Fault-tolerant hamiltonian connectivity of the WK-recursive
networks
Tung-Yang Ho
a,⇑, Cheng-Kuan Lin
b, Jimmy J.M. Tan
c, Lih-Hsing Hsu
d aDepartment of Tourism Management, Ta Hwa University of Science and Technology, Hsinchu 30740, Taiwan, ROC
b
School of Computer Science and Technology, Soochow University, Suzhou 215006, China
c
Department of Computer Science, National Chiao Tung University, Hsinchu 30010, Taiwan, ROC
d
Department of Computer Science and Information Engineering, Providence University, Taichung 43301, Taiwan, ROC
a r t i c l e
i n f o
Article history: Received 31 March 2009 Accepted 13 February 2014 Available online 22 February 2014
Keywords: Hamiltonian Hamiltonian connected WK-recursive network
a b s t r a c t
Many research on the WK-recursive network has been published during the past several years due to its favorite properties. In this paper, we consider the fault-tolerant hamiltonian connectivity of the WK-recursive network. We use Kðd; tÞ to denote the WK-recursive net-work of level t, each of which basic modules is a d-vertex complete graph, where d > 1 and t P 1. The fault-tolerant hamiltonian connectivity Hj
fðGÞ is defined to be the maximum integer k such that G is k fault-tolerant hamiltonian connected if G is hamiltonian connected and is undefined otherwise. In this paper, we prove that Hj
fðKðd; tÞÞ ¼ d 4 if d P 4.
Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction
As is customary in structure studies of parallel architectures, we restrict our attention to a set of identical processors, and we view the architectures of the underlying interconnection networks as graphs. The vertices of a graph represent the pro-cessors of an architecture, and the edges of the graph represent the communication links between propro-cessors. There are many mutually conflicting requirements in designing the topology of interconnection networks. It is almost impossible to design a network which is optimum from all aspects. One has to design a suitable network depending on the requirements of its properties. The hamiltonian property is one of the major requirements in designing the topology of a network. Fault-tolerance is also desirable in massive parallel systems.
In this paper, a network is represented as a loopless undirected graph. For graph definitions and notations we follow[1]. G ¼ ðV; EÞ is a graph if V is a finite set and E is a subset of {ðu;
v
Þjðu;v
Þ is an unordered pair of V}. We say that V is the vertex set and E is the edge set. Two vertices u and v are adjacent if ðu;v
Þ 2 E. Let S be a subset of V. The subgraph of G induced by S is the graph G½S with VðG½SÞ ¼ S and EðG½SÞ ¼ fðu;v
Þjðu;v
Þ 2 E; and fu;v
g Sg. The complement G of a graph G with the same ver-tex set VðGÞ defined by ðu;v
Þ 2 EðGÞ if and only if ðu;v
Þ R EðGÞ. We use e to denote jEðGÞj. The degree of a vertex u of G; degGðuÞ, is the number of edges incident with u. A graph G is k-regular if degGðxÞ ¼ k for any vertex x in G. A path,h
v
0;v
1;v
2; . . . ;v
ki, is an ordered list of distinct vertices such thatv
i andv
iþ1 are adjacent for 1 6 i 6 k 1. A path is ahamiltonian path if its vertices are distinct and span V.
In[5], the performance of the hamiltonian property in faulty networks is discussed. In[10], Huang et al. define a param-eter on fault-tolerant hamiltonicity. A hamiltonian graph G is k fault-tolerant hamiltonian if G F remains hamiltonian for
http://dx.doi.org/10.1016/j.ins.2014.02.087
0020-0255/Ó 2014 Elsevier Inc. All rights reserved.
⇑ Corresponding author. Tel.: +886 3 5927700 3208; fax: +886 3 5927700. E-mail address:hoho@tust.edu.tw(T.-Y. Ho).
Contents lists available atScienceDirect
Information Sciences
In this paper, we consider the fault-tolerant hamiltonian connectivity of the WK-recursive network. The WK-recursive network is proposed by[15]. We use Kðd; tÞ to denote the WK-recursive network of level t, each of which basic modules is a d-vertex complete graph, where d > 1 and t P 1. It offers a high degree scalability, which conforms very well to a mod-ular design and implementation of distributed systems involving a large number of computing elements. A transputer imple-mentation of a 15-vertex WK-recursive network has been realized at the Hybrid Computing Center, Naples, Italy. In this implementation, each vertex is implemented with the IMS T414 Transputer[12]. Recently, the WK-recursive network has received much attention due to its many favorable properties. In particular, it is proved that Kðd; tÞ is hamiltonian connected
[2]and HfðKðd; tÞÞ ¼ d 3[4]. In this paper, we prove that HjfðKðd; tÞÞ ¼ d 4.
In the following section, we give the definition of WK-recursive network. In Section3, we give some preliminaries for the discussion on the fault-tolerant hamiltonian connectivity of the WK-recursive network. In Section 4, we prove that HjfðKðd; tÞÞ ¼ d 4.
2. WK-recursive networks
The WK-recursive network can be constructed hierarchically by grouping basic modules. A complete graph of any size d can serve as the basic modules. We use Kðd; tÞ to denote a WK-recursive network of level t, each of whose basic modules is a d-vertex complete graph, where d > 1 and t P 1. The structures of Kð5; 1Þ; Kð5; 2Þ, and Kð5; 3Þ are shown inFig. 1. Kðd; tÞ is defined in terms of a graph as follows:
Each vertex of Kðd; tÞ is labeled as a t-digit radix d number. Vertex at1at2. . .a1a0is adjacent to (1) at1at2. . .a1b, where
b – a0and (2) at1at2. . .ajþ1aj1ðajÞj1if aj–aj1 and aj1¼ aj2¼ . . . ¼ a0, where ðajÞj1denotes j 1 consecutive ajs. An
open edge is incident with at1at2. . .a0if at1¼ at2¼ ¼ a0. The open edge is reserved for further expansion. Hence,
its other end vertex is unspecified. The open vertex set Ovof Kðd; tÞ is the set fat1at2. . .a0jai¼ aiþ1for 0 6 i 6 t 2g. In other
words, Ovcontains those vertices with open edges.
Obviously, Kðd; 1Þ is a d-vertex complete graph augmented with d open edges. For t P 1; Kðd; t þ 1Þ consists d copies of Kðd; tÞ, say K1ðd; tÞ; K2ðd; tÞ; . . . ; Kdðd; tÞ. Thus, we consider Kiðd; tÞ as the ith component of Kðd; t þ 1Þ. Let I ¼ fw1; . . . ;wqg
be any q subset of f1; 2; . . . ; dg, we define graph KIðd; tÞ is the subgraph of Kðd; t þ 1Þ induced by Sqi¼1VðKwiðd; tÞÞ. For
t P 2, the open vertices of Kiðd; tÞ can be labeled as oi;0 and oi;j for 1 6 i – j 6 d where oi;0 is the only open vertex of
Kðd; t þ 1Þ in Kiðd; tÞ and oi;jis the vertex in Kiðd; tÞ joining with the vertex oj;i in Kjðd; tÞ with an open edge. Note that
ðoi;j;oj;iÞ is the only edge joining Kiðd; tÞ to Kjðd; tÞ.
Now, we define the extended WK-recursive network eKiðd; tÞ as Vð eKiðd; tÞÞ ¼ VðKðd; tÞÞ [ fxg and Eð eKiðd; tÞÞ ¼ EðKðd; tÞÞ[
fðoa;0;xÞj
a
2 f1; 2; . . . ; dg figg. For example, eK2ð5; 1Þ and eK3ð5; 2Þ are illustrated inFig. 2. Obviously, eKiðd; tÞ is isomorphicto eKjðd; tÞ for 1 6 i – j 6 d.
3. Preliminaries
The following theorem is proved by Ore[13].
Theorem 1 [13]. Assume that G is an n-vertex graph with n P 4. Then G is hamiltonian if e 6 n 3, and is hamiltonian connected if e 6 n 4.
Corollary 1. Assume that n P 4. Then Knis ðn 3Þ fault-tolerant hamiltonian and ðn 4Þ fault-tolerant hamiltonian connected.
Proof. Let F be any subset of VðKnÞ [ EðKnÞ. We use Fvto denote F \ VðKnÞ. Then Kn F is isomorphic to KnjFv j F0where F0
is a subset of edges in the subgraph of Kninduced by f1; 2; . . . ; ng Fv. Obviously, jF0j 6 jFj jFvj 6 n 3 jFvj. Since n jFvj
is the number of vertices of KnjFv j F0, the lemma follows fromTheorem 1. h
Let F VðKðd; t þ 1ÞÞ [ EðKðd; t þ 1ÞÞ with jFj 6 d 4. For 1 6 q 6 d, we use Fqto denote F \ ðVðKqðd; tÞÞ [ EðKqðd; tÞÞÞ. Note
that it is possible F [d
q¼1Fq–;. For example, it is possible ðo1;2;o2;1Þ 2 F but ðo1;2;o2;1Þ R [dq¼1Fq.
Now, we construct another graph HðFÞ from the complete graph Kdwith vertex set f1; 2; . . . ; dg by considering vertex i
corresponds to the i-component of Kðd; t þ 1Þ for every i. Let F0
¼ fð
a
;bÞjoa;b2 F; ob;a2 F, or ðoa;b;ob;aÞ 2 Fg. We setpath between any two vertices in Kðd; t þ 1Þ F. However, there are several problems need to be conquered. Let us consider the following example.
Assume that u is a vertex in Kiðd; tÞ and v is a vertex in Kjðd; tÞ with 1 6 i – j 6 d. Let hi ¼ w1;w2; . . . ;wd¼ ji be a
hamil-tonian path of HðFÞ. Let Pibe a hamiltonian path of Kiðd; tÞ Fijoining u to oi;w2, let Pqbe a hamiltonian path of Kwqðd; tÞ Fwq
(a)
(b)
(c)
Fig. 1. The graphs (a) Kð5; 1Þ, (b) Kð5; 2Þ, and (c) Kð5; 3Þ.
x
x
1
2
3
4
5
(a)
(b)
joining owq;wq1 to owq;wqþ1 for 2 6 q 6 d 1, and let Pj be a hamiltonian path of Kjðd; tÞ Fj joining oj;wd1 to v. Obviously,
hPi;P2; . . . ;Pji forms a hamiltonian path of Kðd; t þ 1Þ F joining u to v. SeeFig. 3for illustration.
Yet, we need to guarantee the existence of required paths in each component. Later, we will prove Kðd; tÞ is ðd 4Þ fault-tolerant hamiltonian connected by induction. In the induction step, we assume Kðd; tÞ is ðd 4Þ fault-fault-tolerant hamiltonian connected and prove that Kðd; t þ 1Þ is ðd 4Þ fault-tolerant hamiltonian connected. With the assumption, the required ham-iltonian path Pqexists for 2 6 q 6 d 1. However, we cannot find Piif u ¼ oi;w2. Similarly, we cannot find Pjif oj;wd1¼
v
. Tosolve the problem, we can find another hamiltonian path hi ¼ z1;z2; . . . ;zd¼ ji of HðFÞ to meet the boundary conditions that
u – oi;z2and
v
–oj;zd1. As a conclusion, we have the following lemma.Lemma 1. Assume that Kðd; tÞ is ðd 4Þ fault-tolerant hamiltonian connected. Let F VðKðd; t þ 1ÞÞ [ EðKðd; t þ 1ÞÞ with jFj 6 d 4. Let u be a vertex in Kiðd; tÞ and let v be a vertex in Kjðd; tÞ with 1 6 i – j 6 d. Suppose that hi ¼ w1;w2; . . . ;wd¼ ji be a
hamiltonian path of HðFÞ that satisfies the boundary conditions: u – oi;w2and
v
–oj;wd1. Then there exists a hamiltonian path ofKðd; t þ 1Þ F joining u and v.
From the above discussion, we have three problems to prove that Kðd; t þ 1Þ F is hamiltonian connected; i.e., there ex-ists a hamiltonian path of Kðd; t þ 1Þ F between any two vertices u and v. First, assume that u is a vertex in Kiðd; tÞ and v is a
vertex in Kjðd; tÞ with 1 6 i – j 6 d. We need to find a hamiltonian path in HðFÞ that meets the boundary conditions. Second,
find a hamiltonian path of Kðd; t þ 1Þ F joining u and v if we cannot find a hamiltonian path in HðFÞ that meets the bound-ary conditions. Finally, find a hamiltonian path of Kðd; t þ 1Þ F joining u and v if both u and v are in Kiðd; tÞ for some i.
Now, we face the first problem. Let P1¼ hu1;u2; . . . ;uni and P2¼ h
v
1;v
2; . . . ;v
ni be any two hamiltonian paths of G. Wesay that P1and P2are orthogonal if u1¼
v
1;un¼v
n, and uq–v
qfor q ¼ 2 and q ¼ n 1. We say a set of hamiltonian pathsfP1;P2; . . . ;Psg of G are mutually orthogonal if any two distinct paths in the set are orthogonal. Suppose there are three
mutu-ally orthogonal hamiltonian paths between any two vertices of HðFÞ. By pigeon-hole principle, we can easily find a hamil-tonian path with the desired boundary conditions. For this reason, we would like to know all the cases that there are at most two orthogonal hamiltonian paths in HðFÞ. As mentioned above, HðFÞ is isomorphic to a graph G with n vertices and
e 6 n 4.
However, we need some background. Let G and H be two graphs. We use G þ H to denote the disjoint union of G and H. We use G _ H to denote the graph obtained from G þ H by joining each vertex of G to each vertex of H. For 1 6 m < n=2, let Cm;nbe the graph ðKmþ Kn2mÞ _ Km. SeeFig. 4for illustration.
The following theorem is proved in[3].
Theorem 2. Assume that G is an n-vertex graph with n P 4 and e 6 n 4. Let s and t be any two vertices of G. Then there are at least two orthogonal hamiltonian paths of G between s and t. Moreover, there are at least three mutually orthogonal hamiltonian paths of G between s and t except the following cases:
C1: G is isomorphic to K4where s and t are any two vertices of G.
C2: G is isomorphic to K5 ð1; 2Þ where s and t are any two vertices except fs; tg ¼ f1; 2g.
C3: The subgraph H induced by VðGÞ fs; tg is a complete graph with n P 6 where s is adjacent VðGÞ fsg and t is adjacent to s and exactly two vertices in H.
C4: The subgraph induced by VðGÞ fs; tg is isomorphic to C2;5where s is adjacent VðGÞ fsg and t is adjacent to VðGÞ ftg.
C5: The subgraph induced by VðGÞ fs; tg is isomorphic to C1;n2with n P 6 where s is adjacent VðGÞ fsg and t is adjacent to
VðGÞ ftg.
To solve the remaining problems, we need some path patterns.
Lemma 2. Let d P 4; t P 1, and I ¼ fw1; . . . ;wng be any n subset of f1; 2; . . . ; dg. Let u be a vertex in Kw1ðd; tÞ and v be a vertex in
Kwnðd; tÞ such that u – ow1;w2and
v
–own;wn1. Let F be any subset of VðKðd; t þ 1ÞÞ [ EðKðd; t þ 1ÞÞ such that (1) there exists anedge between Kwqðd; tÞ Fqand Kwqþ1ðd; tÞ Fqþ1 for 1 6 q 6 n 1 (2) Kwqðd; tÞ Fqis hamiltonian connected for 1 6 q 6 n.
Then there is a hamiltonian path P of KIðd; tÞ F joining u to v.
Proof. Since there exists an edge between Kwqðd; tÞ Fqand Kwqþ1ðd; tÞ Fqþ1for 1 6 q 6 n 1, the edge ðowq;wqþ1;owqþ1;wqÞ
and the vertices owq;wqþ1;owqþ1;wq are not in F for 1 6 q 6 n 1. Since Kwqðd; tÞ Fq is hamiltonian connected for 1 6 q 6 n,
there exists a hamiltonian path P1of Kw1ðd; tÞ F1joining u to ow1;w2, there exists a hamiltonian path Pqof Kwqðd; tÞ Fq
join-ing owq;wq1 to owq;wqþ1 for 2 6 q 6 n 1, there exists a hamiltonian path Pnof Kwnðd; tÞ Fnjoining own;wn1 to v. Therefore
P ¼ hu; P1;P2; . . . ;Pn;
v
i is a hamiltonian path of KIðd; tÞ F joining u to v. hLemma 3. Let d P 4 and t P 1. Assume that u and v are two vertices of Kðd; tÞ. Let r and s be any two open vertices such that jfu;
v
g \ fr; sgj ¼ 0. Then there exist two disjoint paths R and S such that (1) R joins u to one of the vertex in fr; sg, say r, (2) S joins v to s, and (3) R [ S spans Kðd; tÞ.Proof. We prove this lemma by induction on t. The lemma is obviously true for t ¼ 1 because Kðd; 1Þ is isomorphic to the complete graph Kdwith Ov¼ VðKðd; 1ÞÞ. Thus, we assume that this lemma holds for Kðd; nÞ for every 1 6 n 6 t. We claim
the statement holds for Kðd; t þ 1Þ.
Let u 2 VðKiðd; tÞÞ;
v
2 VðKjðd; tÞÞ; r 2 VðKkðd; tÞÞ, and s 2 VðKlðd; tÞÞ. Since there is only one open vertex in eachcomponent, we have k – l. Now, we consider the following cases. Case 1: i – j.
Subcase 1.1: jfi; jg \ fk; lgj ¼ 0. Thus, there exists an index in fk; lg, say k, such that u – oi;k. Let I1¼ fi; kg and
I2¼ fw1;w2; . . . ;wd2g ¼ f1; 2; . . . ; dg I1such that w1¼ j; wd2¼ l, and
v
–oj;w2. ByLemma 2, there exists a hamiltonianpath R of KI1ðd; tÞ joining u to r; there exists a hamiltonian path S of KI2ðd; tÞ joining v to s. Obviously, R and S are the required
paths.
Subcase 1.2: jfi; jg \ fk; lgj ¼ 1. Without loss of generality, we assume that i ¼ k. Let R be a hamiltonian path of Kiðd; tÞ
joining u to r. Let I ¼ fw1;w2; . . . ;wd1g ¼ f1; 2; . . . ; dg fig such that w1¼ j and wd1¼ l. By Lemma 2, there exists a
hamiltonian path S of KIðd; tÞ joining v to s. Obviously, R and S are the required paths.
Subcase 1.3: jfi; jg \ fk; lgj ¼ 2. Without loss of generality, we assume that i ¼ k and j ¼ l. By assumption, jfu;
v
g \ fr; sgj ¼ 0. Let I ¼ fw2; . . . ;wd1g ¼ f1; 2; . . . ; dg fi; jg. By induction, we can find two disjoint paths Sj1 and Sj2such that (1) Sj1joins v to oj;w2, (2) Sj2 joins oj;wd1to s, and (3) Sj1[ Sj2spans Kjðd; tÞ. Let R be a hamiltonian path of Kiðd; tÞ
joining u and r. By Lemma 2, there exists a hamiltonian path S0 of K
Iðd; tÞ joining ow2;j to owd1;j. Obviously, R and
S ¼ h
v
;Sj1;S0;S1j2;si are the required paths.4. Fault-tolerant hamiltonian connectivity
Lemma 4. Both Kðd; tÞ and eKiðd; tÞ are ðd 4Þ fault-tolerant hamiltonian connected for d P 4; t P 1, and 1 6 i 6 d.
Proof. Since eKiðd; tÞ is isomorphic to eKjðd; tÞ for 1 6 i – j 6 d, we consider eK1ðd; tÞ in the following.
Suppose t ¼ 1. Note that Kðd; 1Þ is isomorphic to Kdand eK1ðd; 1Þ is isomorphic to Kdþ1 e where e is any edge in Kdþ1. By Corollary 1, Kdand Kdþ1 e are ðd 4Þ fault-tolerant hamiltonian connected.
Assume that this lemma holds for Kðd; qÞ and eK1ðd; qÞ for every 1 6 q 6 t. We will claim that Kðd; t þ 1Þ and eK1ðd; t þ 1Þ
are also ðd 4Þ fault-tolerant hamiltonian connected.
First, we show that Kðd; tÞ is ðd 4Þ fault-tolerant hamiltonian connected. Since Kðd; tÞ is hamiltonian connected, Kð4; tÞ is 0 fault-tolerant hamiltonian connected. Thus, we assume that d P 5. Let u be a vertex in Kiðd; tÞ;
v
be a vertex in Kjðd; tÞ withu –
v
and F be the faulty set with jFj 6 d 4. We need to find a hamiltonian path of Kðd; t þ 1Þ F joining u to v. Case A1: i – j. Assume that there exists a hamiltonian path hi ¼ w1;w2; . . . ;wd¼ ji of HðFÞ joining i and j that meet theboundary conditions: u – oi;w2and oj;wd1–
v
. ByLemma 1, there exists a hamiltonian path of Kðd; t þ 1Þ F joining u and v.ByTheorem 2, such hamiltonian path in HðFÞ that meets the boundary conditions except the cases C2, C3, C4, and C5 of
Theorem 2. We will show that this lemma holds for HðFÞ is isomorphic to K5 ð1; 2Þ, the subgraph N of HðFÞ induced by
VðHðFÞÞ fi; jg is a complete graph; isomorphic to C2;5; isomorphic to C1;n2. Since the proof of this part is tedious, we leave
this part in Appendix A.
Case A2: i ¼ j. Without loss of generality, we assume that i ¼ j ¼ 1. Let A ¼ K1ðd; tÞ [ fðo1;r;or;1Þj2 6 r 6 dg;
B ¼ for;1j2 6 r 6 dg, and C ¼ Kðd; t þ 1Þ A B. We set FA¼ F \ A; FB¼ F \ B, and FC¼ F \ C.
Suppose that jFAj > 0 or jFBj > 0. We consider the graph eK11ðd; tÞ. Let F 0
¼ FA[ fðo1;r;xÞjor;12 F or ðo1;r;or;1Þ 2 F for
2 6 r 6 dg. Obviously, jF0
j 6 d 4. By induction on eK1
1ðd; tÞ, there exists a hamiltonian path P1of eK11ðd; tÞ F
0joining u to v.
Thus, path P1 can be written as hu; P11;o1;a;x; o1;b;P12;
v
i. Since jFAj > 0 or jFBj > 0; jFCj 6 d 5. Therefore, HðFÞ f1g ishamiltonian connected. There exists a hamiltonian path ha ¼ w1; . . . ;wd1¼ bi of HðFÞ f1g joining vertex a to vertex b. By Lemma 2, there is a hamiltonian path Q of KIðd; t þ 1Þ F joining oa;1to ob;1where I ¼ fw1; . . . ;wd1g. Hence, hamiltonian
path hu; P11;o1;a;oa;1;Q ; ob;1;o1;b;P12;
v
i be the required path.Suppose that jFAj ¼ 0 and jFBj ¼ 0. Hence, jFCj 6 d 4. Therefore, HðFÞ f1g is hamiltonian. Let hw1; . . . ;wd1;w1i be the
hamiltonian cycle in HðFÞ f1g with jfo1;w1;o1;wd1g \ fu;
v
gj ¼ 0. ByLemma 3, there exist two disjoint paths R and S suchthat (1) R joins u to one of the vertex in fo1;w1;o1;wd1g, say o1;w1, (2) S joins v to o1;wd1, and (3) R [ S spans K1ðd; tÞ. ByLemma 2, there is a hamiltonian path P of KIðd; tÞ F joining ow1;1 to owd1;1 where I ¼ fw1; . . . ;wd1g. Hence, hamiltonian path
hu; R; o1;w1;P; o1;wd1;S;
v
i be the required path.Second, we show that eK1ðd; tÞ is ðd 4Þ fault-tolerant hamiltonian connected for d P 4 and t P 2. Let u and v be any two
distinct vertices in eK1ðd; tÞ and F be the faulty set with jFj 6 d 4. We want to show that there exists a hamiltonian path of
e
K1ðd; tÞ F joining u to v.
We construct graph eH1ðFÞ by setting Vð eH1ðFÞÞ ¼ VðHðFÞÞ [ f0g and Eð eH1ðFÞÞ ¼ EðHðFÞÞ [ fðr; 0Þjo
r;0 R F where 2 6 r 6 dg.
Obviously, eH1ðFÞ is hamiltonian connected.
Case B1: u 2 Kiðd; tÞ and
v
2 Kjðd; tÞ with i – j. Assume that there exists a hamiltonian path hi ¼ w1;w2; . . . ;wdþ1¼ ji ofe
H1ðFÞ joining i and j that meet the boundary conditions: u – o
i;w2and oj;wd–
v
. ByLemma 1, there exists a hamiltonian pathof Kðd; t þ 1Þ F joining u and v. ByTheorem 2, such hamiltonian path in eH1ðFÞ that meets the boundary conditions except
the cases C2, C3, C4, and C5 ofTheorem 2. We will show that this lemma holds for eH1ðFÞ is isomorphic to K
5 e where e is
any edge in K5, the subgraph N of eH1ðFÞ induced by Vð eH1ðFÞÞ fi; jg is a complete graph; isomorphic to C2;5; isomorphic to
C1;n2. Since the proof of this part is tedious, we leave this part in Appendix B.
Case B2: u 2 Kiðd; tÞ and v is the vertex x. Since jFj 6 d 4; jF \ Kðd; t þ 1Þj 6 d 4 and there exists at least one vertex or;x
with for;x;ðor;x;xÞg \ F ¼ ;. With above proof, Kðd; t þ 1Þ F is hamiltonian connected. There exists a hamiltonian path P1of
Kðd; t þ 1Þ F joining u to or;x. Hence, hamiltonian path hu; P1;or;x;x ¼
v
i be the required path.Case B3: u;
v
2 Kiðd; tÞ. Let A ¼ Kiðd; tÞ [ fðoi;r;or;iÞj1 6 r – i 6 dg; B ¼ for;ij1 6 r – i 6 dg, and C ¼ Kðd; t þ 1Þ fðoi;x;xÞgSuppose that jFAj > 0 or jFBj > 0. Consider the graph eKiiðd; tÞ. Let F 0
¼ FA[ fðoi;r;xÞjor;i2 F or ðoi;r;or;iÞ 2 F for 1 6 r – i 6 dg.
Obviously, jF0
j 6 d 4. By induction on eKi
iðd; tÞ, there exists a hamiltonian path Piof eKiiðd; tÞ F
0joining u to v. Path P ican be
written as hu; Pi1;oi;a;x; oi;b;Pi2;
v
i. Since jFAj > 0 or jFBj > 0; jFCj 6 d 4. Therefore, eH1ðFÞ fig is hamiltonian connected.There exists a hamiltonian path ha ¼ w1; . . . ;b ¼ wdi of eH1ðFÞ fig joining vertex a to vertex b. By Lemma 2, there is a
hamiltonian path Q of KIðd; t þ 1Þ F joining oa;i to ob;i where I ¼ fw1; . . . ;wdg. Hence, hamiltonian path
hu; Pi1;oi;a;oa;i;Q ; ob;i;oi;b;Pi2;
v
i be the required path.Suppose that jFAj ¼ 0 and jFBj ¼ 0. Hence, jFCj 6 d 3. Therefore, we know that eH1ðFÞ fig is hamiltonian. Let
hw1; . . . ;wd;w1i be the hamiltonian cycle in eH1ðFÞ fig with jfoi;w1;oi;wdg \ fu;
v
gj ¼ 0. ByLemma 3, there exist two disjointpaths R and S such that (1) R joins u to one of the vertex in foi;w1;oi;wdg, say oi;w1, (2) S joins v to oi;wd, and (3) R [ S spans
Kiðd; tÞ. By Lemma 2, there is a hamiltonian path P of KIðd; tÞ F joining ow1;i to owd;i where I ¼ fw1; . . . ;wdg. Hence,
hamiltonian path hu; R; oi;w1;P; oi;wd;S;
v
i be the required path. hTheorem 3. Hj
fðKðd; tÞÞ ¼ d 4 for d P 4 and t P 1.
Proof. Since dðKðd; tÞÞ ¼ d 1; HjfðKðd; tÞÞ 6 d 4. By Lemma 4, Kðd; tÞ is ðd 4Þ fault-tolerant hamiltonian connected.
Therefore, Hj
fðKðd; tÞÞ ¼ d 4 for d P 4 and t P 1. h
Appendix A. Remaining part of Case A1 inLemma 4
Subcase A1.1: HðFÞ is isomorphic to the complete graph K5 ð1; 2Þ and fi; jg – f1; 2g. Obviously, d ¼ 5 and jFj ¼ 1. Thus,
exactly one of o1;2;o2;1, or ðo1;2;o2;1Þ is fault. By the symmetric property of HðFÞ, we may assume that ði; jÞ ¼ ð1; 3Þ or
ði; jÞ ¼ ð5; 3Þ.
(i) ði; jÞ ¼ ð1; 3Þ. Obviously, hi; 5; 4; 2; ji and hi; 4; 2; 5; ji form two orthogonal hamiltonian paths of HðFÞ. ByLemma 1, we can construct a hamiltonian path between u and v of Kðd; t þ 1Þ F unless (1) (u ¼ oi;4and
v
¼ oj;2) or (2) (u ¼ oi;5and
v
¼ oj;5).Suppose that u ¼ oi;4and
v
¼ oj;2. Obviously, hi; 5; 2; 4; ji is a hamiltonian path in HðFÞ satisfying the boundarycondi-tions: u – oi;5 and
v
–oj;4. Suppose that u ¼ oi;5 andv
¼ oj;5. Since exactly one of o1;2;o2;1, or ðo1;2;o2;1Þ is fault,Kjðd; tÞ foj;5g is hamiltonian connected. Let Pj be the hamiltonian path of Kjðd; tÞ foj;5g joining oj;4 to oj;2. By
induction, Kqðd; tÞ F is hamiltonian connected for q 2 fi; 5; 4; 2g. Let Pibe the hamiltonian path of Kiðd; tÞ F joining
u to oi;4; let P5be the hamiltonian path of K5ðd; tÞ joining o5;2to o5;j; let P4be the hamiltonian path of K4ðd; tÞ joining o4;i
to o4;j; let P2be the hamiltonian path of K2ðd; tÞ joining o2;jto o2;5. Therefore, path hu; Pi;P4;Pj;P2;P5;
v
i is the requiredpath.
(ii) ði; jÞ ¼ ð5; 3Þ. Obviously, hi; 1; 4; 2; ji and hi; 2; 4; 1; ji form two orthogonal hamiltonian paths of HðFÞ. ByLemma 1, we can construct a hamiltonian path between u and v of Kðd; t þ 1Þ F unless (1) (u ¼ oi;1and
v
¼ oj;1) or (2) (u ¼ oi;2and
v
¼ oj;2).The case (1) is similar to (2). We consider (1) only. Let u ¼ oi;1and
v
¼ oj;1. Since exactly one of o1;2, o2;1, or ðo1;2;o2;1Þ isfault, Kjðd; tÞ foj;1g is hamiltonian connected. Let Pjbe the hamiltonian path of Kjðd; tÞ foj;1g joining oj;ito oj;2. By
induction, Kqðd; tÞ F is hamiltonian connected for q 2 fi; 1; 2; 4g. Let Pibe the hamiltonian path of Kiðd; tÞ joining u
to oi;j; let P1be the hamiltonian path of K1ðd; tÞ joining o1;4 to o1;j; let P4be the hamiltonian path of K4ðd; tÞ joining
o4;2 to o4;1; let P2be the hamiltonian path of K2ðd; tÞ joining o2;jto o2;4. Therefore, path hu; Pi;Pj;P2;P4;P1;
v
i is therequired path.
Subcase A1.2: The subgraph N of HðFÞ induced by VðHðFÞÞ fi; jg is a complete graph; vertex i is adjacent to j and all the vertices in N; j is adjacent to i and exactly two vertices 1 and 2 in N. We label the remaining vertices in N as 3; . . . ; d 2. See
Fig. 5(a) for illustration. It is easy to see that hi; 2; 3; . . . ; d 2; 1; ji and hi; 3; . . . ; d 2; 1; 2; ji form two orthogonal hamiltonian paths of HðFÞ between i and j. ByLemma 1, we can construct a hamiltonian path of Kðd; t þ 1Þ F joining u to v unless (1) (u ¼ oi;2and
v
¼ oj;2) or (2) (u ¼ oi;3andv
¼ oj;1).Suppose that u ¼ oi;2and
v
¼ oj;2. Obviously, hi; 3; 2; 4; . . . ; d 2; 1; ji is a hamiltonian path in HðFÞ satisfying the boundaryconditions: u – oi;3and
v
–oj;1. Suppose that u ¼ oi;3andv
¼ oj;1. Thus, the hamiltonian path hi; 1; d 2; . . . ; 3; 2; ji in HðFÞsatisfying the boundary conditions: u – oi;1 and
v
–oj;2. By Lemma 1, we can construct a hamiltonian path ofKðd; t þ 1Þ F joining u to v
Subcase A1.3: The subgraph N of HðFÞ induced by VðHðFÞÞ fi; jg is isomorphic to C2;5; vertex i is adjacent to j and all the
vertices in N; j is adjacent to i and all the vertices in N. We label the vertices in N as 1; 2; . . . ; 5. SeeFig. 5(b) for illustration. Thus, d ¼ 6 and jFj ¼ 3. Obviously, jFij ¼ jFjj ¼ 0. Moreover, hi; 1; 2; 3; 4; 5; ji and hi; 5; 4; 3; 2; 1; ji form two orthogonal
hamil-tonian paths of HðFÞ between i and j. ByLemma 1, we can construct a hamiltonian path of Kðd; t þ 1Þ joining u to v unless (1) (u ¼ oi;1and
v
¼ oj;1) or (2) (u ¼ oi;5andv
¼ oj;5). By the symmetric property of HðFÞ, we only consider the case u ¼ oi;1andSince jFij ¼ jFjj ¼ 0; Kjðd; tÞ ðFj[ foj;1gÞ is hamiltonian connected. Let Pjbe the hamiltonian path of Kjðd; tÞ ðFj[ foj;1gÞ
joining oj;5to oj;3. By induction, Kqðd; tÞ Fq is hamiltonian connected for q 2 fi; 1; . . . ; 5g. Let Pibe the hamiltonian path of
Kiðd; tÞ Fijoining u to oi;2; let P1be the hamiltonian path of K1ðd; tÞ F1joining o1;4to o1;j; let P2be the hamiltonian path
of K2ðd; tÞ F2joining o2;ito o2;5; let P3be the hamiltonian path of K3ðd; tÞ F3joining o3;jto o3;4; let P4be the hamiltonian
path of K4ðd; tÞ F4joining o4;3 to o4;1; let P5 be the hamiltonian path of K5ðd; tÞ F5joining o5;2 to o5;j. Therefore, path
hu; Pi;P2;P5;Pj;P3;P4;P1;
v
i is the required path.Subcase A1.4: The subgraph N of HðFÞ induced by VðHðFÞÞ fi; jg is isomorphic to C1;n2; vertex i is adjacent to j and all
the vertices in N; j is adjacent to i and all the vertices in N. We label the vertices in N as 1; 2; . . . ; d 2. SeeFig. 5(c) for illus-tration. Obviously, hi; 1; 2; . . . ; d 2; ji and hi; d 2; d 3; . . . ; 1; ji form two orthogonal hamiltonian paths of HðFÞ between i and j. ByLemma 1, we can construct a hamiltonian path of Kðd; t þ 1Þ joining u to v unless (1) (u ¼ oi;d2and
v
¼ oj;d2) or (2)(u ¼ oi;1and
v
¼ oj;1).Suppose that u ¼ oi;d2 and
v
¼ oj;d2. Obviously, hi; 3; d 2; . . . ; 4; 2; 1; ji is a hamiltonian path in HðFÞ satisfying theboundary conditions: u – oi;3and
v
–oj;1. ByLemma 1, we can construct a hamiltonian path of Kðd; t þ 1Þ joining u to v.Suppose that u ¼ oi;1and
v
¼ oj;1. Since jFij ¼ jFjj ¼ 0; Kjðd; tÞ foj;1g is hamiltonian connected. Let Pjbe the hamiltonianpath of Kjðd; tÞ foj;1g joining oj;3to oj;4. By induction, Kqðd; tÞ Fqis hamiltonian connected for q 2 fi; 2; . . . ; d 2g. Let Pibe
the hamiltonian path of Kiðd; tÞ joining u to oi;3; let P3be the hamiltonian path of K3ðd; tÞ F3joining o3;ito o3;j; let P4be the
hamiltonian path of K4ðd; tÞ F4joining o4;jto o4;5if d P 7 and let P4be the hamiltonian path of K4ðd; tÞ F4joining o4;jto
o4;2if d ¼ 6; let Pqbe a hamiltonian path of Kqðd; tÞ Fqjoining oq;q1to oq;qþ1for 4 6 q 6 d 3; let Pd2be the hamiltonian
path of Kd2ðd; tÞ Fd2joining od2;d3to od2;2; let P2be a hamiltonian path of K2ðd; tÞ F2joining o2;d2to o2;1; let P1be a
hamiltonian path of K1ðd; tÞ F1joining o1;2to o1;j. Therefore, hu; Pi;P3;Pj;P4; . . . ;Pd2;P2;P1;
v
i is the required path.Appendix B. Remaining part of Case B1 inLemma 4
In these cases, jFj ¼ jEð eH1ðFÞÞj ¼ d 4. Thus, F contains exactly one of oa
;b, ob;a, or ðoa;b;ob;aÞ if ð
a
;bÞ R Eð eH1ðFÞÞ.Subcase B1.1: eH1ðFÞ is isomorphic to the complete graph K
5 e for any edge e in K5. We label the vertices in K5 as
0; 1; . . . ; 4. SeeFig. 6(a) for illustration. By the definition on eH1ðFÞ; fi; jg – f0; 1g. Obviously, d ¼ 4 and jFj ¼ 0. By the
symmet-ric property of eH1ðFÞ, we may assume that ði; jÞ ¼ ð1; 2Þ or ði; jÞ ¼ ð4; 2Þ.
(a)
(b)
(c)
Fig. 5. Illustrations for Appendix A, (a) Subcase A1.2, (b) Subcase A1.3, and (c) Subcase A1.4.
(a)
(b)
(c)
(d)
(i) ði; jÞ ¼ ð1; 2Þ. Obviously, hi; 4; 0; 3; ji and hi; 3; 0; 4; ji form two orthogonal hamiltonian paths of eH1ðFÞ. ByLemma 1, we
can construct a hamiltonian path between u and v of eK1ðd; t þ 1Þ F unless (1) (u ¼ o
i;4and
v
¼ oj;4) or (2) (u ¼ oi;3andv
¼ oj;3).Suppose that u ¼ oi;4 and
v
¼ oj;4. Obviously, hi; 3; 4; 0; ji is a hamiltonian path in eH1ðFÞ satisfying the boundaryconditions: u – oi;3 and
v
–oj;0. Suppose that u ¼ oi;3 andv
¼ oj;3. Obviously, hi; 4; 3; 0; ji is a hamiltonian path ine
H1ðFÞ satisfying the boundary conditions: u – o
i;4and
v
–oj;0.(ii) ði; jÞ ¼ ð4; 2Þ. Obviously, hi; 0; 3; 1; ji and hi; 1; 3; 0; ji form two orthogonal hamiltonian paths of eH1ðFÞ. ByLemma 1, we
can construct a hamiltonian path between u and v of eK1ðd; t þ 1Þ F unless (1) (u ¼ o
i;0and
v
¼ oj;0) or (2) (u ¼ oi;1andv
¼ oj;1).Suppose that u ¼ oi;0 and
v
¼ oj;0. Let Pi be the hamiltonian path of Kiðd; tÞ fug joining oi;3 to oi;1; let P3 be thehamiltonian path of K3ðd; tÞ joining o3;0 to o3;2; let P1be the hamiltonian path of K1ðd; tÞ joining o1;2to o1;4;Pjbe the
hamiltonian path of Kjðd; tÞ joining oj;1 to v. Therefore, path hu; x; P3;Pi;P1;Pj;
v
i is the required path. Suppose thatu ¼ oi;1and
v
¼ oj;1. Let Pibe the hamiltonian path of Kiðd; tÞ fug joining oi;3to oi;0; let P3be the hamiltonian pathof K3ðd; tÞ joining o3;1to o3;2; let P1be the hamiltonian path of K1ðd; tÞ joining o1;ito o1;3; Pjbe the hamiltonian path
of Kjðd; tÞ joining oj;0to v. Therefore, path hu; P1;P3;Pi;x; Pj;
v
i is the required path.Subcase B1.2: The subgraph N of eH1ðFÞ induced by Vð eH1ðFÞÞ fi; jg is a complete graph; vertex i is adjacent to j and all
vertices in N; j is adjacent to i and exactly two vertices, say x1and x2, in N. Since dege
H1ðFÞð0Þ < d; x1or x2is not vertex 0. We
label the remaining vertices in N as x3; . . . ;xd1. SeeFig. 6(b) for illustration. It is easy to see that hi; x2;x3; . . . ;xd1;x1;ji and
hi; x3; . . . ;xd1;x1;x2;ji form two orthogonal hamiltonian paths of eH1ðFÞ between i and j. ByLemma 1, we can construct a
ha-miltonian path between u and
v
of eK1ðd; t þ 1Þ F unless (1) (u ¼ oi;x2and
v
¼ oj;x2) or (2) (u ¼ oi;x3andv
¼ oj;x1).Suppose that u ¼ oi;x2 and
v
¼ oj;x2. Obviously, hi; x3;x2;x4; . . . ;xd1;x1;ji is a hamiltonian path in eH1ðFÞ satisfying the
boundary conditions: u – oi;x3and
v
–oj;x1. Suppose that u ¼ oi;x3 andv
¼ oj;x1. The hamiltonian path hi; x1;xd1; . . . ;x3;x2;jiin eH1ðFÞ satisfying the boundary conditions: u – o
i;x1and
v
–oj;x2. ByLemma 1, we can construct a hamiltonian pathsbe-tween u and
v
of eK1ðd; t þ 1Þ F.Subcase B1.3: The subgraph N of eH1ðFÞ induced by Vð eH1ðFÞÞ fi; jg is isomorphic to C
2;5; vertex i is adjacent to j and all
the vertices in N; j is adjacent to i and all the vertices in N. We label the vertices of C2;5as indicated inFig. 6(c). Obviously,
d ¼ 6 and jFj ¼ 2. Thus, jEðNÞj ¼ 3 and degNðx1Þ ¼ degNðx3Þ ¼ degNðx5Þ ¼ 2. It is easy to see that hi; x1;x2;x3;x4;x5;ji and
hi; x5;x4;x3;x2;x1;ji form two orthogonal hamiltonian paths of eH1ðFÞ between i and j. ByLemma 1, we can construct a
hamil-tonian paths between u and
v
of eK1ðd; t þ 1Þ F unless (1) (u ¼ oi;x1and
v
¼ oj;x1) or (2) (u ¼ oi;x5andv
¼ oj;x5). By thesym-metric property of eH1ðFÞ, we only consider the case u ¼ o
i;x1and
v
¼ oj;x1.Since jFij ¼ jFjj ¼ 0; Kjðd; tÞ ðFj[ foj;x1gÞ is hamiltonian connected. Let Pjbe the hamiltonian path of Kjðd; tÞ ðFj[ foj;x1gÞ
joining oj;x5 to oj;x3. Let l be the index that xl is vertex 0 in N. By induction, Kqðd; tÞ Fq is hamiltonian connected for
q 2 fi; x1; . . . ;x5g fxlg. Let Pi be the hamiltonian path of Kiðd; tÞ Fi joining u to oi;x2; let P1 be the hamiltonian path of
K1ðd; tÞ F1 joining ox1;x4 to ox1;j if l – 1 and P1¼ fxg if otherwise; let P2 be the hamiltonian path of K2ðd; tÞ F2 joining
ox2;ito ox2;x5; let P3be the hamiltonian path of K3ðd; tÞ F3joining ox3;j to ox3;x4if l – 3 and P3¼ fxg if otherwise; let P4be
the hamiltonian path of K4ðd; tÞ F4joining ox4;x3 to ox4;x1; let P5 be the hamiltonian path of K5ðd; tÞ F5 joining ox5;x2 to
ox5;jif l – 5 and P5¼ fxg if otherwise. Therefore, path hu; Pi;P2;P5;Pj;P3;P4;P1;
v
i is the required path.Subcase B1.4: The subgraph N of eH1ðFÞ induced by Vð eH1ðFÞÞ fi; jg is isomorphic to C
1;n2; vertex i is adjacent to j and all
the vertices in N; j is adjacent to i and all the vertices in N. We label the vertices of C1;n2as indicated inFig. 6(d). Obviously,
EðNÞ ¼ d 3 and degNð1Þ ¼ d 3. It is easy to see that hi; x1;x2; . . . ;xd1;ji and hi; xd1;xd2; . . . ;x1;ji form two orthogonal
ha-miltonian paths of eH1ðFÞ between i and j. By Lemma 1, we can construct a hamiltonian paths between u and
v
ine
K1ðd; t þ 1Þ F unless (1) (u ¼ o
i;xd1 and
v
¼ oj;xd1) or (2) (u ¼ oi;x1andv
¼ oj;x1).Suppose that u ¼ oi;xd1 and
v
¼ oj;xd1. Obviously, hi; x3;xd1; . . . ;x4;x2;x1;ji is a hamiltonian path in eH1ðFÞ satisfying the
boundary conditions: u – oi;x3 and
v
–oj;x1. By Lemma 1, we can construct a hamiltonian paths between u andv
ine
K1ðd; t þ 1Þ F.
Suppose that u ¼ oi;x1and
v
¼ oj;x1. Since jFij ¼ jFjj ¼ 0; Kjðd; tÞ foj;x1g is hamiltonian connected. Let Pjbe the hamiltonianpath of Kjðd; tÞ foj;x1g joining oj;x3to oj;x4. Let l be the index that xlis vertex 0 in N. By induction, Kqðd; tÞ Fqis hamiltonian
connected for q 2 fi; x2; . . . ;xd1g fxlg. Let Pibe the hamiltonian path of Kiðd; tÞ joining u to oi;x3; let P3be the hamiltonian
path of K3ðd; tÞ F3joining ox3;ito ox3;j if l – 3 and P3¼ fxg if otherwise. Suppose l – 4, let P4be the hamiltonian path of
K4ðd; tÞ F4joining ox4;jto ox4;x5if d P 7 and let P4be the hamiltonian path of K4ðd; tÞ F4joining ox4;jto ox4;x2if d ¼ 6.
Sup-pose l ¼ 4, let P4¼ fxg. Let Pq be a hamiltonian path of Kqðd; tÞ Fq joining oxq;xq1to oxq;xqþ1 for 4 6 q 6 d 2 if l – q and
Pq¼ fxg if otherwise; let Pd1 be the hamiltonian path of Kd1ðd; tÞ Fd1 joining oxd1;xd2 to oxd1;x2 if l – d 1 and
Pd1¼ fxg if otherwise; let P2be a hamiltonian path of K2ðd; tÞ F2joining ox2;xd1to ox2;x1; let P1be a hamiltonian path of
K1ðd; tÞ F1 joining ox1;x2 to ox1;j if l – 1 and P1¼ fxg if otherwise. Therefore, path hu; Pi;P3;Pj;P4; . . . ;Pd1;P2;P1;
v
i is theE85-A (2002) 1359–1370.
[10] W.T. Huang, J.M. Tan, C.N. Hung, L.H. Hsu, Fault-tolerant hamiltonicity of twisted cubes, J. Paral. Distrib. Comput. 62 (2002) 591–604. [11]C.N. Hung, H.C. Hsu, K.Y. Liang, L.H. Hsu, Ring embedding in faulty pancake graphs, Inform. Process. Lett. 86 (2003) 271–275. [12] INMOS Limited, Transputer Reference Manual, Prentice-Hall, Upper Saddle River, NJ, 1988.
[13]O. Ore, Coverings of graphs, Ann. Mat. Pura Appl. 55 (1961) 315–321.
[14]C.H. Tsai, Y.C. Chuang, J.M. Tan, L.H. Hsu, Hamiltonian properties of faulty recursive circulant graphs, J. Interconnect. Netw. 3 (2002) 273–289. [15]G.D. Vecchia, C. Sanges, Recursively scalable networks for message passing architectures, in: E. Chiricozzi, A. DAmico (Eds.), Parallel Processing and
