A,,k i f n - k

>

3 and either

JD,,-,

1

= n - k and ID,,

I

= n - k - 2 or

I

=

n - k , IDuk-,

I

=

n .- k - 1, and ID,,

I

=

n - k - 2. above does not exist. By Lemma 2.5, t

i!

e resulting

2

lDa21

2

2

IDuk

1.

Since constructing a

An

R,

is said t o be healthy if after a partition its every r-edge contains at least three ( r

-

1)-edges t h a t are not faulty.

Lemma 3.7 Suppose IF‘,( = le(. - k ) - 2 , and let

S

= a l a 2 ... a k be a permutation of 1 , 2 ,

...,

C

so that lLl,,l

2

lDa21

2

. . .

2

lDak1. W h e n k:

2

3 and n

2

v,

there as a fault-free hamiltonian cycle for A n , k ifjD,,

I

5

n

-

II.

-

3.

Proof: We first prove by induction t h a t healthy rings R k - 1 ,

Rk-2

,.._,

RI

can b e generated

if

a

(UI, 122,

...,

Uk-l)-partition is applied t o An,k. To be- gin, an al-partition is applied t o A n , k , and so a

I<:-’ results. Since

IF,I

= k ( n - k ) - 2 , we have

IDal

I

5

k ( n

-

k )

-

2. Let S k - 1 denote the set of those

( k - 1)-edges in A n , k t h a t each contain n - 4 or more faulty edges. We have ISk-11

5

n

-

3 for otherwise which is a contradiction. I t is assured by Lemma 2.4 t h a t I<$-’ with the set Sk-1 of

( k

- 1)-edges removed is hamiltonian. T h a t is, an R k - 1 can be obtained whose (IC - 1)-edges each contain a t most n

-

5 faulty edges. Since each ( k - 1)-edge of the R k - 1 comprises

n

-

2 ( k - 2)-edges, the R k - 1 is healthy.

Now we assume a healthy R k - j + l =

[Ao, A I , ..., A , ( n _ l ) ( n _ 2 ) . . . ( n - j + 2 ) - 1 ] is obtained after

applying a ( a l , a 2 ,

...,

aj-l)-partition t o & , k , where 2

5

j

5

k - 1. Then an aj-partition is applied to the R k - j + l , and so each Ai (0

5

i

5

n ( n - l ) ( n - 2 ) . . . ( n - j

+

2)

-

1) forms a We note t h a t l D a 3 1

I

1

WJ.

3 Let S k - j denote the set of those

( k

-

j)-edges in all Ai’s t h a t each contain n

-

j - 3 or more faulty edges. We have 1Sk-j

I

5

n

-

j - 3 for oth- erwise

IDai]

2

(n-j-3)lSk-j

I

>

(n-j-3)(n-j-3)

>

[vi,

which is a contradiction. Then Lemma 2.6 assures t h a t each resulting with the set S k - j

of ( k - j)-edges removed is hamiltonian-connected. Since the R k - j + l is healthy, there are a t least three fault-free

( k

-

j)-edges between each pair of Ai and A ( i + l ) m o d l l ( n - l ) ( n - 2 ) . . . ( n - j + 2 ) . AS a result, we can de-

termine the entry and exit ( k - j)-vertices, say

Xi

and yi, respectively, for each Ai so t h a t

Xi

# Y ,

and

I&! 2,

( n - 4)lSk+

>

( n 4)(n -

3)

>

k ( n -

k),

the ( k - j)-edges (Y(i-l)modn(n-l)(n-2)...(n-j+~)

,

Xi)

and

(E ,

X ( i + ~ ) m o d n ( n - l ) ( n - a ) . ( n - j + 2 ) ) are not faulty. An R k - j can be generated thus for An,k whose each k

-

j)-edge contains a t most n

-

j - 4 faulty edges

i

and therefore a t most n - j - 4 faulty ( k

-

j - 1)- edges). Since each ( k - j)-edge of the R k - j compries n

-

j - 1 ( k

-

j - 1)-edges, the R k - j is healthy.

According t o the discussion above, a healthy RI can be obtained after applying a ( a l , a2,

...,

a k - 1 -partition

t o An,k. Then an ak-partiton is applied t o t

h

e

RI,

and its each l-vertex forms a KE-k+l. Since there are to- tally

ID,,

1

<

n - k - 3 faulty edges, Lemma 2.6 assures t h a t each resulting K:-k+l with faulty edges removed is hamiltonian-connceted. Since RI is healthy, a fault- free

Ro

can be generatedfor An,k with the same argu-

ments as above. Q.E.D.

W i t h similar arguments t o prove the above lemmas, we have the following results.

Lemma 3.8 Suppose IF,] = k ( n

-

k ) - 2 , and let

s

=

a l a 2

...

a k be

a

permutation of 1 , 2 ,

...,

k s o that ID,,[

_>

IDa2!

2

. . .

2

ID,,

1.

When k

_>

3 , there is a

fault-free h.amiltonian cycle f o r A n 3 k if n

2

and

I

=

I

= P a ,

I

= n - k - 1.

Lemma 3.9 Suppose

IFe/ =

k ( n -

k)

- 2 , and let

s

= a l a 2 ... a k be a permutatzon of 1 , 2 ,

_..,

k

so thnt IDal[

2

IDa2[

2

. . .

2

lDak1. When

k

2

3, there zs

a fault-free hamaltonzan cycle for A n , k zf n

2

%j@

and ezther lD,k-l

I

= ID,,

1

= n - k - 2.

I

=

p,k-ll

= n - k - 1 and

lDak1

= n - k - 2 .

Theorem 3.10 When k = 2 and n - k

2

4,

or

IC

2

3 and n

2

(or n - k

2

4

+

[$I),

we have H e ( A n , k )

=

k ( n - k ) - 2 , whzch zs the best.

Proof: Since the degree of An,k is k ( n -

k),

H e ( A n , k )

5

k ( n -

k)

- 2. With the aid of Lem- mas 3.5, 3.6, 3.7, 3.8, 3.9, it can be shown t h a t

&(A,&)

2

q n - k ) - 2. Q.E.D.

T h e following results can be obtained similarly.

Theorem 3.11 When

IC

2

2 and n

2

k

2

2

+

IS]),

we have H e ( A n , k ) = k ( n - k - 3) - 1.

Theorem 3.12 When k

2

2 and n -

k

2

3,

we have (or n -

He(&&)

L

k .

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