On: 24 April 2014, At: 18:27 Publisher: Taylor & Francis
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
International Journal of Computer
Mathematics
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/gcom20
Fault-free mutually independent
Hamiltonian cycles of faulty star graphs
Tzu-Liang Kung a , Cheng-Kuan Lin b , Tyne Liang b , Jimmy J.M. Tan b & Lih-Hsing Hsu ca
Department of Computer Science and Information Engineering , Asia University , 500 Lioufeng Road, Taichung, 41354, Taiwan, Republic of China
b
Department of Computer Science , National Chiao Tung University , 1001 University Road, Hsinchu, 30010, Taiwan, Republic of China
c
Department of Computer Science and Information Engineering , Providence University , 200 Chung Chi Road, Taichung, 43301, Taiwan, Republic of China
Published online: 23 Dec 2010.
To cite this article: Tzu-Liang Kung , Cheng-Kuan Lin , Tyne Liang , Jimmy J.M. Tan & Lih-Hsing Hsu (2011) Fault-free mutually independent Hamiltonian cycles of faulty star graphs, International Journal of Computer Mathematics, 88:4, 731-746, DOI: 10.1080/00207161003786614
To link to this article: http://dx.doi.org/10.1080/00207161003786614
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content.
substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions
Vol. 88, No. 4, March 2011, 731–746
Fault-free mutually independent Hamiltonian cycles of faulty
star graphs
Tzu-Liang Kunga*, Cheng-Kuan Linb, Tyne Liangb, Jimmy J.M. Tanband Lih-Hsing Hsuc aDepartment of Computer Science and Information Engineering, Asia University, 500 Lioufeng Road, Taichung 41354, Taiwan, Republic of China;bDepartment of Computer Science, National Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan, Republic of China;cDepartment of Computer
Science and Information Engineering, Providence University, 200 Chung Chi Road, Taichung 43301, Taiwan, Republic of China
(Received 24 April 2009; revised version received 08 February 2010; accepted 15 March 2010)
The star graph interconnection network has been recognized as an attractive alternative to the hypercube for its nice topological properties. Unlike previous research concerning the issue of embedding exactly one Hamiltonian cycle into an injured star network, this paper addresses the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty star network. To be precise, let SGndenote an n-dimensional star network in which f≤ n − 3 edges may fail accidentally. We show that there exist
(n− 2 − f )-mutually independent Hamiltonian cycles rooted at any vertex in SGnif n∈ {3, 4}, and there
exist (n− 1 − f )-mutually independent Hamiltonian cycles rooted at any vertex in SGnif n≥ 5. Keywords: Hamiltonian; interconnection network; star graph; fault tolerance
2000 AMS Subject Classifications: 05C38; 05C45; 05C75; 05C90; 68M10
1. Introduction
The problem of finding Hamiltonian cycles in a graph is well known to be NP-complete and has been discussed in many areas. In 1969, Lovasz [27] asked whether every finite connected vertex-transitive graph has a Hamiltonian path, that is, a simple path that traverses every vertex exactly once.
Definition1 [4] A graph is said to be vertex-transitive if for every pair u, v of vertices, there
exists an automorphism of the graph that maps u into v. A graph is said to be edge-transitive if for any two edges a and b, there exists an automorphism of the graph that maps a into b.
All known vertex-transitive graphs have a Hamiltonian path, but only four vertex-transitive graphs without any Hamiltonian cycle are known to exist. Since none of these four graphs
*Corresponding author. Email: tlkung@asia.edu.tw
ISSN 0020-7160 print/ISSN 1029-0265 online © 2011 Taylor & Francis
DOI: 10.1080/00207161003786614 http://www.informaworld.com
is a Cayley graph, there is a folklore conjecture [6] that every Cayley graph with more than two vertices has a Hamiltonian cycle. In the last decades, this problem was extensively studied [2,3,5–7,10,11,17–19,28–30,35]. For those Cayley graphs for which the existence of Hamiltonian cycles has already been proved, more advanced properties, such as edge-Hamiltonicity, Hamil-tonian connectivity, and HamilHamil-tonian laceability, etc., are investigated [2,22]. In this paper, we address one of such properties, the concept of mutually independent Hamiltonian cycles [36,37], which is related to the number of Hamiltonian cycles in a given graph. Since its introduction, this topic has gained many researchers’ attention [12,15,16,25,26,33]. In particular, Lin et al. [25] showed that the maximum number of mutually independent Hamiltonian cycles rooted at any vertex can be constructed recursively in the star graph interconnection network (for the detailed definitions, see Sections 2 and 3).
The interconnection network is of great interest in the area of parallel and distributed com-puter systems. Because it is usually multi-objected and complicated to design an interconnection network, its underlying topology can be modelled as a graph, whose vertices correspond to pro-cessors and whose edges correspond to connections/communication links. Hence, we use the terms graphs and networks interchangeably. Among various kinds of network topologies, the star graph is attractive for its high degree of symmetry. However, when some edges are removed at random from the star graph, the symmetry will be broken. Hence, we wonder, in a theo-retical point of view, how many mutually independent Hamiltonian cycles can be formed in such an injured network. In this paper, the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty star graph will be studied. To be precise, let SGndenote an n-dimensional star graph with f ≤ n − 3 faulty edges. Then we aim at proving the following result: SGn has (n− 2 − f )-mutually (respectively, (n − 1 − f )-mutually) independent Hamiltonian cycles rooted at any vertex if n∈ {3, 4} (respectively, n ≥ 5).
The rest of this paper is organized as follows. In Section 2, graph-theoretic notations and the definition of mutually independent Hamiltonian cycles are introduced. In Section 3, the star graph and its basic properties are presented. Section 4 consists of the proof of our main result. Finally, directions for future research are discussed in Section 5.
2. Preliminaries
Throughout this paper, graphs are simple, loopless, and undirected. For definitions and notations not defined here, see [4]. A graph G is an ordered pair (V (G), E(G)), where V (G) is a non-empty set, and E(G) is a subset of{{u, v}|{u, v} is a two-element subsets of V (G)}. The set V (G) is called the vertex set of G, and the set E(G) is called the edge set of G. Two vertices u and v of
Gare adjacent if{u, v} ∈ E(G). The degree of a vertex u in G is the number of edges incident to u. A graph G is k-regular if all its vertices have the same degree k. A graph G is bipartite if its vertex set can be partitioned into two disjoint subsets, denoted by V0(G)and V1(G), such that every edge joins a vertex of V0(G)to a vertex of V1(G).
A graph H is a subgraph of a graph G if V (H )⊆ V (G) and E(H ) ⊆ E(G). Let S be a non-empty subset of V (G). The subgraph of G induced by S is a subgraph of G with vertex set S, whose edge set consists of all the edges joining any two vertices in S. We use G− S to denote the subgraph of G induced by V (G)− S. Let F be any subset of E(G). Then we use G − F to denote the subgraph of G with vertex set V (G) and edge set E(G)− F . For any S ⊆ V (G) and
F ⊆ E(G), graph G − (S ∪ F ) is defined to be the graph (G − F ) − S.
A walk of length k≥ 1 in a graph is a sequence of vertices, W := v1v2· · · vk+1, such that viand vi+1are adjacent for i= 1, 2, . . . , k. If v1= x and vk+1 = y, we refer to W as an xy-walk. The notation xWy is also used simply to signify an xy-walk W . Moreover, we use W−1to denote the reversed walk vk+1vk· · · v1. For any three vertices x, y, z in a graph, if xW1yand yW2zare walks,
the sequence x W1y W2z, obtained by concatenating W1and W2at y, is a walk. A walk of length 0 consists of a single vertex. A path is a walk in which no vertex is repeated. For convenience, the
ith vertex of a path P is denoted by P (i). For any two vertices u and v in a graph G, the distance between u and v, denoted by dG(u, v), is the length of the shortest path between u and v. A cycle is a walk v1v2· · · vn+1in which n≥ 3, v1 = vn+1, and the n vertices v1, v2, . . . , vnare distinct.
A path (or cycle) in a graph G is a Hamiltonian path (or Hamiltonian cycle) of G if it spans G. A graph is Hamiltonian if it has a Hamiltonian cycle. A bipartite graph is Hamiltonian laceable [34] if there exists a Hamiltonian path between any two vertices that are in different partite sets. A Hamiltonian laceable graph G is said to be hyper-Hamiltonian laceable [22] if, for any i∈ {0, 1} and for any vertex v∈ Vi(G), there exists a Hamiltonian path in G− {v} between any two vertices of V1−i(G).
Let G be a graph with N vertices. A rooted Hamiltonian cycle C in G can be described as v1 v2· · · vNv1to emphasize the order of vertices on C. Accordingly, v1is seen as the root vertex, and viis seen as the ith vertex on C. Two Hamiltonian cycles rooted at a given vertex s of G, namely C1:= v1v2· · · vNv1and C2:= u1u2· · · uNu1 with v1= u1= s, are independent if vi = ui for 2≤ i ≤ N. A collection of m Hamiltonian cycles C1, . . . , Cmin G, rooted at the same vertex, are said to be m-mutually independent if Ci and Cj are independent whenever i= j. Moreover, the
mutually independent Hamiltonicity of G, denoted byIHC(G), is defined to be the maximum
integer m such that for any vertex v of G, there exists a set of m-mutually independent Hamiltonian cycles rooted at v in G. The concept of mutually independent Hamiltonian cycles can be applied in many different areas [12,15,16,25,26,33].
3. The star graph
The hypercube has long been one of the most popular network topologies [21] because of its nice topological properties. The star graph, proposed by Akers and Krishnamurthy [1], is an attractive alternative to the hypercube for interconnecting processors in parallel computers. Since then, star networks have received many researchers’ attention. For example, the diameter and fault diameter were computed in [1,20,32]. Moreover, Fragopoulou and Akl [8,9] studied how to embed directed edge-disjoint spanning trees into the star graph. The Hamiltonian properties of star graphs are addressed in [13,14,23,38]. In particular, because processors or links may fail accidentally to affect network performance, Tseng et al. [38] addressed fault-tolerant ring embedding in an injured star network if no more than n− 3 edge faults occur.
The definition of star graphs is described as follows. Let n be any positive integer. For conve-nience, we useInto denote the set{1, 2, . . . , n}. A permutation u1u2· · · unonInis a sequence of all elements ofIn. Every permutation can be written as a product of transpositions. An even
permutation (respectively, odd permutation) is a permutation that can be written as a product
of an even (respectively, odd) number of transpositions. The n-dimensional star graph SGnis a graph whose vertex set is the set of all permutations onIn. Two vertices, u1· · · ui· · · un and v1· · · vi· · · vn, are adjacent through an edge of dimension i with 2≤ i ≤ n if u1= vi, v1= ui, and uj = vj for j ∈ In− {1, i}. Clearly, SGnis (n− 1)-regular with n! vertices. Moreover, it is precisely a Cayley graph of the symmetric group with edge set consisting of all the transpositions of form (1 i), where 2≤ i ≤ n. So it is vertex-transitive and edge-transitive [1]. The star graphs SG2, SG3, and SG4are illustrated in Figure 1.
For the sake of clarity, we use boldface letters to denote vertices of SGn. Moreover, we use e to denote the vertex 12, . . . , n. It is known that SGnis a bipartite graph with one partite set V0(SGn) consisting of all the even permutations and the other partite set V1(SGn)consisting of all the odd permutations. Let u= u1u2· · · unbe a vertex in SGn. Then uiis the ith coordinate of u, denoted
1423 2413 4213 3214 2134 1234 1342 3142 4132 c d f g 2314 1324 3124 a b e 4123 2143 1243 4231 3241 2431 3412 4312 1432 b c e f 2341 4321 3421 a d g 123 213 321 312 231 132 12 21 SG2 SG3 SG4
Figure 1. Illustrations for SG2, SG3, and SG4.
by (u)i, for 1≤ i ≤ n. For any 2 ≤ i ≤ n, the i-neighbour of vertex u, denoted by (u)i, is a vertex adjacent to u through an edge of dimension i. Obviously, ((u)i)i = u.
For any 1≤ i ≤ n, let SG{i}n denote the subgraph of SGn induced by the set of ver-tices {u ∈ V (SGn)|(u)n= i}. Then SGn can be partitioned into n vertex-disjoint subgraphs SG{1}n , aSG{2}n , . . . ,SGn{n}, and every of them is isomorphic to SGn−1. Let I ⊆ In. We use SGIn to denote the subgraph of SGninduced by∪i∈IV (SG{i}n ). For any pair i, j of distinct integers in
In, we use Ei,j to denote the set of edges between SG{i}n and SG{j}n . In the rest of this section, we introduce some results to be used later.
Theorem1 [38] Let F ⊂ E(SGn)with|F | ≤ n − 3 for n ≥ 3. Then SGn− F is Hamiltonian.
Li et al. [23] introduced the edge-fault-tolerant Hamiltonian laceability of a bipartite graph G, which is the integer f such that for any F ⊆ E(G) with |F | ≤ f , G − F is still Hamiltonian lace-able and there exists a subset F of E(G) with|F | = f + 1 such that G − F is not Hamiltonian laceable. Moreover, they also defined the edge-fault-tolerant hyper-Hamiltonian laceability of a graph G as the integer f such that for any F ⊆ E(G) with |F | ≤ f , G − F is hyper-Hamiltonian laceable and there exists a subset F of E(G) with|F | = f + 1 such that G − F is no longer Hyper-Hamiltonian laceable.
Theorem2 [23] The star graph SGnis (n− 3)-edge-fault-tolerant Hamiltonian laceable and (n− 4)-edge-fault-tolerant hyper-Hamiltonian laceable for n ≥ 4.
Lemma1 [31] Assume that n≥ 3. Then |Ei,j| = (n − 2)! for any 1 ≤ i = j ≤ n. Moreover,
there are (n− 2)!/2 pairwise disjoint edges joining vertices of Vt(SG{i}n )to vertices of V1−t(SG{j}n )
for any t∈ {0, 1}.
Lemma2 For n≥ 3, let u and v be two distinct vertices of SGnwith dSG
n(u, v)≤ 2. Then
(u)1= (v)1.
Lemma3 Let n≥ 5 and F ⊂ E(SGn)with|F | ≤ n − 4. Assume that I = {a1, . . . , ar} is an
r-element subset ofInfor any r∈ In. Suppose that u∈ Vt(SGn{a1})and v∈ V1−t(SG{anr})for any
t ∈ {0, 1}. Then there exists a Hamiltonian path H := x1P1y1x2P2y2· · · xrPryrin SGIn− F such
that x1= u, yr = v, and Piis a Hamiltonian path of SGn{ai}− F joining xito yifor every 1≤ i ≤ r.
Proof Without loss of generality, we can assume that t= 0. Since SG{a1}
n is isomorphic to SGn−1, this statement holds for r= 1 by Theorem 2. Thus, suppose that r ≥ 2 and set x1= u and yr= v. By Lemma 1, there are ((n − 2)!/2) > n − 4 pairwise disjoint edges joining vertices of V1(SG{ani})to vertices of V0(SGn{ai+1})for every i∈ Ir−1. Therefore, we can choose{yi,xi+1} ∈ Eai,ai+1− F with y
i ∈ V1(SG{ani})and xi+1∈ V0(SG{ani+1})for i∈ Ir−1. By Theorem 2, SG{ani}− F has a Hamiltonian path Pijoining xi to yifor every i∈ Ir. As a result, the sequence of vertices, x1P1y1x2P2y2· · · xrPr yr, forms a desired Hamiltonian path of SGIn− F joining u to v.
Lemma4 Let n≥ 5. Assume that F ⊂ E(SGn)with|F | ≤ n − 4, and |F ∩ SG{i}n | ≤ n − 5 for
every i∈ In. Moreover, assume that I = {a1, . . . , ar} is an r-element subset of Infor any 2≤ r ≤
n. Suppose that u∈ Vt(SG{an1}), w∈ V1−t(SGn{a1}), and v∈ Vt(SG{anr})for any t∈ {0, 1}. Then
there exists a Hamiltonian path H of (SGIn− F ) − {w} joining u to v.
Proof Without loss of generality, we can assume that t = 0. By Lemma 1, there are (n − 2)!/2 >
n− 3 pairwise disjoint edges joining vertices of V0(SG{an1})to vertices of V1(SG{an2}). Thus, we can choose a vertex x of V0(SG{an1})− {u} with (x)1= a2and{x, (x)n} /∈ F . By Theorem 2, there exists a Hamiltonian path P of (SG{a1}
n − F ) − {w} joining u to x. By Lemma 3, there exists a Hamiltonian path Q of SGI−{a1}
n − F joining (x)
nto v. As a result, the sequence of vertices, u P
x (x)nQv, forms a desired Hamiltonian path.
Lemma5 [24] Let w and b denote two adjacent vertices of SGnwith n≥ 4. For any vertex u
in Vt(SGn)− {w, b}, t ∈ {0, 1}, and for any i ∈ In, there exists a Hamiltonian path P of SGn−
{w, b} joining u to some vertex v in V1−t(SGn)− {w, b} with (v)1= i.
Lemma6 Let i∈ Inand F ⊂ E(SGn)with|F | ≤ n − 4 for n ≥ 4. Suppose that w and b are
two adjacent vertices of SGn, and u∈ Vt(SGn)− {w, b} for any t ∈ {0, 1}. Then there exists a
Hamiltonian path of (SGn− F ) − {w, b} joining u to some vertex v of V1−t(SGn)− {w, b} with
(v)1= i.
Proof Without loss of generality, we can assume that t = 0. Since SGnis vertex-transitive, we
can assume that w= e and b = (e)j with some j∈ In− {1}. We set Fk= F ∩ E(SG{k}n )for every k∈ In. The proof is done by induction on n. The induction basis, that is, the case n= 4, follows from Lemma 5. Suppose that this statement holds for SGn−1with n≥ 5. We consider the dimensions of all edges in F∪ {{e, (e)j}}. If there is an edge in F whose dimension, say j , is different from j , then SGncan be partitioned into n vertex-disjoint subgraphs with the j th coordinate of each vertex (that is, the subgraph of SGninduced by the vertices with the same j th coordinate is SGn−1). Otherwise, every edge of F has the same dimension j .
Case 1 The dimension j exists. Without loss of generality, we can assume that j = n. Thus,
we have{e, (e)j} ∈ E(SG{n}n )and|Fk| ≤ n − 5 for every k ∈ In.
Subcase 1.1 Suppose that u∈ V0(SG{n}n ). Since|F | ≤ n − 4, we can choose an integer r ∈
In−1 such that|F ∩ Er,n| = 0. By the induction hypothesis, there exists a Hamiltonian path P of (SG{n}n − Fn)− {e, (e)j} joining u to a vertex x ∈ V1(SG{n}n )with (x)1= r. We can choose a vertex v in V1(SGInn−1−{r})with (v)1= i. By Lemma 3, there exists a Hamiltonian path Q of SGIn−1
n − F joining (x)nto v. Then the sequence of vertices, uP x(x)nQv, is a desired path.
Subcase 1.2 Suppose that u∈ V0(SG{k}n )for some k∈ In−1. By Lemma 1, there are (n− 2)!/2 > n − 3 pairwise disjoint edges joining vertices of V1(SG{k}n )to vertices of V0(SG{n}n ). We can pick out a vertex y of V1(SG{k}n )such that (y)n∈ V0(SG{n}n )− {e} and {y, (y)n} /∈ F . By Theorem 2, there exists a Hamiltonian path H of SG{k}n − Fk joining u to y. We can choose an integer r ofIn−1− {k} such that |F ∩ Er,n| = 0. By the induction hypothesis, there exists a Hamiltonian path P of (SG{n}n − Fn)− {e, (e)j} joining (y)nto a vertex x of V1(SG{n}n )− {(e)j} with (x)1= r. Besides, we choose a vertex v of V1(SGInn−1−{k,r})with (v)1= i. By Lemma 3, there exists a Hamiltonian path Q of SIn−1−{k}
n − F joining (x)nto v. Then the sequence of vertices, u
Hy (y)nP x (x)nQv, turns out to be a desired path.
Case 2 Every edge in F has the same dimension j . Without loss of generality, we may assume
that j= n. Thus, we have |Ft| = 0 for every t ∈ In.
Subcase 2.1 Suppose that u∈ V0(SG{k}n )for some k∈ In−1− {1}. By Lemma 1, there are
(n− 2)!/2 > n − 4 pairwise disjoint edges joining vertices of V1(SG{k}n )to vertices of V0(SG{1}n ). Thus, we can choose a vertex x of V1(SG{k}n )with (x)1= 1 and {x, (x)n} /∈ F . By Theorem 2, there exits a Hamiltonian path H of SG{k}n joining u to x. Similarly, we can choose a vertex y of V0(SG{1}n ) with (y)1 = n, {y, (y)n} /∈ F , and y = (x)n. This can be done because there are ((n− 2)!)/2 ≥ n− 2 pairwise disjoint edges between the sets V0(SG{1}n )and V1(SG{k}n ). By Theorem 2, SG{1}n −
{(e)n} has a Hamiltonian path P joining (x)n to y. Let v be a vertex in V
1(SGInn−1−{1,k})with (v)1= i. By Lemma 4, there exists a Hamiltonian path Q of (SGInn−{1,k}− F ) − {e} joining (y)
n to v. Then the sequence of vertices, u H x (x)nP y (y)nQv, turns out to be a desired path.
Subcase 2.2 Suppose that u∈ V0(SG{1}n ). By Lemma 1, there are (n− 2)!/2 > n − 4 pairwise
disjoint edges joining vertices of V0(SG{1}n )to vertices of V1(SG{n}n ). Thus, we can choose a vertex
x of V0(SG{1}n )− {u} with (x)1= n and {x, (x)n} /∈ F . By Theorem 2, there exists a Hamiltonian
path H of SG{1}n − {(e)n} joining u to x. Furthermore, we choose a vertex v of V1(SGInn−1−{1})with (v)1= i. By Lemma 4, there exists a Hamiltonian path Q of (SGInn−{1}− F ) − {e} joining (x)
n to v. Then the sequence of vertices, u H x (x)nQv, forms a desired path.
Subcase 2.3 Suppose that u∈ V0(SG{n}n ). Since |F | ≤ n − 4, we can choose two
inte-gers k1 and k2 in In−1− {1} such that {(e)k1, ((e)k1)n} /∈ F and {(e)k2, ((e)k2)n} /∈ F . Let X= {{e, (e)t}|t ∈ In−1− {1, k1, k2}}. Obviously, |X| = n − 4. Moreover, we can choose a ver-tex x∈ V1(SG{n}n ) such that (x)1∈ In−1− {1, k1, k2} and {x, (x)n} /∈ F . Since (x)1= k1 and (x)1= k2, we have x= (e)k1 and x= (e)k2. By Theorem 2, there exists a Hamiltonian path Hof SG{n}n − X joining u to x. Because vertex e has precisely two neighbours, that is, (e)k1and
(e)k2, in SG{n}
n − X, the edges {e, (e)k1} and {e, (e)k2} must be consecutive on H . Thus, with no loss of generality, we can write H = u H1(e)k1e(e)k2H2x. Let y= (e)k2. Since (y)1= (x)1, we have i= (x)1or i= (y)1.
Subcase 2.3.1 Suppose that i = (x)1. Let k3= (x)1. We choose a vertex v of V1(SG{kn3})with
(v)1= i. By Lemma 1, there are (n − 2)!/2 > n − 4 pairwise disjoint edges joining vertices of V1(SG{kn1})to vertices of V0(SGn{1}). Thus, we can choose a vertex z of V1(SG{kn1})with (z)1= 1 and{z, (z)n} /∈ F . By Theorem 2, there exists a Hamiltonian path T of SG{k3}
n joining (x)nto v. Similarly, there exist a Hamiltonian path P of SG{k1}
n joining ((e)k1)nto z. By Lemma 4, there exists a Hamiltonian path Q of (SGIn−1−{k1,k3}
n − F ) − {(e)n} joining (z)nto (y)n. Then the sequence of vertices, u H1(e)k1((e)k1)nP z (z)nQ (y)ny H2x (x)nT v, is a desired one.
Subcase 2.3.2 Suppose that i= (y)1. Let k3= (y)1. Then the proof of this case happens to
be similar to that of Subcase 2.3.1. Thus, we omit the details.
Lemma7 Let{a, b} ⊂ Inwith a < b, and let F ⊂ E(SGn)with|F | ≤ n − 4 for n ≥ 4. Suppose
that x∈ V0(SGn), and x1and x2are two distinct neighbours of x. Then there exists a Hamiltonian
path of (SGn− F ) − {x, x1,x2} between two vertices u and v in V0(SGn)− {x} such that (u)1= a
and (v)1= b.
Proof Since SGnis vertex-transitive, we can assume that x= e, x1= (e)i1, and x2= (e)i2with
some{i1, i2} ⊂ {2, 3, . . . , n}. Then this lemma is proved by induction on n.
Suppose that n= 4. Thus, we have |F | = 0. Since SG4is edge-transitive, we can assume that x1= (e)2= 2134 and x2= (e)3= 3214. The required paths of SG4− {1234, 2134, 3214} are listed in Table 1.
Suppose that the statement holds for SGn−1 with n≥ 5. Let Fk = F ∩ E(SG{k}n ) for every k∈ In. Without loss of generality, suppose that F contains at least one edge of dimension n. Thus, we have|Fk| ≤ n − 5 for every k ∈ In. Because a < b, we have a= n and b = 1. Since
|F | ≤ n − 4, we can choose an integer c in In−1− {1, a} such that |F ∩ Ec,n| = 0. Moreover, we can choose a vertex v of V0(SG{1}n )with (v)1= b.
Case 1 Suppose that i1= n and i2= n. By the induction hypothesis, there exists a
Hamil-tonian path H of (SG{n}n − Fn)− {e, (e)i1, (e)i2} joining a vertex u of V0(SG{n}n )with (u)1= a to a vertex y of V0(SG{n}n )with (y)1= c. By Lemma 3, there exists a Hamiltonian path R of SGIn−1
n − F joining (y)
nto v. As a result, the sequence of vertices, u H y (y)nRv, forms a desired path in (SGn− F ) − {e, (e)i1, (e)i2}.
Case 2 Either i1= n or i2= n. Without loss of generality, we can assume that i2= n. We
choose a vertex u∈ V0(SG{n}n )with (u)1= a. By Lemma 6, there exists a Hamiltonian path H of (SG{n}n − Fn)− {e, (e)i1} joining a vertex u to some vertex y of V1(SG{n}n )with (y)1= c. By Lemma 4, there exists a Hamiltonian path Q of (SGIn−1
n − F ) − {(e)
n} joining (y)n to v. As a
result, the sequence of vertices, u H y (y)nQv, is a desired path.
4. Mutually independent Hamiltonian cycles in faulty star graphs
Lin et al. [25] showed the next theorem.
Theorem3 [25] IHC(SG3)= 1, IHC(SG4)= 2, and IHC(SGn)= n − 1 if n ≥ 5.
For the sake of clarity, our main result, Theorem 4, will be divided into three lemmas (Lemma 8–10).
Lemma8 Let f ∈ E(SG4). ThenIHC(SG4− {f }) = 1.
Proof Since SG4is edge-transitive, we can assume that f = {1234, 4231}. By Theorem 1, there
exists a Hamiltonian cycle in SG4− {f }. Thus, we have IHC(SG4− {f }) ≥ 1. To show that IHC(SG4− {f }) ≤ 1, it suffices to point out that there will be no two-mutually independent Hamiltonian cycles rooted at vertex 1234. In Table 2, we list all Hamiltonian cycles of SG4− {f } rooted at 1234. By brute force, we can check that there do not exist two-mutually independent
Hamiltonian cycles. Hence, the proof is completed.
T .-L. K ung et al.
Table 1. The required Hamiltonian paths in SG4− {1234, 2134, 3214}.
a= 1 and b = 2 1324 3142 4132 1432 3412 4312 2314 1324 3124 4123 2143 1243 4213 2413 1423 3421 4321 2341 3241 4231 2431 a= 1 and b = 3 1423 2413 4213 1243 2143 4123 3124 1324 2314 4312 3412 1432 4132 3142 1342 2341 4321 3421 2431 4231 3241 a= 1 and b = 4 1324 3142 4132 1432 3412 4312 2314 1324 3124 4123 2143 1243 4213 2413 1423 3421 2431 4231 3241 2341 4321 a= 2 and b = 3 2314 1324 3124 4123 2143 1243 4213 2413 1423 3421 4321 2341 3241 4231 2431 1432 4132 3142 1342 4312 3412 a= 2 and b = 4 2314 1324 3124 4123 2143 1243 4213 2413 1423 3421 4321 2341 3241 4231 2431 1432 3412 4312 1342 3142 4132 a= 3 and b = 4 3124 1324 2314 4312 3412 1432 4132 3142 1342 2341 4321 3421 2431 4231 3241 1243 2143 4123 1423 2413 4213
Table 2. All Hamiltonian cycles rooted at 1234 in SG4− {{1234, 4231}}.
1234 2134 3124 1324 2314 4312 3412 1432 4132 3142 1342 2341 4321 3421 2431 4231 3241 1243 2143 4123 1423 2413 4213 3214 1234 1234 2134 3124 1324 4321 2341 3241 4231 2431 3421 1423 4123 2143 1243 4213 2413 3412 1432 4132 3142 1342 4312 2314 3214 1234 1234 2134 3124 4123 1423 2413 4213 1243 2143 3142 4132 1432 3412 4312 1342 2341 3241 4231 2431 3421 4321 1324 2314 3214 1234 1234 2134 4132 1432 2431 4231 3241 1243 2143 3142 1342 2341 4321 3421 1423 4123 3124 1324 2314 4312 3412 2413 4213 3214 1234 1234 2134 4132 3142 1342 4312 3412 1432 2431 4231 3241 2341 4321 3421 1423 2413 4213 1243 2143 4123 3124 1324 2314 3214 1234 1234 2134 4132 3142 2143 4123 3124 1324 2314 4312 1342 2341 4321 3421 1423 2413 3412 1432 2431 4231 3241 1243 4213 3214 1234 1234 3214 2314 1324 3124 4123 2143 1243 4213 2413 1423 3421 4321 2341 3241 4231 2431 1432 3412 4312 1342 3142 4132 2134 1234 1234 3214 2314 1324 4321 3421 2431 4231 3241 2341 1342 4312 3412 1432 4132 3142 2143 1243 4213 2413 1423 4123 3124 2134 1234 1234 3214 2314 4312 1342 3142 4132 1432 3412 2413 4213 1243 2143 4123 1423 3421 2431 4231 3241 2341 4321 1324 3124 2134 1234 1234 3214 4213 2413 1423 4123 2143 1243 3241 4231 2431 3421 4321 2341 1342 3142 4132 1432 3412 4312 2314 1324 3124 2134 1234 1234 3214 4213 2413 3412 4312 2314 1324 3124 4123 1423 3421 4321 2341 1342 3142 2143 1243 3241 4231 2431 1432 4132 2134 1234 1234 3214 4213 1243 3241 4231 2431 1432 3412 2413 1423 3421 4321 2341 1342 4312 2314 1324 3124 4123 2143 3142 4132 2134 1234
Lemma9 Suppose that n≥ 5 and F ⊂ E(SGn)with|F | = n − 3. Let u ∈ V (SGn). Then there
exist two-mutually independent Hamiltonian cycles rooted at u in SGn− F .
Proof Because SGn is edge-transitive, there exists an automorphism φ1 of SGn mapping any
edge in F into an edge of dimension n. For convenience, let w= φ1(u). Moreover, let φ2: V (SGn)→ V (SGn)be a function defined as follows: φ2(v)= h((v)1)h((v)2), . . . , h((v)n)for any v∈ V (SGn), where h: In→ Inis a function such that h((w)j)= j for each j ∈ In. Clearly, φ2is also an automorphism of SGn. It is easy to check that the composition of φ1and φ2, namely φ2◦ φ1, is an automorphism of SGnsuch that φ2◦ φ1(u)= e. For this reason, we can assume that u= e, and F contains at least one edge of dimension n. Let Fk= F ∩ E(SG{k}n )for every k∈ In. As a result, we have|Fk| ≤ n − 4 for every k ∈ In.
Case 1 Suppose that{e, (e)n} /∈ F . Let B = (b
i,j)be a 2× n matrix with
bi,j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ j if i= 1, n if i= 2 and j = 1, j+ 1 if i = 2 and 2 ≤ j ≤ n − 2, 2 if i= 2 and j = n − 1, 1 if i= 2 and j = n.
By Lemma 3, there exists a Hamiltonian path P of SG
n j=1{b1,j}
n − F joining (e)nto e. Similarly, there exists a Hamiltonian path H of SG
n j=1{b2,j}
n − F joining e to (e)n. Then we set C1 := e
(e)nP e and C
2 := e H (e)n e. Obviously,{C1, C2} forms a set of two-mutually independent Hamiltonian cycles rooted at e in SGn− F (see Figure 2(a) for illustration).
Case 2 Suppose that {e, (e)n} ∈ F and |F
n| = n − 4. Obviously, we have |Fk| = 0 for every k∈ In−1. By Theorem 1, there exists a Hamiltonian cycle H = e R q p e of
(b) (p)5p e q (q)5 P 1 P2 {p} -1 e)i4 ( (a) e 1 e (e)5 (e)5 C1 P SG{ }1 5-F1 2 25 26 49 50 73 74 97 98 1 H 1 24 e 25 48 49 72 73 96 97 120 e 1 C2 R { }5 5- F5 1 23 e C1 ( U ) 24 47 4849 50 71 7273 74 95 96 119120 1 (p)5 Q 4 { }i4 5 3 26 e R 99 1 q (y3) 5 Q 1 75 98 (y2) 5 y 3 Q3 51 72 7374 (q)5 y2 Q2 27 48 4950 1 e 2 p C2 (c) ( )5 e w (w)5 { } R { }5 5- F5 1 23 e C1 ( U ) 24 47 4849 50 71 7273 74 95 96 119120 1 Q4 3 26 e R 99 1 w Q1 75 98 Q3 51 72 7374 (w)5 Q2 27 48 4950 1 e 2 C2 e)i4 ( e)i4 ( ( )5 e)i4 ( e)i4 ( -1 P3 P4 -Fi4 { }2 5-F2 { }3 5-F3 { }4 5-F4 { }5 5-F5 { }5 5-F5 { }3 5-F3 { }4 5-F4 { }2 5-F2 { }1 5-F1 { }i4 5 { }i1 5 { }i2 5 { }i3 5 { }i4 5 { }i1 5 { }i3 5 { }i2 5 -Fi1 { }i1 5 -Fi2 { }i2 5 -Fi3 { }i3 5 -Fi 4 { }i4 5 -Fi1 { }i1 5 -Fi3 { }i3 5 -Fi2 { }i2 5 e) i4 ( { } { }5 5- F(5U (y4) 5 y4 (x1) 5 x1 (x2) 5 x2 (x3) 5 x3 {p} { }5 5- F(5U ) (x1) 5 x1 (x2) 5 x2 (x3) 5 x3 (y3) 5 y3 (y2) 5 y2 (y4) 5 y4 ) P1 P2 P3 P4 SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG
Figure 2. The two-mutually independent Hamiltonian cycles in SG5− F for Lemma 9.
SG{n}n − Fn. Accordingly, we have that {p, (p)n} /∈ F and {q, (q)n} /∈ F . By Lemma 2, (p)1= (q)1. We set (p)1= in−1 and (q)1= i1. Let i2i3· · · in−2 be an arbitrary permutation ofIn−1− {i1, in−1}.
For 1≤ k ≤ n − 2, let xkbe a vertex of V0(SG{ink})such that (xk)1= ik+1and{xk, (xk)n} /∈ F . By Theorem 2, there exists a Hamiltonian path P1of SG{in1}joining (q)nto x1. Similarly, there is a Hamiltonian path Pkof SG{ink}joining (xk−1)nto xkfor 2≤ k ≤ n − 2, and there is a Hamiltonian path Pn−1 of SG{inn−1}joining (xn−2)nto (p)n. Then we set C1:= e R q (q)nP1 x1 (x1)nP2x2 (x2)n· · · xn−2(xn−2)nPn−1(p)np e.
We can pick out a vertex yn−1of V1(SG{inn−1})such that (yn−1)1= i2and{yn−1, (yn−1) n} /∈ F . For 2≤ k ≤ n − 3, we have |{u ∈ V1(SG{ink})|(u)1= ik+1and dSGn(u, (xk−1)n)= 2}| = n − 3 < (n− 2)!/2 if n ≥ 5. Thus, we can choose a vertex ykof V1(SGn{ik})such that dSGn(yk, (xk−1)n) > 2, (yk)1 = ik+1, and {yk, (yk)n} /∈ F for 2 ≤ k ≤ n − 3. Since |{u ∈ V1(SG{inn−2})|(u)1= i1 and dSGn(u, (xn−3)n)= 2}| = n − 3 < (n − 2)!/2 if n ≥ 5, we can choose a vertex yn−2 of V1(SG{inn−2}) such that dSGn(yn−2, (xn−3)n) >2, (yn−2)1= i1, and {yn−2, (yn−2)n} /∈ F . By Theorem 2, there exists a Hamiltonian path Q1 of SG{in1}joining (yn−2)nto (q)n. Again, there exists a Hamiltonian path Q2of SG{in2}joining (yn−1)nto y2, there exists a Hamiltonian path Qn−1 of SG{in−1}
n joining (p)nto yn−1, and there exists a Hamiltonian path Qkof SGn{ik}joining (yk−1)n to ykfor each 3≤ k ≤ n − 2. Then we set C2:= e p (p)nQn−1yn−1(yn−1)nQ2y2(y2)nQ3y3 (y3)n. . . (y
n−2)nQ1(q)nq R−1e.
In summary,{C1, C2} forms a set of 2-mutually independent Hamiltonian cycles rooted at e in SGn− F . Figure 2(b) illustrates C1and C2in S5.
Case 3 Suppose that {e, (e)n} ∈ F and |Fn| ≤ n − 5. Since |F | = n − 3, there exists an
integer ofIn−1− {1}, say in−1, such that|F ∩ Ein−1,n| = 0. Assume that i1and i2 are two inte-gers of In−1− {in−1} such that |F ∩ Ei1,i2| = max{|F ∩ Es,t||s, t ∈ In−1− {in−1}}. Moreover, let i3i4· · · in−2be an arbitrary permutation ofIn−1− {i1, i2, in−1}. Since {e, (e)n} ∈ F , we have
|F ∩ Ei1,i2| ≤ n − 4. Therefore, we have |F ∩ Ein−2,i1| ≤ n − 5 and |F ∩ Eik,ik+1| ≤ n − 5 for
2≤ k ≤ n − 3.
By Lemma 1, there are (n− 2)!/2 > n − 3 pairwise disjoint edges joining vertices of V0(SG{n}n ) to vertices of V1(SG{in1}). Thus, we can choose a vertex w∈ V0(SG{n}n )− {e} such that (w)1= i1 and{w, (w)n} /∈ F . By Theorem 2, there exists a Hamiltonian path R of (SG{n}
n − Fn)− {(e)in−1} joining e to w. For each 1≤ k ≤ n − 2, let xk be a vertex of V0(SGn{ik})such that (xk)1= ik+1 and{xk, (xk)n} /∈ F . By Theorem 2, there exists a Hamiltonian path P1 of SG{in1}− Fi1 joining
(w)nto x
1. Similarly, there exists a Hamiltonian path Pkof SG{ink}− Fik joining (xk−1) nto x
kfor each 2≤ k ≤ n − 2, and there exists a Hamiltonian path Pn−1of SGn{in−1}− Fin−1joining (xn−2)n to ((e)in−1)n. Then we set C
1:= e R w (w)nP1x1(x1)nP2x2(x2)n. . . (xn−2)nPn−1((e)in−1)n (e)in−1e.
Next, we can pick out a vertex yn−1of V1(SG{inn−1})such that (yn−1)1= i2and{yn−1, (yn−1) n} /∈ F. For any 2≤ k ≤ n − 3, we have |{u ∈ V1(SG{ink})|(u)1= ik+1and dSGn(u, (xk−1)n)= 2}| = n− 3. By Lemma 1, there are (n − 2)!/2 pairwise disjoint edges joining vertices of V1(SG{ink}) to vertices of V0(SG{ink+1}). It is noticed that (n− 2)!/2 > (n − 3) + (n − 5) = 2n − 8 if n ≥ 5. Thus, we can choose a vertex yk of V1(SG{ink})such that dSGn(yk, (xk−1)n) >2, (yk)1= ik+1, and{yk, (yk)n} /∈ F for each 2 ≤ k ≤ n − 3. Since (n − 2)!/2 > |{u ∈ V1(SGn{in−2})|(u)1= i1and dSGn(u, (xn−3)
n)= 2}| + (n − 5) = (n − 3) + (n − 5) = 2n − 8 if n ≥ 5, we can choose a ver-tex yn−2of V1(SGn{in−2})such that dSGn(yn−2, (xn−3)n) >2, (yn−2)1= i1, and{yn−2, (yn−2)n} /∈ F. By Theorem 2, there exists a Hamiltonian path Q1 of SG{in1}− Fi1 joining (yn−2)n to (w)n. Again, there exists a Hamiltonian path Q
2 of SG{in2}− Fi2 joining (yn−1)n to y2, there exists a Hamiltonian path Qn−1 of SG{inn−1}− Fin−1 joining ((e)in−1)n to yn−1, and there exists
a Hamiltonian path Qk of SG{ink}− Fik joining (yk−1)n to yk for 3≤ k ≤ n − 2. We set C2:= e(e)in−1((e)in−1)nQn−1yn−1(yn−1)nQ2y2(y2)nQ3y3(y3)n· · · (yn−2)nQ1(w)nwR−1e.
As a result,{C1, C2} turns out to be a set of two-mutually independent Hamiltonian cycles
rooted at e in SGn− F . Figure 2(c) illustrates C1and C2in S5.
Lemma10 Let f be any integer ofIn−4for n≥ 5. Suppose that F ⊂ E(SGn)with|F | = f , and
u is any vertex of SGn. Then there exist (n− 1 − f )-mutually independent Hamiltonian cycles
rooted at u in SGn− F .
Proof As explained in the proof of Lemma 9, there exists an automorphism of SGnthat can map
any edge in F into an edge of dimension n and map u to e simultaneously. Hence, we can assume that u= e, and F contains at least one edge of dimension n. Let Fk= F ∩ E(SG{k}n )for every k∈ In. Thus, we have|Fk| ≤ n − 5 for every k ∈ In. Moreover, let A1= E1,n− {{e, (e)n}}, and let Ai = Ei,n∪ {{e, (e)i}} for 2 ≤ i ≤ n − 1.
Case 1 Suppose that{e, (e)n} ∈ F . It is noticed that there are at least n − 1 − f elements of
|F ∩ A2|, |F ∩ A3|, . . . , |F ∩ An−1| equal to 0. Without loss of generality, we can assume that
|F ∩ (∪n−1
i=f +1Ai)| = 0. Thus, at least one of |F ∩ A1|, . . . , |F ∩ Af| equals to 0.
Subcase 1.1 Suppose that|F ∩ A1| = 0. Let B = (bi,j)be an (n− 1 − f ) × n matrix with
bi,j =
f+ i + j if f+ i + j ≤ n,
f+ i + j − n otherwise.
It is noticed that bi,n−f −i= n for every 1 ≤ i ≤ n − 1 − f . Then we will construct a set of (n − 1− f )-mutually independent Hamiltonian cycles {C1, C2, . . . , Cn−1−f} rooted at e in SGn− F .
Let i ∈ In−2−f. We set ti= n − f − i. By Lemma 7, there exists a Hamiltonian path Qi of (SG{bi,ti}
n − Fbi,ti)− {e, (e)bi,1, (e)bi,n} joining two vertices xiand yiin V0(SG
{bi,ti}
n )− {e} such that (xi)1= bi,ti−1and (yi)1 = bi,ti+1. By Lemma 3, there exists a Hamiltonian path Piof SG
∪ti −1 j=1{bi,j}
n −
F joining ((e)bi,1)nto (x
i)n. Similarly, there exists a Hamiltonian path Ri of SG
∪n j=ti +1{bi,j}
n − F
joining (yi)nto ((e)bi,n)n. Then we set C
i:= e (e)bi,1((e)bi,1)nPi(xi)nxiQi yi(yi)nRi((e)bi,n)n (e)bi,ne.
By Lemma 6, (SG{bn−1−f,1}
n − Fbn−1−f,n)− {e, (e)
bn−1−f,n} has a Hamiltonian path T joining (e)b1,n
to a vertex z of V0(SG{bnn−1−f,1})− {e} with (z)1= bn−1−f,2. By Lemma 3, there exists a Hamiltonian path W of SG∪
n
j=2{bn−1−f,j}
n − F joining (z)nto ((e)bn−1−f,n)n. Then we set Cn−1−f := e (e)b1,n T z (z)nW ((e)bn−1−f,n)n(e)bn−1−f,ne.
As a result,{C1, . . . , Cn−2−f, Cn−1−f} turns out be a set of (n − 1 − f )-mutually independent Hamiltonian cycles rooted at e in SGn− F . Figure 3 illustrates {C1, C2, C3, C4} in SG6− F with
|F | = f = 1.
Subcase 1.2 Suppose that|F ∩ A1| > 0. It is noticed that f ≥ 2 in this subcase. Thus, at
least one of|F ∩ A2|, . . . , |F ∩ Af| equals to 0. Without loss of generality, we can assume that
(e)3 (e)2 (e)2 (e)5 z(z)6 T ((e)5)6 ((e)3)6 ((e)2)6 Q1 x1 (x1)6 y1(y1)6 P1 R1 W C1 C4 e e e e
(e)4((e)4)6 P2 (x2)6x2 Q2 y2(y2)6 R2 ((e)3)6(e)3
(y3)6
e e
C2
(e)5((e)5)6 ((e)4)6(e)4
e e C3 (x3)6x3 Q3 y3 R3 P3 - (F6U{e,(e) 5} SG{ }3 6-F3 {6 6 } ) { }4 6-F4 { }5 6-F5 - (F6U{e,(e) 2 } {6 6 } ) ,(e)3 { }1 6-F1 { }2 6-F2 { }4 6-F4 { }5 6-F5 { }1 6-F1 { }2 6-F2 { }3 6-F3 - (F6U{e,(e) 3 } {6 6 } ) ,(e)4 { }5 6-F5 { }1 6-F1 { }2 6-F2 { }3 6-F3 { }4 6-F4 - (F6U{e,(e) 4 } {6 6 } ) ,(e)5 { }1 6-F1 { }2 6-F2 { }3 6-F3 { }4 6-F4 { }5 6-F5 SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG
Figure 3. Mutually independent Hamiltonian cycles in SG6− F with |F | = 1 for Subcase 1.1 of Lemma 10. |F ∩ A2| = 0. Let B = (bi,j)be an (n− 1 − f ) × n matrix with
bi,j = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ f + i + j if f + i + j ≤ n, 2 if f + i + j = n + 1, 1 if f + i + j = n + 2, f + i + j − n otherwise.
Using a similar manner to that of Subcase 1.1, we can construct a set of (n− 1 − f )-mutually independent Hamiltonian cycles{C1, C2, . . . , Cn−1−f} rooted at e in SGn− F .
Case 2 Suppose that{e, (e)n} /∈ F . It is noticed that there are at least n − 2 − f elements
of|F ∩ A2|, |F ∩ A3| . . . , |F ∩ An−1| equal to 0. Without loss of generality, we can assume that
|F ∩ (∪n−1
i=f +2Ai)| = 0. Thus, at least one of |F ∩ A1|, . . . , |F ∩ Af+1| is 0.
Subcase 2.1 Suppose that|F ∩ A1| = 0. Let Bn= (bi,j)be an (n− 1 − f ) × n matrix with
B5= ⎡ ⎣14 25 31 42 53 5 4 2 3 1 ⎤ ⎦ , and for n≥ 6 bi,j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ j if i= 1, f + i + j if 2≤ i ≤ n − 2 − f and f + i + j ≤ n, f + i + j − n if 2 ≤ i ≤ n − 2 − f and f + i + j > n, n if i= n − 1 − f and j = 1, 3 if i= n − 1 − f and j = 2, 2 if i= n − 1 − f and j = 3, n− 1 if i= n − 1 − f and j = 4, j− 1 if i= n − 1 − f and 5 ≤ j ≤ n − 1, 1 if i= n − 1 − f and j = n.
Then we construct a set of (n− 1 − f )-mutually independent Hamiltonian cycles
{C1, C2, . . . , Cn−1−f} rooted at e in SGn− F as follows.
We can pick out a vertex v of V1(SG{bn1,n})− {(e)bn−2−f,1} with (v)1= b1,n−1. By Theorem 2, there exists a Hamiltonian path W of (SG{b1,n}
n − Fb1,n)− {e} joining v to (e)
bn−2−f,1. By Lemma 3,
there exists a Hamiltonian path D of SG∪ n−1
j=1{b1,j}
n − F joining (e)nto (v)n. We set C1:= e (e)nD (v)nv W (e)bn−2−f,1 e.
Let i∈ In−2−f − {1}. We set ti = n − f − i. By Lemma 7, there exists a Hamiltonian path Qi of (SG{bni,ti}− Fbi,ti)− {e, (e)bi,1, (e)bi,n} joining two vertices xi and yi in V0(SG{bni,ti})− {e} such that (xi)1 = bi,ti−1 and (yi)1= bi,ti+1. By Lemma 3, there exists a Hamiltonian path Pi of SG∪
ti −1 j=1{bi,j}
n − F joining ((e)bi,1)n to (xi)n. Similarly, there exists a Hamiltonian path Ri of S∪
n j=ti +1{bi,j}
n − F joining (yi)nto ((e)bi,n)n. Then we set Ci := e (e)bi,1 ((e)bi,1)nPi (xi)nxi Qi yi (yi)nR
i ((e)bi,n)n(e)bi,ne.
By Lemma 1, there are (n− 2)!/2 > n − 3 pairwise disjoint edges joining vertices of
V0(SG{bnn−1−f,k}) to vertices of V1(SG{bnn−1−f,k−1}) for 3≤ k ≤ n − 1. Thus, we can choose a
vertex zk of V0(SG{b
n−1−f,k}
n ) such that (zk)1= bn−1−f,k−1, {zk, (zk)n} /∈ F , and zk= C1((k−
1)(n− 1)! + 1). By Lemma 4, there exists a Hamiltonian path T of (SG∪
2
j=1{bn−1−f,j}
n −
F )− {e} joining (e)b2,n to (z
3)n. By Theorem 2, there exists a Hamiltonian path Hk of
SG{bn−1−f,k}
n − Fbn−1−f,k joining zk to (zk+1)n for 3≤ k ≤ n − 2. By Lemma 3, there exists a
Hamiltonian path Hn−1 of SG
∪n
j=n−1{bn−1−f,j}
n − F joining zn−1 to (e)n. Then we set Cn−1−f :=
e(e)b2,nT (z
3)nz3H3(z4)n· · · zn−2Hn−2(zn−1)nzn−1Hn−1(e)ne.
Consequently, {C1, C2, . . . , Cn−2−f, Cn−1−f} is a set of (n − 1 − f )-mutually independent Hamiltonian cycles rooted at e in SGn− F . Figure 4(a) illustrates {C1, C2, C3, C4} in SG6− F with|F | = f = 1.
Subcase 2.2 Suppose that |F ∩ A1| > 0. Thus, at least one of |F ∩ A2|, . . . , |F ∩ Af+1|
equals to 0. Without loss of generality, we can assume that |F ∩ A2| = 0. Let Bn= (bi,j)be an (n− 1 − f ) × n matrix with bi,j = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ n if i = 1 and j = 1, j + 1 if i = 1 and 2 ≤ j ≤ n − 2, 2 if i = 1 and j = n − 1, 1 if i = 1 and j = n, f + i + j if 2≤ i ≤ n − 2 − f and f + i + j ≤ n, 2 if 2≤ i ≤ n − 2 − f and f + i + j = n + 1, 1 if 2≤ i ≤ n − 2 − f and f + i + j = n + 2, f + i + j − n if 2 ≤ i ≤ n − 2 − f and f + i + j ≥ n + 3, j if i = n − 1 − f.
By Lemma 1, there are (n− 2)!/2 > n − 3 pairwise disjoint edges joining vertices of
V0(SG{bn1,2})to vertices of V1(SGn{b1,1}). Thus, we can choose a vertex z of V0(SG{bn1,2})such that (z)1= b1,1,{z, (z)n} /∈ F , and (z)n= (e)b2,n. By Theorem 2, there exists a Hamiltonian path T of (SG{b1,1}
n − Fb1,1)− {e} joining (e)
b2,n to (z)n. By Lemma 3, there exists a Hamiltonian path H
of SG∪ n j=2{b1,j}
n − F joining z to (e)n. Then we set C1:= e (e)b2,nT (z)nz H (e)ne.
Let i∈ In−2−f − {1}. We set ti = n − f − i. By Lemma 7, there exists a Hamiltonian path Qi of (SG
{bi,ti}
n − Fbi,ti)− {e, (e)bi,1, (e)bi,n} joining two vertices xi and yi in V0(SG
{bi,ti}
n )− {e}
such that (xi)1 = bi,ti−1 and (yi)1= bi,ti+1. By Lemma 3, there exists a Hamiltonian path Pi of SG∪
ti −1 j=1{bi,j}
n − F joining ((e)bi,1)n to (xi)n. Similarly, there exists a Hamiltonian path Ri of SG∪
n j=ti +1{bi,j}
n − F joining (yi)nto ((e)bi,n)n. Then we set Ci := e (e)bi,1((e)bi,1)nPi(xi)nxiQiyi (yi)nR
i ((e)bi,n)n(e)bi,ne.
By Lemma 1, there are (n− 2)!/2 > n − 3 pairwise disjoint edges joining vertices of
V0(SG{b
n−1−f,2}
n )to vertices of V1(SG{b
n−1−f,3}
n ). Thus, we can pick out a vertex w of V0(SG{b
n−1−f,2}
n )
(e)6 (e)3 e)5 (e)6 T W D C1 C4 e e e e (e)4 P2 Q R2 (e)3 2 x2 (x2)6 y2(y2)6 ((e)3)6 ((e)4)6 (y3)6 e e C2
(e)5((e)5)6 ((e)4)6(e)4
e e C3 (x3)6 x3 Q3 y3 R3 P3 { }1 6-F1 { }2 6-F2 { }3 6-F3 - (F6U{e {6 6 } ) { }4 6-F4 { }5 6-F5 { }4 6-F4 { }5 6-F5 { }1 6-F1 { }2 6-F2 { }3 6-F3 -(F6U{e,(e) 3 } {6 6 } ) ,(e)4 { }5 6-F5 { }1 6-F1 { }2 6-F2 { }3 6-F3 { }4 6-F4 -(F6U{e,(e) 4 } {6 6 } ) ,(e)5 { }3 6-F3 { }2 6-F2 { }5 6-F5 { }4 6-F4 { }1 6-F1 (a) ( (v)6 v } (z3)6 z3 -(F6U{e {6 6 } ) } (z4)6 z4 H3 H4 (z5)6 z5 H5 C1 C4
(e)4((e)4)6 P2 (x2)6 x2 Q2 y2(y2)6 R2 ((e)3)6(e)3
(y3)6
e e
C2
(e)5((e)5)6 ((e)4)6(e)4
e e C3 (x3)6 x3 Q3 y3 R3 P3 { }4 6-F4 { }5 6-F5 { }2 6-F2 { }1 6-F1 { }3 6-F3 -(F6U{e,(e) 3 } {6 6 } ) ,(e)4 { }5 6-F5 { }2 6-F2 { }1 6-F1 { }3 6-F3 { }4 6-F4 -(F6U{e,(e) 4 } {6 6 } ) ,(e)5 (b) (e)6 W e)5 e e { }1 6-F1 { }2 6-F2 { }3 6-F3 - (F6U{e {6 6 } ) { }4 6-F4 { }5 6-F5 ( (v)6 v } (w)6 w D1 D2 (e)3 T (e)6 e e { }3 6-F3 { }4 6-F4 { }5 6-F5 { }2 6-F2 { }1 6-F1 (z)6z -(F6U{e {6 6 } ) } H SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG SG
Figure 4. Mutually independent Hamiltonian cycles in SG6− F with |F | = 1 for Case 2 of Lemma 10.
such that (w)1= bn−1−f,3,{w, (w)n} /∈ F , and dSGn(w, (yn−2−f)n) >1. Moreover, we choose a vertex v of V1(SG{bnn−1−f,n})such that (v)1= bn−1−f,n−1and{v, (v)n} /∈ F . By Lemma 3, there exists a Hamiltonian path D1of SG
∪2
j=1{bn−1−f,j}
n − F joining (e)nto w. Similarly, there exists a
Hamiltonian path D2 of SG
∪n−1
j=3{bn−1−f,j}
n − F joining (w)n to (v)n. By Theorem 2, there exists
a Hamiltonian path W of (SG{bn−1−f,n}
n − Fbn−1−f,n)− {e} joining v to (e)bn−2−f,1. Then we set
Cn−1−f := e (e)nD1w (w)nD2(v)nv W (e)bn−2−f,1e.
Hence, {C1, C2, . . . , Cn−2−f, Cn−1−f} forms a set of (n − 1 − f )-mutually independent Hamiltonian cycles rooted at e in SGn− F . Figure 4(b) illustrates {C1, C2, C3, C4} in SG6− F
with|F | = f = 1.
Combining Theorem 3 and Lemmas 8–10, we summarize those results as follows.
Theorem4 Let F ⊂ E(SGn)with|F | ≤ n − 3 for n ≥ 3, and let u ∈ V (SGn). Then there exist (n− 2 − |F |)-mutually independent Hamiltonian cycles rooted at u in SGn− F if n ∈ {3, 4},
and there exist (n− 1 − |F |)-mutually independent Hamiltonian cycles rooted at u in SGn− F
if n≥ 5.
5. Conclusion
In this paper, we study the problem of finding mutually independent Hamiltonian cycles in a faulty star graph. That is, given a set of faulty edges F ⊂ E(SGn)with|F | ≤ n − 3, we show that SGn− F has a set of (n − 2 − |F |)-mutually (respectively, (n − 1 − |F |)-mutually) independent Hamiltonian cycles rooted at any vertex if n∈ {3, 4} (respectively, n ≥ 5). We believe that a similar result could be obtained for the graph generated by any transposition tree of order n [1] when at most n− 3 edges fail. On the other hand, we also believe that our current result can be further refined; to be precise, we would like to show thatIHC(SGn− F ) = δ(SGn− F ), where δ(SGn− F ) denotes the minimum degree of graph SGn− F .
The edge faults considered in this work are random and independent. To guarantee that such a faulty star graph remains Hamiltonian in this situation, the maximum number of faulty edges
cannot exceed n− 3. For this reason, we can make a more general condition on the nature of faulty edges such that every vertex still has at least two neighbours in a faulty star graph. This kind of sets of faulty edges is called conditionally faulty. Let F ⊂ E(SGn)be conditionally faulty. Then we believe that mutually independent Hamiltonian cycles can be constructed in SGnif|F | ≤ 3n − 10. These results convince us that the star graph is really robust enough to interconnect computing units in parallel and distributed systems.
Acknowledgements
The authors express the most immense gratitude to Editor-in-Chief, Editor, and the anonymous referees for their careful reading and constructive comments. They greatly improve the quality of the paper. This work was supported in part by the National Science Council of the Republic of China under Contract NSC 98-2218-E-468-001-MY3. J.J.M. Tan was supported in part by the National Science Council of the Republic of China under Contract NSC 96-2221-E-009-134-MY3 and in part by the Aiming for the Top University and Elite Research Center Development Plan.
References
[1] S.B. Akers and B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks, IEEE Trans.Comput. 38 (1989), pp. 555–566.
[2] B. Alspach and Y.S. Qin, Hamilton-connected Cayley graphs on Hamiltonian groups, Eur. J. Combin. 22 (2001), pp. 777–787.
[3] B. Alspach and C.Q. Zhang, Hamilton cycles in cubic Cayley graphs on dihedral groups, Ars Combin. 28 (1989), pp. 101–108.
[4] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, London, 2008.
[5] Y.-Q. Chen, On hamiltonicity of vertex-transitive graphs and digraphs of order p4, J. Combin. Theory Ser. B 72 (1998), pp. 110–121.
[6] S.J. Curran and J.A. Gallian, Hamiltonian cycles and path in Cayley graphs and digraphs - A survey, Discrete Math. 156 (1996), pp. 1–18.
[7] E. Dobson, H. Gavlas, J. Morris, and D. Witte, Automorphism groups with cyclic commutator subgroup and Hamilton
cycles, Discrete Math. 189 (1998), pp. 69–78.
[8] P. Fragopoulou and S.G. Akl, Optimal communication algorithms on the star graphs using spanning tree
constructions, J. Parallel Distrib. Comput. 24 (1995), pp. 55–71.
[9] P. Fragopoulou and S.G. Akl, Edge-disjoint spanning trees on the star networks with applications to fault tolerance, IEEE Trans. Comput. 45 (1996), pp. 174–185.
[10] H. Glover and D. Marušiˇc, Hamiltonicity of cubic Cayley graphs, J. Eur. Math. Soc. 9 (2007), pp. 775–787. [11] R.J. Gould, Advances on the Hamiltonian problem – a survey, Graphs Combin. 19 (2003), pp. 7–52.
[12] S.-Y. Hsieh and P.-Y. Yu, Fault-free mutually independent Hamiltonian cycles in hypercubes with faulty edges, J. Combin. Optim. 13 (2007), pp. 153–162.
[13] S.-Y. Hsieh, G.-H. Chen, and C.-W. Ho, Hamiltonain-laceability of star graphs, Networks 36 (2000), pp. 225–232. [14] J.S. Jwo, S. Lakshmivarahan, and S.K. Dhall, Embedding of cycles and grids in star graphs, J. Circuits Syst. Comput.
1 (1991), pp. 43–74.
[15] T.-L. Kueng, T. Liang, and L.-H. Hsu, Mutually independent Hamiltonian cycles of the binary wrapped butterfly
graphs, Math. Comput. Modelling 48 (2008), pp. 1814–1825.
[16] T.-L. Kueng, C.-K. Lin, T. Liang, J.J.M. Tan, and L.-H. Hsu, A note on fault-free mutually independent Hamiltonian
cycles in hypercubes with faulty edges, J. Combin. Optim. 17 (2009), pp. 312–322.
[17] K. Kutnar and D. Marušiˇc, Hamiltonicity of vertex-transitive graphs of order 4p, Eur. J. Combin. 29 (2008), pp. 423–438.
[18] K. Kutnar and D. Marušiˇc, Hamilton cycles and paths in vertex-transitive graphs-Current directions, Discrete Math. 309 (2009), pp. 5491–5500.
[19] K. Kutnar and P. Šparl, Hamilton paths and cycles in vertex-transitive graphs of order 6p, Discrete Math. 309 (2009), pp. 5444–5460.
[20] S. Latifi, On the fault-diameter of the star graph, Inform. Process. Lett. 46 (1993), pp. 143–150.
[21] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays·Trees · Hypercubes, Morgan Kaufmann, San Mateo, 1992.
[22] M. Lewinter and W. Widulski, Hyper-Hamilton laceable and caterpillar-spannable product graphs, Comput. Math. Appl. 34 (1997), pp. 99–104.
[23] T.-K. Li, J.J.M. Tan, and L.-H. Hsu, Hyper-Hamiltonian laceability on edge fault star graph, Inform. Sci. 165 (2004), pp. 59–71.
[24] C.-K. Lin, H.-M. Huang, D.F. Hsu, and L.-H. Hsu, On the spanning w-wide diameter of the star graph, Networks 48 (2006), pp. 235–249.
[25] C.-K. Lin, H.-M. Huang, J.J.M. Tan, and L.-H. Hsu, Mutually independent Hamiltonian cycles for the pancake
graphs and the star graphs, Discrete Math. 309 (2009), pp. 5474–5483.
[26] C.-K. Lin, Y.-K. Shih, J.J.M. Tan, and L.-H. Hsu, Mutually independent Hamiltonian cycles in some graphs, Ars Combin., accepted for publication.
[27] L. Lovasz, Combinatorial structures and their applications, in Proc. Calgary Internat. Conf. Calgary, Alberta (1969), Gordon and Breach, New York, 1970, pp. 243–246. Problem 11.
[28] D. Marušiˇc, Hamiltonian cycles in vertex symmetric graphs of order 2p2, Discrete Math. 66 (1987), pp. 169–174.
[29] Š. Miklaviˇc and P. Šparl, On hamiltonicity of circulant digraphs of outdegree three, Discrete Math. 309 (2009), pp. 5437–5443.
[30] Š. Miklaviˇc and W. Xiao, Connected graphs as subgraphs of Cayley graphs: Conditions on hamiltonicity, Discrete Math. 309 (2009), pp. 5426–5431.
[31] J.H. Park and H.C. Kim, Longest paths and cycles in faulty star graphs, J. Parallel Distrib. Comput. 64 (2004), pp. 1286–1296.
[32] Y. Rouskov, S. Latifi, and P.K. Srimani, Conditional fault diameter of star graph networks, J. Parallel Distrib. Comput. 33 (1996), pp. 91–97.
[33] Y.-K. Shih, C.-K. Lin, D.F. Hsu, J.M. Tan, and L.-H. Hsu, The construction of mutually independent Hamiltonian
cycles in bubble-sort graphs, Int. J. Comput. Math., 87 (2010), pp. 2212–2225.
[34] G. Simmons, Almost all n-dimensional rectangular lattices are Hamilton laceable, Congr. Numer. 21 (1978), pp. 649–661.
[35] L. Stacho and D. Szeszlér, On a generalization of Chvatals condition giving new Hamiltonian degree sequences, Discrete Math. 292 (2005), pp. 159–165.
[36] C.-M. Sun, C.-K. Lin, H.-M. Huang, and L.-H. Hsu, Mutually independent Hamiltonian paths and cycles in
hypercubes, J. Interconnect. Netw. 7 (2006), pp. 235–255.
[37] Y.-H. Teng, T.-Y. Ho, J.J.M. Tan, and L.-H. Hsu, On mutually independent Hamiltonian paths, Appl. Math. Lett. 19 (2006), pp. 345–350.
[38] Y.-C. Tseng, S.-H. Chang, and J.-P. Sheu, Fault-tolerant ring embedding in a star graph with both link and node
failures, IEEE Trans. Parallel Distrib. Syst. 8 (1997), 1185–1195.
