Hamiltonian-laceability of star graphs

Hamiltonian-Laceability

of Star Graphs

Sun-yuan Hsieh

Gen-Huey Chen

Dept.

of

Computer Science & Info. Engg.

National Taiwan University, Taiwan

Dept.

of

Computer Science & Info. Engg.

National Taiwan University, Taiwan

e-mail:

[email protected]~

e-mail: [email protected]

Chin-Wen Ho

Dept.

of

Computer Science & Info. Engg.

National Central University, Taiwan

e-mail: [email protected]

Abstract

Suppose G is a bipartite graph with two partite sets

of equal size. G is said t o be strongly hamiltonian- laceable i f there is a hamiltonian path between ev- ery two vertices that belong to diflerent partite sets, and there is a path of (maximal) length N - 2 be-

tween every two vertices that belong t o the same par-

tite set, where N is the order of G. The star graph is known t o be bipartite. I n this paper, we show that the n-dimensional star graph, where n

2

4 is strongly

hamilt orxian-laceable.

1

Introduction

Usually when the hamiltonicity of a graph G is con- cerned, it is investigated whether

G

is hamiltonian or hamiltonian-connected. A cycle (path) in G is called

a hamiltonian cycle (path) if it contains every ver- tex of G exactly once. G is said to be hamiltonian if it contains a hamiltonian cycle, and hamiltonian- connected if there exists a hamiltonian path between every two vertices of G. Since a bipartite graph is not hamiltonian-connected, Wong [5] has introduced the concept of hamiltonian-laceability for the class of bipart.ite graphs. A bipartite graph G = ( V l , V 2 , E )

with lVll = IV21 is ha,miltonian-laceable if there is a hamiltonian path between every vertex of VI and every vertex of Vz, where VI and V2 are the two partite sets of G. We note that any path between two vertices of

the same partite set has length at most IVll+/V21-2. It is meaningful to extend the concept of hamiltonian-laceability so that the lengths of the paths between two vertices of the same partite set are specified and the edge faults are considered. We say that a hamiltonian-laceable graph G = ( V l , V2, E ) is strongly if G additionally owns the property that there is a path of length 1Vl)+IV21-2 between every two vertices of the same partite set. Further, G is le edge fault-tolerant strongly hamiltonian-laceable if it remains strongly hamiltonian-laceable after removing at most L edges. In other words, there is a longest path between every two vertices of a le edge fault-tolerant

strongly hamiltonian-laceable graph G, even if at most

L edges of G are removed. The longest path has length

lVll+ 1V21 - 1 if the two vertices belong t o different

partite sets, and 1Vl1

+

1V21

-

2 if the two vertices

belong to the same partite set.

The star graph [ l ] , which belongs t o the class of Cayley graphs, has been recognized, as an attractive alternative to the hypercube. It possesses many nice topological properties, e.g., recursiveness, vertex and edge symmetry, maximal fault tolerance, sublo arith- mic degree and diameter, and strong resilience f l ]

[a],

which are desirable when we are building an intercon- nection topology for parallel and didributed systems. In [3], Jwo, Lakshmivarahan, and Dhall have shown

that the star graph is bipartite. Besides, its two par- tite sets have equal size. In this paper we show that the n-dimensional star graph is strongly hamiltonian- laceable when n

2

4.

2

Prelimiaries

The n-dimensional star graph, denoted by

S,,

is defined as follows.

Definition 1 The vertex set of

S,

is denoted b y

{alaz...anl a l a 2 ... a , is a permutation of { I , 2 ,

...,

n } } .

Vertex adjacency is defined as follows: ala2

...

a , is

adjacent to aia2

...

a i - l a l a i + l

...

a , for all 2

5

i

5

n . T h e vertices of

S, are n! permutations

of { 1 , 2 , ..., n},,

and there is an edge between two vertices of

S,

if

and only if they can be obtained from each other

b y swapping the leftmost number with one of the other n - 1 numbers. For convenience we refer to the position of ai in a l a 2 ... a , as the ith dimen-

sion, and ( a l a 2

...

a,,aiaz ... a i - l a l a i t l

...

a,) as the ith- dimensional edge.

Definition 2 There are embedded S,’s contained in

S,, where 1

5

r

5

n . A n embedded

S,.

can be con- veniently represented b y

<

s1s2 ... s,

>,.,

where s1 =

*,

si E {*, 1 , 2 ,

...,

n} f o r all 2

5

i 5 n , and exactly r of

SI, sa,

...,

s, are

*

(* denotes a ”don’t care” symbol).

Definition 3 A n i-partition on

<

s1sz ... s,

>,.

parti- tions

<

s1s2 ... s,

>,.

into r embedded S:-ls, denoted b y

<

S1sz...si-1qsi+l...sn >,.-l, where 2

5

i

5

n ,

Definition 4 A n

partition on

<

s1s2

...

s,

>,

performs an il-partition,

an iz-partition,

.

..,

an, ir,-partition, sequentially, on

<

s1s2

...

sn

>,.,

where 2 1 2 2 ... ,2 is a permutation of m

elements from { 2 , 3 ,

...,

n ) .

Definition 5 T w o embedded S,.'s

<

s1sz ... s,

>,.

and

<

tltz

...

t n

>,

are said t o be adjacent if s j

#

*,

t j

#

*,

and sj

#

t j for some 2

5

j

5

n, a n d si = ti f o r all

1

5

i

5

n and i

#

j . Moreover, the position j is denoted b y d i f ( < s1sz

...

s,

>,, <

t l t 2 ... t n >,.).

Definition 6 Let A I , A z ,

...,

An(n-l)(n-2)...(,.tl) represent those embedded Sr's that are obtained b y executing an ( i l , i z ,

...,

i,-,)-partition on

S,,

where

1

5

r

5

n - 1. They form an r-path, denoted b y P,. = [A1

,

Az,

... ,

An(,-1)(,-2)...(,+l)], if Ai is adjacent

t o Ai+l f o r all 1

5

i

5

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

. .

( r

+

1) - 1.

Each vertex of P,. i.e., Ai, is called an r-vertex, and each edge of P,., i.e., ( A i , Aa+l), is called an r-edge.

Definition 7 A n i- par t i t ion on Pr = [Ai, Az,

...,

A n ( n - l ) ( , - 2 ) . . . ( T t l ) ] performs an

i-partition on A I , A2, ..., A,(,-l)(,-2).,.(T+1), respec-

tively, where 2

5

i

5

n and r

2

2. After an i-

partition, each Aj is partitioned into r ( r - 1)-vertices, where 1

5

j

5 n ( n

- l ) ( n - 2 ) .

. .

( r

+

1). Since every

two of the r (r - 1)-vertices are joined with an ( r - 1)-

edge, each Aj can be viewed as a complete graph of r

( r - 1)-vertices. Throughout this paper, we refer to the complete graph as I<:-'. We note that each vertex of

( r - 1)-edge.

( i l l i 2 ,

...,

im)-

I<'-1

,.

is an ( r - 1)-vertex and each edge of I<:-' is an

3

Hamiltonian-Laceability

of

Star

In this section we show

S,,

with n

2

4 is strongly hamiltonian-laceable.

Lemma 3.1 Suppose U =

<

u1uz ... U ,

>,,

V =

<

211212

...

v n

>,.,

and W =

<

w1wz

...

w,

>,

are arbi-

trary three consecutive r-vertices in a P,., where r

2

2 .

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

#

w q , then after executing a partition on the P,. each ( r - 1)-

vertex of V is connected to U or W .

Proof: Without loss of generality, we assume that a j-partition is executed on the P,, where 2

5

j

5

n.

Hence, uj = v j = wj = *. Since p = d i f ( U , V )

#

1 and q = dif(V, W )

#

1 , we have U

#

u p , v q

#

w q , ui = U ; for all 1

5

i

5

n and i p p , and vi = wi

for all 1

5

i

5

n and i

#

q. Suppose conversely

U

#

w q and there exists an ( r - 1)-vertex, say

J:

=

<

211212 ... ~ j - i ~ U j + i ... U , > r - l J of V which is not

connected to either of U and W . Thus, z = u p , for

otherwise VI is adjacent to some ( r - 1)-vertex of U . Similarly, z = wq. This implies up = w q , which con- tradicts our assumption. Q.E.D. Lemma 3.2 Suppose U and v are arbitrary two dis-

tinct vertices of S, with n

2

4. There exists a Pn-l whose first ( n

-

1)-vertex contains U and whose last

( n - 1)-vertex contains U .

Graphs

Proof: Suppose U == u1u2 ... u, and U = v1v2 ... vn.

Without loss of generality, we assume u,j # vj for some 2

5

j

5

n. After a j-partlition, S, is partitioned into n ( n

-

1)-vertices, which form a I<:-'. Clea.rly, U and U belong to two different vertices, say U and V , of

the I<,"-1. The desired Pn-l can be constructed as a

hamiltonian path from U to V in the I<:-'. Q.E.D. In the rest of this paper, we suppose U and v are the beginning vertex and the ending vertex, respectively, of a path. We call an r-vertex the beginning r-vertex (ending r-irertex) if it, contains U (U). Besides, a path from U to V is abbreviated t o a U - V path.

Lemma 3.3 A PT-l whose first ( r

-

1)-vertex is the beginning ( r - l)-veriex and whose last (r - 1)-vertex is the ending ( r - l)-vertex can be obtained from a P,. whose first r-vertiex is the beginning r-vertex and whose last r-vertex is the ending r-vertex, where 4

5

r l n - 1 a n d n 2 5 .

-

Proof: Suppose PT

-

[Ai, Az,

...,

A n ( n - l ) ( n ~ - ~ ) . . . ( ~ t l ) ] , where A1 is the begin-

ning r-vertex and An,~,-~)(,-2)...(,.t1) is the ending r-

vertex. After executing a partition on the P,., each Ai forms a I<:-', where 1 5 i

5 n ( n - l ) ( n - 2 )

Since each Ai contains a t least three ( r - 1)-vertices, we can select two dist,inct ( r

-

1)-vertices, say X i and

x,

from each Ai so tlhat X 1 is the beginning ( r - 1)- vertex, Y,,(,-1)(,-2)...(,.+1) is the ending (r- 1)-vertex, and for 2

5

j

5

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

+

1) - 1,

X j and Yj are adjacent to Yj-1 and X , + l , respec-

tively. Since there exists a hamiltonian

ki

-

path in the I<;-' formed by Ai, the desired P,.-1 can be obtained by concatenating all the hamiltonian paths interleaved with ( r

-

1)-edges (YI, X z ) , ( Y 2 , X 3 j 1

...,

In the rest of this; paper, X i and yi as specified above are referred to as the entry ( r

-

1)-vertex and the exit ( r - 1)-vertex of A i , respectively.

Lemma 3.4 A Ps whose first 5-vertex is the begin-

ning &vertex and whose last 5-vertex is the ending 5-vertex can be obtained in S, with n

>

5.

(Y,(,-l)(,-2)...(r+l)-11 X n ( n - l ) ( n - z ) , , , ( r t l ) ) . Q.E.D.

A Pr = [ A I ~ A z , . . . , A n ( n - l ) ( n - 2 ) . . . ( r + l ) ] in S n , where 2

5

r

5

n - 1, is said to be good if it satis- fies the following three conditions.

(Cond. 1 ) A1 and A;,(,-l)(,-2)...(,+1) are the begin- ning and ending r-vertices, respectively.

(Cond. 2) For arbitrary three consecutive r-vertices

X

<

... 2,

>,,

Y =

<

YIYz...Yn

>,.,

and 2 =

<

z1z2 ... z ,

>,

in the P,., x d i f ( ~ , y )

#

z d i f ( y , ~ ) holds. 3) After executing a &partition on the P,. or some 2

5

IC

5

n , the beginning (ending) ( r - 1)-

vertex in A1 (An(,-l)(,-2),..(,.+l)) is not connected to In the rest of this section we show that a good P3

can be obtained in S,,

.

Given arbitrary two vertices of S,, a longest path between them can be constructed from a good P3.

ond.

r"

Lemma 3.5 A good P4 can be obtaaned from a Ps whose first $-vertex as the begannang 5-vertex and whose last 5 - v e r t e x as the endanq 5-vertex.

Proof: We suppose P5 = [ A I , Az, ..., An(,-,1)(,-2)..,6I, where A1 and A,(,-1)(~-2)...6 are the beginning and

ending 5-vertices. Without loss of generality, we as- sume that the P5 is obtained from

s,

by executing

an ( u l , u2, ...) a,_s)-partition, where ala2

...

an-5 is an

arrangement out of {2,3,

...,

n } . Let j E { 2 , 3 ,

...,

n } -

{ a l , a 2 , ..., a,-5}. First, a j-partition is executed on the Ps, and so each Ai forms a IC:, where 1

5

i

5

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

.

. 6 . In the rest of the proof we con-

struct a good P4 from the P5 by establishing a hamil- tonian path for each IC:.

Suppose U = uluz

...

un and v = v1vz

...

vu, are the be- ginning and ending vertices, respectively. A hamilto- nian path for the IC: formed by A I can be established

as follows. Let XI =

<

2122

...

2, > 4 be the begin-

ning 4-vertex (in A I ) , T be the 4-vertex of AI that

is not connected to Az, and W =

<

W ~ W Z ... w, >4 be

a 4-vert,ex of AI which is different from X I and has wj = uk for some k E { 2 , 3 ,

...,

n } - { j , a l , a 2 , ...) an-5}. Since there are four 4-edges between A1 and Az, there exists a 4-vertex Y1

$

{XI,

W } which is connected

to Az. If X 1 = T or ( X I

#

T and T = W ) ,

a hamiltonian XI - U1 path can be established as

( X I , W )

+

P[W, Y I ] , where P[W,

YI]

denotes a TV - YI path passing all the vertices of the IC: but XI exactly

once. Otherwise, if XI

#

T and T

#

W , a hamilto- nian X I - 1'1 pat8h can be established as ( X I , W )

+

(W, T )

+

P[T,

Y l ] , where

P[T,

Yl] denotes a T - Yl path passing all the vertices of the IS: but X1 and W

exactly once. Then we continue to establish a hamil- tonian path for the IC; formed by A n ( n - ~ ) ( ? - ~ ) ..6.

The construction of the hamiltonian path is simi- lar to the situation of A l . Let Y n ( n - l ) ( n - 2 ) . . . 6 be

the ending 4-vertex (in An(n-l)(n-2)...6, C be the 4-vert3ex of An(,-1)(,-2)...6 that is not connected

to A n ( n - l ) ( n - ~ ) . . . 6 - ~ , and D =

<

dldz ... d, >4 be the 4-vertex of A , ( , - I ) ( ~ - ~ ) . . 6 that is different from

Yn(n-l)(n-z)...6 _{and has} dj = V k (here, k is identical

with that k appearing in the situation of A I ) . There

exists a vertex Xn(,-1)(,-2)...6 @ { D , Yn(,-l)(,-2)...6} which is connected to A,(,-1)(,-2)...6-1. If

Y,,(,-1)(,-2)...6 = C or Y,(,-1)(,-2)...6

#

C and C =

D ) , a hamiltonian Xn(n-l)(,-2)...6 - Yn(n-l)(n-2)...6 path can be established as P [ X n ( n - l ) ( n - z ) ... 6 , D]

+

( D , Y n ( n - l ) ( n - 2 ) - - - 6 ) , where P[dYn(n-l)(n-2)-..6, D] de-

notes an Xn(,-1)(,-z)...6 - D path passing all the vertices of the 11; but Y , ( , - I ) ( ~ - ~ ) 6 exactly once.

Otherwise, if Ya(,-l)(,-2)...6

#

c

and

c

#

D , a

hamiltonian X n ( n , - l ) ( n - 2 ) . . . 6 - Yn(,-1)(,-2)...6 path can be established as P [ X n ( n - 1 ) ( n - 2 ) . . . 6 , C] + ( C , D ) + ( D , yn( n - I ) ( n - 2 ) . . -6 )

,

where P [ X n ( n - 1 )( n - 2 ) . . . 6 ,

c]

de-

notes an & ( n - . l ) ( n - z ) . . . 6 - C path passing all the ver-

tices of the Itr: but D and Yn(n-l)(n-2)...6 exactly once.

In the discussion above, X 1 and y1 (X,(,-I)(,-Z)...~ and Yn(n--l)(n-2)...6 are the entry and exit 4-vertices

of A1 ( A n ( n - l ) ( n - ~ ) . . . 6 ) , respectively. Additionally, we use X i and yi to denote the entry and exit 4-vertices

o f A i , respectively, for 2

5

i

5 n ( n - l ) ( n - 2 ) . . . 6 - 1

.

Let

Li

( Q i ) be the 4-vertex of Ai that is not connected

to Ai-1 ( A i + l ) . A hamiltonian

Xi

- path in the

IC:

formed by Ai can be established according t o the following four cases. Case 1. Qi = Xi and

Li

=

E.

A hamiltonian

Xi

- Y , path can be established

easily. Case 2. Qi

#

Xi and Li =

X .

A hamiltonian

X i -yi path can be established as ( X i , Qi)+P[Qi, y i ] , where P [ Q i , y i ] denotes a Qi - yi path passing all the vertices of the I<: but X i exactly once. Ca.se 3.

Qi = X i and

Li

# Yi. A hamiltonian X i

-x

path can

be established as P[Xi, Li]

+

( L i , x ) , where P [ X ; , Li]

denotes an X i - Li path passing all the vertices of

the I<: but exactly once. Case 4. Qi = X i and

Li

#

y i . If Qi =

Li,

a hamiltonian Xi - yi path can be established a,s ( X i , U i , L i ,

K,x),

where Vi and

V;:

are the other two 4-vertices of Ai than X i , Yi, and

Li.

If Qi

#

L i , a hamiltonian Xi

-

yi path can be established as ( X i , Qi, U ; , L ; ,

x),

where U; is the other 4-vertex of Ai than X i , Y , ,

Li,

and Q i .

Clearly the hamiltonian paths obtained above interleaved with 4-edges (Y1, X Z ) , ( Y z , X s ) ,

...,

we show the P4 good. (Cond. 1) holds because X 1 is the beginning 4-vertex and Y n ( n - l ) ( n - ~ ) . . . 6 is

t,he ending 4-vertex. (Cond. 3 ) holds for the rea-

son as follows. Recall that wj = U L for some

IC

E { 2 , 3 ,

...,

n } - { j , a l , a2, ..., an-5}. After executing a

k-partition on the P4, X1 =

<

~ 1 x 2 ... x , > 4 forms a

IC:. Since xk = U S = wj, the beginning 3-vertex is not connected to W . Similarly, the ending 3-

vertex is not connected to D . In the following, we

show (Cond. 2 ) holds. Let X =

<

~ 1 x 2 ...x, > 4 ,

Y

=

<

y1y2 ...y, >4,. and 2 =

<

z1z2 ... z, > 4 be arbi- trary three consecutive 4-vertices in the P4. Assuming

p = d i f ( X , U ) and q = d i f ( Y ,

Z),

we show xp

#

zq

according to three cases. If X is the exit 4-vertex of Ai for some 1

5

i

5

n ( n - 1)(n

-

2 ) . . . 6 - 1, then

Y is the entry 4-vertex of Ai+l and 2 is the second

4-vertex in the hamiltonian path established for the

11: formed by Ai+l. Besides, p

#

j = q . Suppose conversely x p = z q . Then, 2 is not connected t o A;

(recall that the pair of 4-vertices in Ai and Ai+l that

are not adjacent are

<

x l . . . x q - l z p x q + l . . . x , >4 and

<

z1

...

z q - l x p z q + l

...

zn >4, respectively, where x q = z p

#

x p 1 zq and xi = z; for a.11 1

5

i

5

n and

i

#

{ p , q } ) . According to our construction for the P4,

2 should be the third or fourth or fifth 4-vertex in the hamiltonian path established for the I<: formed

by Ai+l, which is a contradiction. If 2 is the entry 4-vertex of Ai for sorpe 1

<

i

5

n(n, - l ) ( n - 2 ) " ' 6 ,

then II:

#

zq can be shown similarly. Otherwise, if X ,

e,

and 2 belong to the same 4-vertex, then p = d i f ( X , Y = d i f ( X , 2) = d i f ( Y , 2) = q . Since

,Y

Q.E.D. completes the proof.

( y ~ ( n - l ) ( n - Z ) - . . 6 - l , X n ( n - l ) ( n - Z ) . . 6 ) form a p4. Next,

As with similar arguments to prove the above, we can show the following lemmas. Due to space limita- tion, the details are omitted.

Lemma 3.6 A good P3 can be obtained from a good

p4.

Proof: We suppose P4 = [ A I , A2,

...,

An(,-1)(,-z)...5].

Without loss of generality, we assume that the P4 is

obtained from

S,

by executing an ( u 1 , az, ... ,a n - & partition, where a l a 2

...

an-4 is an arrangement out of { 2 , 3 ,

...,

n } . Since the P4 is good, there exists

j E {2,3,

...,

n }

.-.

{ U I , az,

...,

a n - 4 } so that after ex-

ecuting a j-partition on the P4, the beginning (end-

ing) 3-vertex in AI ( A n ( n - 1 ) ( n - 2 ) . . . 5 ) is not adjacent to

A2 (A,(,-l)(,-2)...5-l). Besides, each Ai forms a IC:,

where 1

5

i

5

n ( n - l ) ( n - 2). . ' 5 . In the rest of the proof, we construct a good P3 from the P4 by estab-

lishing a hamiltonian path for each I<:. Suppose U =

u1uz ... U,, and v = v1v2

...

v,

are the beginning and end-

ing vertices, respectively. We establish a hamiltonian path for the I<: formed by A1 as follows. Let X 1 be the beginning 3-vertex (in A I ) and W =

<

w1wz ...

w,

>3 be a 3-vertex in A1 which is different from X 1 and has

wj = U k for some

k

E { 2 , 3 ,

...,

n } - { j , a l , U Z ,

...,

an-4}.

We note that X 1 is not connected to Az. Since there are three 2-edges between A1 and A2, there is another

3-vertex Y1

4

{ X I , W } in A I which is connected to

A2. A hamiltonian XI - Y1 path can be established as

( X I

,

W )

+

P [ W , Y I ] , where P [ W , Yl] denotes a W - Y1

path passing all the vertices of the 11': but X1 exactly

once.

Then we continue t o establish a hamiltonian path

for

the I<: formed by An(n--l)(n-~)...5. Let Y,(,,-l)(,,-z). .5

be the ending 3-vertex (in A , ( , - 1 ) ( ~ - 2 ) . . . 5 ) and D =

<

d l d 2

...

d , >3 be the 3-vertex in An(,,-1)(,,-2)...5

that is different from Y n ( , T 1 ) ( n - 2 ) . . . 5 and has d j =

v k (here,

k

is identical with that

k

appearing in

the situation of A I ) . There exists a 3-vertex

X,,(n-1)(n-2)...5

#

D in A,(,,-1)(~-2)...5 which is connected to A n ( n - I ) ( n - - 2 ) . . . 5 - 1 . A hamiltonian

X n ( n - l ) ( n - 2 ) . . . 5 - Y n ( n - ~ ) ( n - 2 ) . . . 5 path can be estab- lished as P [ X n ( n - l ) ( n - 2 ) . . . 5 , D1+ ( D , Y n ( n - l ) ( n - 2 ) . . . 5 ) ,

where P[X71(n-l)(?l-2)~. 51

Dl

denotes an X n ( n - l ) ( n - 2 ) . . . 5 - D path passing all the

vertices of the I<: but Yn(,,-1)(,-2)...5 exactly once. In

the discussion above, X I and YI (Xn.(,,-1)(,,-2)...5 and

Y n ( n - l ) ( n - 2 ) . , . 5 ) are the entry and exit 3-vertices of A1

( A n ( n - l ) ( n - 2 ) . . . 5 ) r respectively. By X i and Y , we de-

note the entry and exit 3-vertices of Ai, respectively,

for 2

5

i

5

n ( n - l ) ( n - 2 ) . . . 5 - 1 . Let Li ( Q i ) be the 3-vertex in Ai that is not connected to Ai-1 ( A i t l ) . A hamiltonian X i - yi path for the formed by Ai can

be established according to the following four cases. Case 1. Q i = X; and L ; = Y;. A hamiltonian X; -

x

path can be established easily.

Case 2. A hamiltonian

Xi-Yi path can be established as ( X i , Q i ) + P [ Q i , Y , ] ,

Qi

#

Xi

and Li = E:.

where P[Qi,

x ]

denotes a Q; -

Y;

path passing all the vertices of the I<: but X i exactly once.

Case 3. Qi = X i and Li

#

x.

A hamiltonian Xi - path can be established as P [ X i

,

Li]

+

(Li

,

x),

where P [ X i , L i ] denotes an X i

-

Li path passing all

the vertices of the

IC:

but

E

exactly once.

Case 4. Qi

#

X i and Li

#

yi. Since the P4 is good, Lemma 3.1 assures that each %vertex of

Ai is connected to A i - 1 or A i t l . Hence, Qi

#

Li. A hamiltonian X.; -

Y;

path can be established

as ( X i , Q i , L i ,

Y;

.

The hamiltonian paths obtained (Yn(,,-1)(,,:2)...5-l, X n , ; n - l ) ( n - 2 ) . . . 5 ) form a P3. More-

over, the P3 is good, with the same arguments as the

proof of Lemma 3.5. Q.E.D. Lemma 3.7 There is a good P3 in S5.

Proof: Suppose U = C ~ ~ U Z U ~ U ~ U ~ and v = ~ 1 ~ 2 ~ 3 ~ 4 ~ 5 are the beginning and ending vertices, respectively.

We a,ssume U ;

#

vi for i E { a 1 , a 2 , .. . , a k }

{ 1 , 2 , 3 , 4 , 5 } and ui = vi otherwise, where 2

5

k

5

5

and a1

< a2

< . . .

<:

U k . First, an ak-partition is

executed on 5'5, and so a I<: results. We use U4 and

V4 to denote the beginning and ending 4-vertices, re-

spectively. In the following, we construct a good P3

according to the values of 6.

Case 1.

k

= 2. We assume a1

#

1. The

discussion for al' = I. is very similar. For ease of

explanation, we assume, without loss of generality,

a1 = 2 and a2 = 3. VVe then arbitrarily select 1 = 4 from the set {2,3,4,51}

-

{ a 1 , a 2 } = {4,5}, and let S =

<

S ~ S Z S ~ S ~ S ~ >4

=<

* *

s3 t t >4 be the vertex of the I<: with ( s a z -=)ss = u4(= U,). Since there are five vertices in the I<:, we can find a 4-vertex with (za2 = ) z 3

#

V I . ]Let T be the other vertex than

U 4 , S,

2,

and V4 in the I<;. A hamiltonian path for the

Kt

can be establiished as ( U 4 , S, T , 2, V4), which constitutes a P4 = [U',, SIT, 2, V4]. An 1-partition is then executed on the P 4 , and so each 4-vertex of the P4 forms a I<:. By establishing a hamiltonian path for each I<:, a good P:3 can be obtained as follows.

First we establish a hamiltonian path for the

K:

formed by

&.

Let V3 =

<

* *

v3v4* > 3 be the end- ing 3-vertex (in V4) and D =

<

d l d 2 d 3 d 4 d 5 >3=

<

*

*

v3d4* >3 be the 3-vertex of V4 that is not con-

nected to 2 . Since sa,2 = U( = 01 =)v4

#

z3(= fa,),

V3 is connected to 2 . So, D

#

V3. Moreover, since

there are three 3-edges between

2

and V4, there ex-

ists a 3-vertex X # V3 in V4 which is connected to 2 .

A hamiltonian path for the I<: can be established as P [ X , D]

+

( D , V3), where P [ X , D] denotes an X

-

D path passing all the vertices of the I<: but V3 exactly

once.

We then continue to establish a hamiltonian path for the I<: formed by U4. We have dl = U,. for some T E { 2 , 3 , 4 , 5 }

-

{ a z l l } = { 2 , 5 } . We note

T

#

1 because D is .the 3-vertex in V4 that is not connected to 2 (which implies dl = z,,

#

w1). Let

U3 =

<

t

*

u3'114* > 3 be the beginning 3-vertex (in

above interleave

d

with. 3-edges ( Y I , X z ) , (Y2, X 3 ) ,

...,

U4) and W =

<

w1w2w3w4w5 > 3 = <

* *

u3w4* > 3 be

the 3-vertex in U4 that is different from U3 and has

(wf =)w4 = U,. We note that U3 is not connected to

S because ( s a z =)s3 = u4(= u f ) . So, there exists an- other %vertex Y @ {Us, W } in U4 which is connected

to

S.

A hamiltonian path for the 21. can be estab- lished as (Us, W )

+

P [ W , Y ] , where P[W, Y ] denotes a

W - Y path passing all the vertices of the K: but U3

exactly once.

Since there are three 3-edges between every two ad- jacent 4-vertices of the P 4 , distinct entry and exit 3-

vertices can be determined for S, T, and 2. Then, a hamiltonian path from the entry 3-vertex to the exit 3-

vertex can be established for each formed by them, similar to the proof of Lemma 3.6, in order to satisfy (Cond. 2). The obtained hamiltonian paths inter- leaved with used 3-edges form a P3 = [AI, A2,

...,

Am],

where

AI

= U 3 , A2 = W , A19 = D , and A20 = Vi. In

the following we show that the P3 is good.

1) holds, and with the same ar- guments as the proof of Lemma 3.5, (Cond 2) also holds. After executing an r-partition on the P3, each

A, forms a K z , where 1

5

i

5

20. Without loss of gen- erality, we assume r = 2. Let U2 =

<

* U Z U ~ U ~ * > 2

(in AI) and

%

=

<

*v2v3?4*

> z

(in Azo) be the

beginning and ending 2-vertices1 respectively. Since

(U, =)u2 = w4(= uII = W d , f ( A 1 , A z ) ) , U2 is not con-

nected to W = A2. Similarly, since (U,. =)v2 = d4(=

dl = d d z f ( A 1 9 , ~ z o ) ) , V2 is not connected to D = A19.

Thus, (Cond. 3) holds.

Case 2. IC = 3. The method for constructing a good

P3 is almost the same as Case 1, but

k

is changed t o 3

and 1 is selected from the set { 2 , 3 , 4 , 5 ) - { a l l a2, a3).

Case 3.

IC

= 4. We assume u f = w f , where

If U t

#

U t , U t

#

v u 4 , and ut

#

U,, for some t E { a i , a 2 , a 3 } - {l}, then two 4-vertices Q =

<

qlq2q3q4q5 > 4 and H =

<

h l h ~ h 3 h 4 h 5 > 4 with qar = ut and ha, = ut are determined.

A

hamiltonian path for the K: can be established as (U4, Q , T, H , V4), where T is the other

4-vertex than U4, Q, H, and V4. The hamiltonianpath forms a good P4 = [U4, Q,T,

H ,

V4] for the following

reasons. (Cond. 1) and (Cond 2) hold with the same reasons as Case 1. (Cond. 3) holds as a consequence of executing a t-partition on the P4. By Lemma 3.6, a good P3 can be obtained from the P4.

Otherwise, if there exists no t E {all a2, u3) - (1) satisfying ut

#

u t , ut $3 v a 4 , and ut

#

t i a r t then a l . = 1, which implies 1

#

1. The method for constructing a good P3 is almost the same as Case 1, but IC is changed

to 4 and 1 is unique.

Case 4. IC = 5. There exists a number t E

{ai, az, a3, a4) - { 1) satisfying ut

#

u t , ut

#

va,, and vt

#

u a 5 . A good P3 can be obtained similar to Case

3 . Q.E.D.

We note that S3 forms a cycle of length six. The following two lemmas have been shown in [4]. Lemma 3.8

[4]

Suppose X and Y are t w o adjacent 3-vertzces zn a P3, and let ( c o , c ~ ,

...,

c5) denote the cycle f o r m e d by X . T h e n , t h e vertaces of X t h a t are connected t o Y are c, and C ( ~ + 3 ) m o d 6 f o r s o m e 0 5 j

5

Clearly, (Cond.

E { 1 , 2 , 3 , 4 , 5 } - { a l r a 2 , a 3 , a 4 } .

5.

Lemma 3.9

[4]

Suppose X =

<

21x2

...

x, >3, Y =

<

y1y2 ...y, > 3 r and Z =

<

z1z2

...

z, > 3 are arbztrary

three consecutzve 3-vertaces zn a P3. I f x d i f ( x , y )

#

z d , f ( Y , Z ) ! t h e n t h e t w o vertzces of Y t h a t are connected t o X are dasjoznt f r o m t h e t w o of Y t h a t are connected

t o 2.

Lemma 3.10 Suppose U and v are arbztrary t w o dzs-

tznct vertzces of S, wzth n

2

4. A longest U

-

v p a t h can be constructed f r o m a good P3. T h e longest p a t h

h a s length n! - 1 zf d i s t ( u , v zs odd, and n! - 2 af

d i s t ( u , v ) as even, where d i s t

?

u , v ) zs t h e dzstance be- t w e e n U and U.

Proof:

It

is not difficult to check that this lemma holds for S4 (recall that

S,

is vertex symmetric).

Hence, we assume n

2

5. According to Lemmas ??

and 3.7, a good P3 = [Al,Aa,...,A,(,-l)(,-2) 41

can be obtained in S,. We use (cz,o, c Z , l ,

...,

?,5)

to denote the cycle formed by A,, where 1

5

z

5

n ( n - l ) ( n - 2) S. .4. According to Lemma 3.8, two

vertices c1,j and Cl,(j+3)mod6 (C,(,-I)(,-Z) 4 , k and

IC

5

5) are connected t o A2 (A,(,-l)(,-Z) 4-1).

We have U

#

{ ~ 1 , ~ , ~ ~ , ( , + 3 ~ ~ ~ ~ ~ } , for otherwise

the beginning 2-vertex must be connected t o Az,

which contradicts (Cond. 3). Similarly, v

#

A1 (An(,-1)(,-2) 4 ) forms a cycle of length 6, U (v)

is adjacent t o or C1,(3+3)mod6 (C,(,-I)(,-~) 4 , k or c,(,.-1)(,-2) 4 , ( ~ + 3 ) ~ ~ d 6 ) . Without loss of generality,

we assume U is adjacent to ~ 1 , ~ . We let 1c1

=

U and

y1 =

, and select

x, and y,

,

sequentially, for i =

2 , 3 , ...

,

n(n

-

l ) ( n

-

2) . . 4

-

1 from each A, so that x,

is adjacent t o both y,-1 and y,, and yn(,-1)(,-z) 4-1

is connected t o An(n-l)(n-2) 4. Lemmas 3.8 and 3.9

assure the existence of x, and yz. Since A1 contains a

hamiltonian U- y1 path and each A, contains a hamil-

tonian ~c,-y, path, a hamiltonian u-y,(,-1)(,-2) 4-1

path (of length n!

-

6) for

S,

- {A,(,-l)(,-Z) 4 ) thus results.

Next we augment the U - yn(,-1)(,-2) 4-1 path

with a longest yn(,-1)(,-2) 4-1 - v path. Without loss of generality, we assume y,(,-1)(,-2) 4-1 is adja-

cent to ~ , ( , - ~ ) ( , - 2 ) 4 , k . If d i s t ( u , v ) is odd, any U - v

path has odd length because

S,

is bipartite. So, v

#

for otherwise there exists a U

-

v path of even length,

which is a contradiction. Since we also have v

#

k ( n - l ) ( n - 2 ) 4,(k+3)mod6) for SOme 0

5

j

5

5 (0

5

l)(n-2) 4 , k 1 %(n- l ) ( n - 2) 4 , ( k+3)mod6}. Since

Icn(n-l)(n-2) 4 , ( k + 2 ) m o d 6 t %(n-l)(n-Z) 4,(k-2)mod6),

{%(?2-1)("-2) 4 , k , c?%(71-l)(n-2) 4,(k+3)mod6),

be '%(fl-l)(,-Z) 4 , ( k + l ) m o d 6

or c,(,-~)(,-z) In either case, there exists a hamiltonian cn(,-1)(,-2) 4 , k - v path (of length 5) for An(,-1)(,-2) 4. Similarly, if d i s t ( u , v ) cn(,-1)(,-z) 4,(k-2)mod6. In either case, there ex- is even, should be c,(,-I)(~-z) 4,(k+2)mod6 or

ists a c ~ ( ~ . - I ) ( ~ . - ~ ) 4,k - w path of length 4 in

An(n-l)(n-a) 4. This completes the proof. Q.E.D. The following theorem holds as an immediate con- sequence of Lemma 3.10.

Theorem 3.11 S, wath n

2

4 as strongly hamaltonaan-laceable.

4

Concluding remarks

In this paper we have introduced the concept of strongly hamiltonian-laceability for star graphs. By extanding our results, we can show that the n- dimensional star graph, where n 2 6, remains strongly hamiltonian-laceable, even if n - 4 random edge faults happen, and show that the n-dimensional star graph, where n

2

6, remains strongly hamiltonian-laceable, even if n - 3 random edge faults happen, exclusive of two exceptions in which there are at most two vertices missing from the longest paths.

References

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IEEE Transactaons on Computers, vol. 38, no. 4,

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