Hyper hamiltonian laceability on edge
fault star graph
Tseng-Kuei Li
a,*, Jimmy J.M. Tan
b, Lih-Hsing Hsu
ba
Department of Computer Science and Information Engineering, Ching Yun University, JungLi 320, Taiwan, ROC
b
Department of Computer and Information Science, National Chiao Tung University, Hsinchu 300, Taiwan, ROC
Received 30 December 2002; received in revised form 17 March 2003; accepted 17 September 2003
Abstract
The star graph possess many nice topological properties. Edge fault tolerance is an important issue for a network since the edges in the network may fail sometimes. In this paper, we show that the n-dimensional star graph is (n 3)-edge fault tolerant hamil-tonian laceable, (n 3)-edge fault tolerant strongly hamiltonian laceable, and (n 4)-edge fault tolerant hyper hamiltonian laceable. All these results are optimal in a sense described in this paper.
Ó 2003 Elsevier Inc. All rights reserved.
Keywords: Star graph; Hamiltonian laceable; Strongly hamiltonian laceable; Hyper hamiltonian laceable; Fault tolerant
1. Introduction
Network topology is a crucial factor for a network since it determines the performance of the network. For convenience of discussing their properties, networks are usually represented by graphs. In this paper, a network topology is represented by a simple undirected graph, which is loopless and without
*Corresponding author.
E-mail address:tealee@ms8.hinet.net(T.-K. Li).
0020-0255/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2003.09.023
Information Sciences 165 (2004) 59–71
multiple edges. For the graph definition and notation we follow [5]. G¼ ðV ; EÞ is a graph if V is a finite set and E is a subset offða; bÞja 6¼ b 2 V g, where (a; b) denotes an unordered pair. We call V the vertex set and E the edge set. a and b are adjacent if and only ifða; bÞ 2 E. A path is a sequence of adjacent vertices, denoted byhv0; v1; . . . ; vki, in which v0; v1; . . . ; vk are distinct except that
pos-sibly v0¼ vk. The length of the path is k. For ease of description, we may use P
or hv0; P ; vki to denote the path. A hamiltonian path of G is a path which
crosses all vertices of G. A graph G is hamiltonian connected if there exists a hamiltonian path joining any two vertices of G.
Hypercubes [12] and stars [1] are bipartite graphs. A graph G¼ ðV0[ V1; EÞ
is bipartite if V0\ V1¼ ; and E fða; bÞja 2 V0 and b2 V1g. Given vertices x
and y, we say that x and y are in the same partite set if x; y2 Vior in different
partite sets if x2 Vi and y2 V1i for i2 f0; 1g. However, the concept of
hamiltonian connectivity does not apply to bipartite graphs because bipartite graphs are definitely not hamiltonian connected except for a few exceptions such as K2 or K1. As such a property is important, Wong [19] introduced the
concept of hamiltonian laceability on bipartite graphs. A bipartite graph G¼ ðV0[ V1; EÞ is hamiltonian laceable if there is a hamiltonian path between
any two vertices x and y which are in different partite sets. It is trivial thatjV0j
must be equal tojV1j. On the condition of jV0j ¼ jV1j, Hsieh et al. [10] extended
this concept and proposed the concept of strongly hamiltonian laceability. G is strongly hamiltonian laceable if it is hamiltonian laceable and there is a path of lengthjV0j þ jV1j 2 between any two vertices in the same partite set. Lewinter
and Widulski [13] introduced another concept, hyper hamiltonian laceability. G is hyper hamiltonian laceable if it is hamiltonian laceable and for any vertex v2 Vi, there is a hamiltonian path of G v between any two vertices in V1i. So
hyper hamiltonian laceability is definitely also strongly hamiltonian laceabil-ity.
Fault tolerance is an important property of network performance. Hsieh, Chen, and Ho [9] proposed the edge fault-tolerant hamiltonicity to measure the performance of the hamiltonian property in the faulty networks. A graph G is k-edge-fault tolerant hamiltonian if G F remains hamiltonian for every F EðGÞ with jF j 6 k. Extending this concept, we introduce the following indicators. The edge fault tolerant hamiltonian laceability of the graph G is the integer value f such that for any F EðGÞ with jF j 6 f , G F is still hamil-tonian laceable and there exits F0 EðGÞ with jF0j ¼ f þ 1 such that G F0is
not hamiltonian laceable. We use eftHLðGÞ to denote this capacity. Similarly, we can define the edge fault tolerant strongly hamiltonian laceability of G, de-noted by eftSHLðGÞ, and the edge fault tolerant hyper hamiltonian laceability of G, denoted by eftHHLðGÞ. eftSHLðGÞ is the integer f such that for any F EðGÞ with jF j 6 f , G F is still strongly hamiltonian laceable and there exits F0 EðGÞ with jF0j ¼ f þ 1 such that G F0is not strongly hamiltonian
G F is still hyper hamiltonian laceable and there exits F0 EðGÞ with
jF0j ¼ f þ 1 such that G F0 is not hyper hamiltonian laceable. We say a
graph G is optimal with respect to eftHL (eftSHL, eftHHL, respectively) if for a fixed number of vertices, G contains the least number of edges among all graphs G0with eftHLðG0Þ¼ eftHLðGÞ (eftSHLðG0Þ¼ eftSHLðGÞ, eftHHLðG0Þ¼
eftHHLðGÞ).
This paper is to study these three indicators of the star graphs. The star graphs [2] are Cayley graphs. They have many nice properties such as recur-siveness, vertex and edge symmetry, maximal fault tolerance, sublogarithmic degree and diameter [2]. These properties are important for designing inter-connection topologies for parallel and distributed systems. Star graphs are able to embed cycles [19], grids [11], trees [3], and hypercubes [16]. Many efficient communication algorithms for shortest-path routing [17], multiple-path rout-ing [6], broadcastrout-ing [15], gossiprout-ing [4], and scatterrout-ing [8] were proposed. And many efficient algorithms designed for sorting and merging [14], selection [17], Fourier transform [7], and computational geometry [18] have been proposed. As a result, star graphs are recognized as an attractive alternative to the hy-percubes.
In this paper, we show that the n-dimensional star graphs are optimal with respect to the edge fault tolerant hamiltonian laceability, the edge fault tolerant strongly hamiltonian laceability, and the edge fault tolerant hyper hamiltonian laceability. In the next section, we introduce the definition of star graphs. And then in Section 3, we show our main result. Finally, we make our conclusion in Section 4.
2. Definition and basic properties
In this section, we introduce the definition and some properties of the star graph.
Definition 1. The n-dimensional star graph is denoted by Sn. The vertex set V of
Sn is fa1. . . anja1. . . an is a permutation of 1; 2; . . . ; ng and the edge set E is
fða1a2. . . ai1aiaiþ1. . . an; aia2. . . ai1a1aiþ1. . . anÞja1. . . an2 V and 2 6 i 6 ng.
By definition, Sncontains n! vertices and each vertex is of degree (n 1). For
example, vertex 1234 in S4connects to 2134, 3214, and 4231. S1, S2, and S3are a
vertex, an edge, and a cycle of length 6, respectively. We show S4in Fig. 1. It is
easy to observe that there are four vertex-disjoint S3’s embedded in S4. The
following proposition states this property. Proposition 1. There aren!
k!vertex-disjoint Sk’s embedded in Sn for k P 1.
Proof. Let B¼ fbkþ1. . . bnjbkþ1. . . bn is a permutation of anyðn kÞ elements
of 1; 2; . . . ; ng. So jBj ¼n!
k!. For any bkþ1. . . bn2 B, let S bkþ1...bn
k denote the
in-duced subgraph of Sn with vertex set fa1. . . anjakþ1. . . an¼ bkþ1. . . bng.
Obvi-ously, Sbkþ1...bn
k and S b0kþ1b0
n
k are vertex-disjoint for bkþ1 bn6¼ b0kþ1 b0n and
VðSnÞ ¼
S
bkþ1bn2BVðS
bkþ1bn
k Þ.
Let u¼ u1. . . ukbkþ1 bn be some vertex in Skbkþ1...bn. Define fukða1. . . anÞ ¼
i1. . . ik for aj¼ uij and 1 6 j 6 k. For example, let u¼ 54 123 be a vertex in S5.
Then f2
uð54 123Þ ¼ 12 and f 2
uð45 123Þ ¼ 21. We can easily check that
ffk
uðvÞjv 2 V ðS bkþ1...bn
k Þg ¼ V ðSkÞ and ðfukðv1Þ; fukðv2ÞÞ is an edge if and only if
(v1; v2) is an edge. So S bkþ1...bn
k ffi Sk and the proposition follows. h
In the following discussion, we will frequently use the notation Sbkþ1...bn
k
de-fined in the proof above. We call Sbkþ1...bn
k a substar of Sn or specifically, a
k-dimensional substar of Sn. Let u be a vertex not in S bkþ1...bn k . We say that u is adjacent to Sbkþ1...bn k if u is adjacent to a vertex in S bkþ1...bn k . And we call S bkþ1...bn k
an adjacent substar of u. The following proposition and corollary are con-cerning adjacent substars:
Proposition 2. Given k with 1 6 k 6 n 1 and bkþ1. . . bn, a vertex u¼ u1. . . unis
adjacent to Sbkþ1...bn
k if and only if ukþ1. . . ui1u1uiþ1. . . un¼ bkþ1bkþ2. . . bn for
some i with kþ 1 6 i 6 n. 1234 3214 2134 2314 3124 1324 4231 3241 2341 2431 3421 4321 3412 4312 1342 1432 4132 3142 2413 4213 1243 2143 1423 4123
Corollary 1. There are ðk 1Þ! edges between Sbkþ1...bn k and S b0 kþ1...b 0 n k if there is
exactly one different bit between bkþ1. . . bn and b0kþ1. . . b0n.
Proof. Without loss of generality, assume that bkþ1 6¼ b0kþ1 and bkþ2. . . bn¼
b0
kþ2. . . b0n. The first bit of all vertices in S bkþ1...bn k being adjacent to S b0 kþ1...b0n k must be b0
kþ1. So the number of these vertices is ðk 1Þ!. And the corollary
follows. h
For example, there areðn 2Þ!-edges between Si
n1and S j
n1for 1 6 i6¼ j 6 n.
We use Ei;jðS
nÞ to denote the set of these edges. And we call these edges
out-going edges of Si n1(or S
j
n1). Particularly, we say (u; v) an outgoing edge of u if
ðu; vÞ 2 Ei;jðS
nÞ for some 1 6 i 6¼ j 6 n.
It has been shown that the star graphs are edge symmetric [2], i.e., for any two edges ðx; yÞ; ðu; vÞ 2 EðSnÞ, there is an automorphism of Sn mapping x; y
into u; v, respectively. For ease of description, we use pðF Þ to denote the edge setfðpðuÞ; pðvÞÞjðu; vÞ 2 F g if p is an automorphism of Snand F ðSnÞ. Thus,
we have following proposition.
Proposition 3. Let F EðSnÞ. Then there is an edge set F0 EðSnÞ and an
automorphism p of Snsuch that pðF Þ ¼ F0 andjF0\ EðSn1i Þj 6 jF j 1 for each
1 6 i 6 n.
Proof. If jF \ EðSi
n1Þj 6 jF j 1 for each 1 6 i 6 n, let F0 be F and p be the
identity mapping. Then the statement follows. Otherwise, choose an arbitrary edgeðx; yÞ 2 F . With the edge symmetric property, there is an automorphism p of Sn such that pðxÞ ¼ 123 . . . ðn 1Þn and pðyÞ ¼ n23 . . . ðn 1Þ1. Let
F0¼ pðF Þ. So ð123 . . . ðn 1Þn; n23 . . . ðn 1Þ1Þ 2 F0. But ð123 . . . ðn 1Þn;
n23 . . .ðn 1Þ1Þ 62 EðSi
n1Þ for all 1 6 i 6 n. Thus, jF0\ EðSn1i Þj 6 jF j 1 for
each 1 6 i 6 n. h
By this proposition, given any edge set F EðSnÞ, we may assume that
jF \ EðSi
n1Þj 6 jF j 1 for each 1 6 i 6 n. This property will help us simplify the
proof a lot.
3. Main result
In this section, we present our main result on the three indicators, which are the edge fault tolerant hamiltonian laceability (eftHL), edge fault tolerant strongly hamiltonian laceability (eftSHL), and edge fault tolerant hyper hamiltonian laceability (eftHHL) of the star graphs. We provide a lemma to give three upper bounds for the bipartite graphs and then three theorems to give the exact values for the three indicators on the star graphs. We will see that
all the values match the upper bounds. So the star graphs are optimal with respect to these properties. Now we show the upper bounds.
Lemma 1. Let G¼ ðV0[ V1; EÞ be a bipartite graph with jV0j ¼ jV1j and let d be
the minimum degree of G among all vertices. We have eftHLðGÞ 6 d 2, eftSHLðGÞ 6 d 2 for d P 2, and eftHHLðGÞ 6 d 3 for d P 3.
Proof. Assume that the degree of vertex u is d. Removing (d 1)-edges con-necting to u. Suppose that v is the remainder vertex concon-necting to u and v0is a
neighbor of v which is not u (see Fig. 2). Then it is easy to check that there is no hamiltonian path from v to v0. So G is at most (d 2)-edge fault tolerant
hamiltonian laceable and obviously, at most (d 2)-edge fault tolerant strongly hamiltonian laceable.
Then consider removing (d 2)-edges which connect to u. Suppose that v1
and v2 are the remainder vertices connecting to u and let u0 be a vertex
con-necting to v1which is not u (see Fig. 3). Then it is easy to check that there is no
hamiltonian path of G u0 from v
1 to v2. So G is at most (d 3)-edge fault
tolerant hyper hamiltonian laceable. Hence, the lemma follows. h
u
v v'
Fig. 2. Upper bound for eftHLðGÞ.
u
v1 u'
v2
Next, we show the capacity of star graphs on these three indicators. First, we use a computer program to check the base case S4 (see Fig. 1) and the case
indeed holds for S4. So we state the results in the following lemma. Then we
prove our results by induction.
Lemma 2. S4 is 1-edge fault tolerant hamiltonian laceable, 1-edge fault tolerant
strongly hamiltonian laceable, and hyper hamiltonian laceable. To make the proofs clear, we introduce the following transform:
Definition 2. Given a fixed n, let V f1; 2; . . . ; ng and F EðSnÞ. Then
STGnðV ; F Þ is the graph GðV ; EÞ such that E¼ fði; jÞji; j 2 V and
Ei;jðS
nÞ \ F <ðn2Þ!2 g. (STG means to transmit a star graph to another graph.)
In fact, STGnmaps the substar Sin1 in (Sn F ) into the vertex i in G for all
i2 V . And for i 6¼ j 2 V , if i and j are adjacent in G, there is a vertex in each partite set of Si
n1 adjacent to S j
n1 in (Sn F ). So we have following lemma:
Lemma 3. Let G¼ STGnðV ; F Þ for V f1; 2; . . . ; ng with jV j P 2 and
F EðSnÞ. And let x 2 Sn1j1 and y2 S j2
n1 with j16¼ j22 V such that x; y are in
different partite sets. Assume that Si
n1 F is hamiltonian laceable for each
i2 V . Then there is a path from x to y crossing all vertices in all Si
n1 for i2 V
without crossing edges in F if there is a hamiltonian path from j1 to j2in G.
Proof. LetjV j ¼ h. And let hj1; j3; j4; . . . ; jh; j2i be a hamiltonian path from j1to
j2 in G. Since j1 and j3 are adjacent in G, we can find a vertex v12 V ðS j1
n1Þ
adjacent to Sj3
n1 such that v1; x are in different partite sets and the outgoing
edge, say (v1; u3), of v1 is not in F (see Fig. 4). Similarly, we can find vertices
v32 Sj3 n1; v42 S j4 n1; . . . ; vh2 S jh n1 adjacent to S j4 n1; S j5 n1; . . . ; S j2 n1, respectively,
such that v1; v3; v4; . . . ; vhare in the same partite set and the outgoing edges of
these vertices are not in F . Assume that the outgoing edges of these vertices are ðv3; u4Þ; ðv4; u5Þ; . . . ; ðvh; v2Þ. Then by assumption that each Si
n1 F is
hamil-tonian laceable, we can construct a path from x to y crossing all vertices in all Si n1 for i2 V as follows: 1 1 j n S 2 1 j n S h j n S 1 4 1 j n S 3 1 j n S - - - - -x v 1 u 3 v 3 u 4 v 4 u h v h u 2 y
Fig. 4. Remaining path.
hx; . . . ; v1; u3; . . . ; v3; u4; . . . ; vh; u2; . . . ; yi
Hence, the lemma follows. h Now we can show our first result:
Theorem 1. Snis (n 3)-edge fault tolerant hamiltonian laceable for n P 4.
Proof. We prove it by induction. By Lemma 2, we know that S4is 1-edge fault
tolerant hamiltonian laceable. In the induction step, we assume that Sn1 is
(n 4)-edge fault tolerant hamiltonian laceable for n P 5. Then consider Sn.
Let F EðSnÞ be arbitrary faulty edge set such that jF j 6 n 3. By
Propo-sition 3, we may assume that jF \ EðSi
n1Þj 6 n 4 for each 1 6 i 6 n. So
Si
n1 F is still hamiltonian laceable for each 1 6 i 6 n. Let x 2 V ðS j1
n1Þ and
y2 V ðSj2
n1Þ such that x; y are in different partite sets. We shall construct a
fault-free hamiltonian path from x to y. Consider the following two cases:
Case 1. j16¼ j2. Let V ¼ f1; 2; . . . ; ng. Since jF j 6 n 3 <ðn2Þ!2 for n P 5,
Ei;jðS
nÞ \ F <ðn2Þ!2 for any i6¼ j 2 V . So G ¼ STGnðV ; F Þ is a complete graph.
It is easy to find a hamiltonian path of G from j1to j2. By Lemma 3, there is a
hamiltonian path of Sn from x to y.
Case 2. j1¼ j2¼ j. There is a hamiltonian path P of Sn1j from x to y. The
length of P isðn 1Þ! 1. So we can find an edge, say (u; v), on path P such that the outgoing edges of u and v are fault-free. (If such (u; v) does not exist, jF j Pðn1Þ!12 > n 3 for n P 5.) Let P ¼ hx; P1; u; v; P2; yi and ðu; v0Þ; ðv; u0Þ are
the outgoing edges of u and v, where v02 Sj3
n1and u02 S j4
n1(see Fig. 5). v0and
u0are in different partite sets of S
nand j36¼ j4. Let V ¼ f1; 2; . . . ; ng j. Then
j S 3 j n -1 n -1 n -1 S Sj4 x y u v v' u'
G¼ STGnðV ; F Þ is a complete graph with (n 1) vertices sinceðn2Þ!2 >jF j for
nP5. Thus, there is a hamiltonian path of G from j3to j4. By Lemma 3, there
is a path P3 crossing all vertices of all Sn1i for i2 V from v0 to u0. So a
ham-iltonian path of Snfrom x to y can be constructed as follows:
hx; P1; u; v0; P3; u0; v; P2; yi
Hence, the theorem follows. h
Since Snis (n 1) regular, by Lemma 1, Sn is optimal with respect to edge
fault tolerant hamiltonian laceability and eftHLðSnÞ ¼ n 3.
Theorem 2. Sn is (n 3)-edge fault tolerant strongly hamiltonian laceable for
nP4.
Proof. We also prove it by induction. S4 is shown to be 1-edge fault tolerant
strongly hamiltonian laceable in Lemma 2. So we need only to consider the induction step. Assume that Sn1 is (n 4)-edge fault tolerant strongly
hamil-tonian laceable for n P 5 and consider Sn.
Given any fault edge set F in SnwithjF j 6 n 3, by Proposition 3, we can
assume thatjF \ EðSi
n1Þj 6 n 4 for each 1 6 i 6 n. So Sn1i F is still strongly
hamiltonian laceable for each 1 6 i 6 n. Let x2 V ðSj1
n1Þ and y 2 V ðS j2
n1Þ such
that x and y are in the same partite set. Consider the following two cases: Case 1. j16¼ j2. Let Ve be the number of vertices which are in the different
partite set from x and which are not adjacent to Sj2
n1. Then Ve is equal to ðn1Þ!
2
ðn2Þ!
2 which is strictly greater than jF j. So there is a fault-free edge
(u1; v3), i.e., not in F , such that u12 V ðSj1
n1Þ, v32 V ðS j3
n1Þ for j362 fj1; j2g, and
x; y; u1are in the same partite set of S
n(see Fig. 6). By the induction hypothesis,
there is a path P1of lengthðn 1Þ! 2 in Sn1j1 from x to u1. Then consider the
remainder subgraphs. Let V ¼ f1; 2; . . . ; ng fj1g. Thus, jV j P 2 and
G¼ STGnðV ; F Þ is a complete graph. There is a hamiltonian path of G from j3
to j2. Since u1and y are in the same partite set, y and v3are in different partite
sets. So there is a path P2crossing all vertices of all Sn1i for i2 V from v3to y.
1 j
S
Sj3 Sj2 x u 1 v 3 y |P1| = (n-1)! -2 |P2| = (n-1)(n-1)! -1 n -1 n -1 n -1Fig. 6. x and y are in different substars.
The length of this path isðn 1Þðn 1Þ! 1. We can construct a path from x to y as follows:
hx; P1; u1; v3; P2; yi
The length of this path is
½ðn 1Þ! 2 þ 1 þ ½ðn 1Þðn 1Þ! 1 ¼ n! 2 So the theorem follows in this case.
Case 2. j1¼ j2¼ j. The proof of this case is similar to that of case 2 in
Theorem 1 except that the path in Sjn1 from x to y is of lengthðn 1Þ! 2. Hence, the theorem follows. h
Since Snis (n 1) regular, by Lemma 1, Snis also optimal with respect to the
edge fault tolerant strongly hamiltonian laceability.
Theorem 3. Snis (n 4)-edge fault tolerant hyper hamiltonian laceable for n P 4.
Proof. The proof is a little more complex than the previous two theorems. Again, S4 is hyper hamiltonian laceable by Lemma 2. So we show that the
statement is true for n P 5. Assume that Sn1is (n 5)-edge fault tolerant hyper
hamiltonian laceable for n P 5.
Let F be a faulty edge set in Sn withjF j 6 n 4. By Proposition 3, we may
assume that jF \ EðSi
n1Þj 6 n 5 for each 1 6 i 6 n. So Sn1i F is still hyper
hamiltonian laceable and obviously, strongly hamiltonian laceable for each 1 6 i 6 n. Given a vertex v, in the following we will construct a hamiltonian path ofðSn F Þ v between any two vertices in the partite set which v is not
in. Let x and y be two such vertices. Consider the following four cases: Case 1. v; x; y are in the same substar, say Sj1
n1 (see Fig. 7(a)). By the
induction hypothesis, there is a hamiltonian path P ofðSj1
n1 F Þ v from x to
y. The length of P isðn 1Þ! 2 > 2jF j for n P 5. So there is an edge (u1; v1) on
P such that the outgoing edges of u1and v1, say (u1; v2) and (v1; u3), are
fault-free. (x; u1are not necessary in the same partite set.) Let P ¼ hx; P
1; u1; v2; P2; yi.
Clearly, v2 and u3are in different partite sets of S
n. Assume that v22 S j2
n1 and
u32 Sj3
n1. So j26¼ j3. Let V ¼ f1; 2; . . . ; ng fj1g. Then STGnðV ; F Þ is a
com-plete graph. There is a hamiltonian path from j2to j3and so a path P3from v2
to u3 crossing all vertices of Si
n1 for all i2 V . Therefore, we can construct a
hamiltonian path ofðSn F Þ v as: hx; P1; u1; v2; P3; u3; v1; P2; yi.
Case 2. v; x2 Sj1
n1and y2 S j2
n1with j16¼ j2(see Fig. 7(b)). Let j36¼ j2. Since ðn2Þ!
2 1 > jF j for n P 5, we can easily find a vertex u
16¼ x 2 Sj1
n1 such that u1
and x are in the same partite set and the outgoing edge of u1, say (u1; v3), is
fault-free. (Note that since u16¼ x, there areðn2Þ!
2 1 choices for u 1in Sj1
n1.) By
the induction hypothesis, there is a hamiltonian path P1ofðSjn11 F Þ v from
xto u1. Let V ¼ f1; 2 . . . ; ng fj
that v3and y are in different partite sets. So there is a hamiltonian path from j 3
to j2and a path P2from v3to y crossing all vertices of all Sin1for i2 V . Hence,
we have a hamiltonian pathhx; P1; u1; v3; P2; yi of ðSn F Þ v.
Case 3. v2 Sj1
n1 and x; y2 S j2
n1with j16¼ j2(see Fig. 7(c)). Sinceðn2Þ!2 >jF j,
there is a vertex v02 V ðSj2
n1Þ adjacent to S j1
n1such that the outgoing edge of v0,
say (v0; u1), is fault-free and v0; vare in the same partite set. By the induction
hypothesis, there is a hamiltonian path P ofðSj2
n1 F Þ v0 from x to y. Since
there are (n 2) neighbors of v0in Sj2
n1 andjF j < ðn 2Þ, there exists an edge
(u2; v2) on P such that u2 is adjacent to v0 and the outgoing edge of v2, say
(v2; u3), is fault-free. Clearly, j
362 fj1; j2g since v2; v0 are neighbors of u2 but
v26¼ v0. Let P¼ hx; P
1; u2; v2; P2; yi. (Note that P may be hx; P1; v2; u2; P2; yi and
the argument of this case is similar to the following discussion.) Let j462 fj1; j2; j3g. Since ðn2Þ!2 1 > jF j for n P 5, there is a vertex w12 Sn1j1
adjacent to Sj4
n1 such that w1; u1 are in the same partite set and the outgoing
1 j S 2 j
S
x y u1v1 v2 3 jS
u3 1 j S 3 j S x y u1 v3 2 j S v (a) (d) (c) (b) v 2 j S 1 j S x y u2 v1 3 j S v 4 j S u4 w1 2 j S 1 j S x u3 u2 u1 3 j S v 4 j S v4 w1 y v' v2 n -1 n -1 n -1 n -1 n -1 n -1 n -1 n -1 n -1 n -1 n -1 n -1 n -1 n -1Fig. 7. Edge fault tolerant hyper hamiltonian laceability of the star.
edge of w1, say (w1; v4), is fault-free. So v42 V ðSj4
n1Þ and v4; u3are in different
partite sets. By the induction hypothesis, there is a hamiltonian path P3 of
ðSj1
n1 F Þ v from u
1to w1. Let V ¼ f1; 2; . . . ; ng fj
1; j2g. Then STGnðV ; F Þ
is a complete graph. There is a hamiltonian path from j4to j3and so a path P4
crossing all vertices of Si
n1 for all i2 V from v
4 to u3. Thus, we have a
ham-iltonian path ofðSn F Þ v as follows:
hx; P1; u2; v0; u1; P3; w1; v4; P4; u3; v2; P2; yi: Case 4. v2 Sj1 n1, x2 S j2 n1, and y2 S j3
n1 for distinct j1, j2, and j3 (see Fig.
7(d)). Sinceðn2Þ!2 >jF j, there is a vertex u22 V ðSj2
n1Þ adjacent to S j1
n1such that
u2; xare in different partite sets and the outgoing edge of u2, say (u2; v1), is
fault-free. By the induction hypothesis, there is a hamiltonian path P1of (Sjn12 F )
from x to u2. Let j
462 fj1; j2; j3g. In Sn1j1 , since ðn2Þ!
2 1 > jF j, there is a vertex
w16¼ v1 adjacent to Sj4
n1 such that w1; v1 are in the same partite set and the
outgoing edge of w1, say (w1; u4), is fault-free. By the induction hypothesis,
there is also a hamiltonian path P2 of ðSn1j1 F Þ v from v1 to w1. For the
remaining substars, let V ¼ f1; 2; . . . ; ng fj1; j2g. Then G ¼ STGnðV ; F Þ is a
complete graph. So there is a hamiltonian path of G from j4to j3and thus, a
path P3from u4to y crossing all vertices of Sn1i for all i2 V . Finally, we have a
hamiltonian pathhx; P1; u2; v1; P2; w1; u4; P3; yi of ðSn F Þ v.
Hence, the theorem follows. h
Since Snis (n 1) regular, by Lemma 1, Snis optimal with respect to the edge
fault tolerant hyper hamiltonian laceability.
4. Conclusion
Fault tolerance is an important research subject of the multi-process com-puter systems. Graphs are usually used to represent the interconnection architecture of these systems, where vertices represent processors and edges represent links between processors. Many researches concerned the vertex-fault tolerant or edge-fault tolerant properties of some specific graphs. In this paper, we study some fault tolerant results of the star graphs. We show that the n-dimensional star graph is (n 3)-edge fault tolerant hamiltonian laceable, (n 3)-edge fault tolerant strongly hamiltonian laceable, and (n 4)-edge fault tolerant hyper hamiltonian laceable.
In particular, we use computer programs to check the base cases. It not only gives us some preliminary intuition but also simplifies our proof. If we did such check by theoretical proof, we would have spent too much effort since there would have been too many subcases to deal with. Apparently, such a method may be applied in other cases nowadays, especially, for those facts which can be proved by induction.
Acknowledgements
This work was supported in part by the National Science Council of the Republic of China under Contract NSC 91-2218-E-231-002.
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