www.elsevier.com/locate/tcs
On the spanning connectivity and spanning laceability of
hypercube-like networks
Cheng-Kuan Lin
a, Jimmy J.M. Tan
a, D. Frank Hsu
b, Lih-Hsing Hsu
c,∗aDepartment of Computer Science, National Chiao Tung University, Hsinchu, 30010, Taiwan, ROC bDepartment of Computer and Information Science, Fordham University, New York, NY 10023, USA cDepartment of Computer Science and Information Engineering, Providence University, Taichung, 43301, Taiwan, ROC
Received 7 November 2006; received in revised form 1 May 2007; accepted 3 May 2007
Communicated by D.-Z. Du
Abstract
Let u and v be any two distinct nodes of an undirected graph G, which is k-connected. For 1 ≤ w ≤ k, a w-container C(u, v) of a k-connected graph G is a set of w-disjoint paths joining u and v. A w-container C(u, v) of G is a w∗
-container if it contains all the nodes of G. A graph G isw∗-connected if there exists aw∗-container between any two distinct nodes. A bipartite graph G isw∗-laceable if there exists aw∗-container between any two nodes from different parts of G. Let G0=(V0, E0) and
G1 = (V1, E1) be two disjoint graphs with |V0| = |V1|. Let E = {(v, φ(v)) | v ∈ V0, φ(v) ∈ V1, andφ : V0 → V1is a
bijection}. Let G = G0⊕G1=(V0∪V1, E0∪E1∪E). The set of n-dimensional hypercube-like graph Hn0 is defined recursively
as (a) H10 = {K2}, K2=complete graph with two nodes, and (b) if G0and G1are in Hn0, then G = G0⊕G1is in Hn+10 . Let
Bn0 = {G ∈ Hn0and G is bipartite} and Nn0 =Hn0 \Bn0. In this paper, we show that every graph in Bn0 isw∗-laceable for everyw, 1 ≤w ≤ n. It is shown that a constructed Nn0-graph H can not be 4∗-connected. In addition, we show that every graph in Nn0 is w∗-connected for everyw, 1 ≤ w ≤ 3.
c
2007 Published by Elsevier B.V.
Keywords: Hamiltonian; Hamiltonian connected; Hamiltonian laceable; Hypercube networks; Hypercube-like networks; w∗-connected; w∗-laceable; Spanning connectivity; Spanning laceability; Graph container
1. Introduction 1.1. Definitions
For graph definitions and notations we follow [4]. G = (V, E) is a graph if V is a finite set and E is a subset of {(u, v) | (u, v) is an unordered pair of V }. We say that V is the node set and E is the edge set. We use n(G) to denote |V |. Two nodes u andv are adjacent if (u, v) is an edge of G. For a node u, NG(u) denotes the neighbourhood
∗Corresponding address: Information Engineering Department, Ta Hwa Institute of Technology, Hsinchu, Taiwan, ROC.
E-mail addresses:cklin@cs.nctu.edu.tw(C.-K. Lin),jmtan@cs.nctu.edu.tw(J.J.M. Tan),hsu@trill.cis.fordham.edu(D.F. Hsu),
lhhsu@pu.edu.tw,lhhsu@cis.nctu.edu.tw(L.-H. Hsu).
0304-3975/$ - see front matter c 2007 Published by Elsevier B.V.
a path as hv1, v2, Q, v2, v3, . . . , vkiif l(Q) = 0. Let I (P) = V (P) − {v1, vk}be the set of the internal nodes of P.
A set of paths {P1, P2, . . . , Pk}are internally node-disjoint (abbreviated as disjoint) if I(Pi) ∩ I (Pj) = ∅ for any
i 6= j. A path is a hamiltonian path if it contains all nodes of G. A graph G is hamiltonian connected if there exists a hamiltonian path joining any two distinct nodes of G [18]. A cycle is a path with at least three nodes such that the first node is the same as the last one. A hamiltonian cycle of G is a cycle that traverses every node of G. A graph is hamiltonianif it has a hamiltonian cycle. A graph G is bipartite if its node set can be partitioned into two subsets V1
and V2such that every edge connects nodes between V1and V2. A bipartite graph G is hamiltonian laceable if there
is a hamiltonian path of G joining any two nodes from distinct bipartition [20]. A bipartite graph G is k-edge fault hamiltonian laceableif G − F is hamiltonian laceable for any edge subset F of G with |F | ≤ k.
A graph G is k-connected if there exists a set of k internally disjoint paths {P1, P2, . . . , Pk}between any two
distinct nodes u andv. A subset S of V (G) is a cut set if G − S is disconnected. A w-container of G between two distinct nodes u andv is a set of w internally disjoint paths between u and v. The concepts of a container and of a wide distance were proposed by Hsu [12] to evaluate the performance of communication for an interconnection network. The connectivity of G,κ(G), is the minimum number of nodes whose removal leaves the remaining graph disconnected or trivial. Hence, a graph G is k-connected ifκ(G) ≥ k. It follows from Menger’s Theorem [17] that there is aw-container for w ≤ k between any two distinct nodes of G if G is k-connected.
1.2. w∗-connected graphs andw∗-laceable graphs
In this paper, we are interested in a specific type of container. We say that aw-container C(u, v) is a w∗-container if every node of G is on some path in C(u, v). A graph G is said to be w∗-connectedif there exists aw∗-container
between any two distinct nodes u andv. Obviously, we have the following remarks:
Remark 1. (1.a) a graph G is 1∗-connected if and only if it is hamiltonian connected [18],(1.b) a graph G is 2∗ -connected if it is hamiltonian, and(1.c) an 1∗-connected graph except K1and K2is 2∗-connected.
Using our definition of aw∗-connected graph, the globally 3∗-connected graphs proposed by Albert et al. [3] are 3-regular 3∗-connected graphs. Assume that the graph G isw∗-connected wherew ≤ κ(G). The spanning connectivity of a graph G,κ∗(G), is the largest integer k such that G is i∗-connected for every i , 1 ≤ i ≤ k. A graph G is super spanning connectedifκ∗(G) = κ(G). In such case, the number κ∗(G) = κ(G) is called the super spanning connectivity of G. In [13,16,15,21], some families of graphs are proved to be super spanning connected.
Let G be a bipartite graph with bipartition V1and V2such that |V1| ≥ |V2|. Suppose that there exists aw∗-container
C(u, v) = {P1, P2, . . . , Pw}in G joining u tov with u, v ∈ V1. Obviously, the number of nodes in Pi is 2ti +1 for
some integer ti. There are ti −1 nodes of Pi in V1other than u andv, and ti nodes of Pi in V2. As a consequence,
|V1| =Pwi =1(ti−1) + 2 and |V2| =Pwi =1ti. Therefore, any bipartite graph G withκ(G) ≥ 3 is not w∗-connected
for anyw, 3 ≤ w ≤ κ(G).
For this reason, a bipartite graph is said to bew∗-laceableif there exists aw∗-container between any two nodes from different partite sets for somew, 1 ≤ w ≤ κ(G). Obviously, any bipartite w∗-laceable graph withw ≥ 2 has the equal size of bipartition. We have the following remarks:
Remark 2. (2.a) an 1∗-laceable graph is also known as hamiltonian laceable graph [20],(2.b) a graph G is 2∗ -laceable if and only if it is hamiltonian, and(2.c) an 1∗-laceable graph except K1and K2are 2∗-laceable.
The spanning laceability of a bipartite graph G,κ∗L(G), is the largest integer k such that G is i∗-laceable for every
i, 1 ≤ i ≤ k. A graph G is super spanning laceable if the numberκ∗L(G) = κ(G). Recently, Chang et al. [5] proved
that the n-dimensional hypercube Qnis superspanning laceable for every positive integer n. It was proved in [15] that
1.3. Hypercube-like graphs Hn0
Graph containers do exist in engineering design information and telecommunication networks or in biological and neural systems ([2,12] and its references). The study of w-container, w-wide distance, and their w∗-versions play a pivotal role in the design and the implementation of parallel routing and efficient information transmission in large-scale networking systems. In bioinformatics and neuroinformatics, the existence as well as the structure of a w∗-container signifies the cascade effect in the signal transduction system and the reaction in a metabolic pathway.
Among all interconnection networks proposed in the literature, the hypercube Qn is one of the most popular
topologies [5,14]. However, the hypercube does not have the smallest diameter for its resources. Various networks are proposed by twisting some pairs of links in hypercubes [1,8,10,11]. Because of the lack of the unified perspective on these variants, results of one topology are hard to be extended to others. To make a unified study of these variants, Vaidya et al. introduced the class of hypercube-like graphs [22]. We denote there graphs as H0-graphs. The class of H0-graphs, consisting of simple, connected, and undirected graphs, contains most of the hypercube variants.
Let G0 =(V0, E0) and G1=(V1, E1) be two disjoint graphs with the same number of nodes. A 1–1 connection
between G0and G1is defined as an edge set E = {(v, φ(v)) | v ∈ V0, φ(v) ∈ V1, andφ : V0 →V1is a bijection}.
We use G0⊕G1to denote G =(V0∪V1, E0∪E1∪E). The operation “⊕” may generate different graphs depending
on the bijectionφ. There are some studies on the operation “⊕” [6,7]. Let G = G0⊕G1and let x be any node in G.
We use ¯xto denote the unique node matched underφ.
Now, we can define the set of n-dimensional H0-graph, Hn0, as follows: (1) H10 = {K2}, where K2is the complete graph with two nodes.
(2) Assume that G0, G1∈ Hn0. Then G = G0⊕G1is a graph in Hn+10 .
Note that some n-dimensional H0-graphs are bipartite. We can define the set of bipartite n-dimensional H0-graph, Bn0, as follows:
(1) B10 = {K2}, where K2is the complete graph defined on {a, b} with bipartition V0= {a}and V1= {b}.
(2) For i = 0, 1, let Gi be a graph in Bn0 with bipartition V0i and V1i. Letφ be a bijection between V00∪V10and
V01∪V11such thatφ(v) ∈ V1−i1 ifv ∈ Vi0. Then G = G0⊕G1is a graph in Bn+10 .
Every graph in Hn0 is an n-regular graph with 2n nodes, and every graph in Bn0 contains 2n−1 nodes in each bipartition. We use Nn0 to denote the set of non-bipartite graphs in Hn0. Clearly, we have Qn∈ Bn0.
Let G be a graph in H0
n+1. Then G = G0⊕G1with both G0and G1in H 0
n. Let u be a node in V(G). Then u is a
node in V(Gi) for some i = 0, 1. We use ¯u to denote the node in V (G1−i) matched under φ. So u = ¯v if ¯u = v.
In the following section, we give some basic properties about Hn0-graphs. In Section3, we prove that every graph in Bn0 is super spanning laceable. In Section4, we show that every graph in Nn0 isw∗-connected for everyw, 1 ≤ w ≤ 3, for n ≥ 3. We also construct an Nn0-graph H and show that H can not be 4∗-connected. In the final section, we give our concluding remark.
2. Preliminaries
Lemma 1. Assume that G is graph in Nn0. Then n ≥3.
Theorem 1 ([19]). Let n ≥ 3. Every graph in Nn0 is hamiltonian connected and hamiltonian.
Theorem 2 ([19]). Every graph in Bn0 is hamiltonian laceable and every graph in Bn0 is hamiltonian if n ≥2. Theorem 3 ([19]). Let n ≥ 2. Suppose that G is a graph in Bn0 with bipartition V0and V1. Suppose that u1and u2
are two distinct nodes in Vi and thatv1andv2are two distinct nodes in V1−i with i ∈ {0, 1}. Then there are two
disjoint paths P1and P2of G such that(1) P1joins u1tov1,(2) P2joins u2tov2, and(3) P1∪P2spans G.
Theorem 4. Let G be a graph in Bn0 with bipartition V0and V1for n ≥2. Suppose that z is a node in Vi and that u
Fig. 1. Illustration forTheorem 4.
Proof. We prove this statement by induction on n. Since Q2 is the only graph in B20, it is easy to check that this
statement holds for n = 2. Thus, we assume that G = G0⊕G1in Bn0 with n ≥ 3. We have Gi ∈ Bn−10 for i = 0, 1.
Let V0i and V1ibe the bipartition of Gi for i = 0, 1. Without loss of generality, we assume that V00∪V01and V10∪V11
form the bipartition of G. Let z be any node in V10∪V11, and let u andv be any two distinct nodes in V00∪V01. We need to show that there is a hamiltonian path of G − {z} joining u tov. Without loss of generality, we assume that z ∈ V10. We have the following cases:
Case 1: u ∈ V00andv ∈ V00. By induction, there is a hamiltonian path Q in G0− {z}joining u tov. Without loss
of generality, we write Q as hu, x, R, vi. Since u ∈ V00, x ∈ V10. ByTheorem 2, there is a hamiltonian path W of G1
joining the node ¯u ∈ V11to the node ¯x ∈ V01. Then hu, ¯u, W, ¯x, x, R, vi is the hamiltonian path of G − {z} joining u tov. SeeFig. 1(a) for an illustration.
Case 2: u ∈ V00andv ∈ V01. Since n ≥ 3, |V00| =2n−1≥2. We can choose a node x in V00− {u}. By induction, there is a hamiltonian path Q in G0− {z}joining u to x. Since x ∈ V00, ¯x ∈ V11. ByTheorem 2, there is a hamiltonian
path W of G1joining ¯xtov. Then hu, Q, x, ¯x, W, vi is the hamiltonian path of G − {z} joining u to v. SeeFig. 1(b)
for an illustration. Case 3: u ∈ V1
0 and v ∈ V01. We can choose a node x in V11. By Theorem 2, there is a hamiltonian
path W in G1 joining u to x. Without loss of generality, we write W as hu, W1, y, v, W2, xi. Since v ∈ V01,
y ∈ V11. By induction, there is a hamiltonian path Q in G0 − {z} joining the node ¯y ∈ V00 to the node
¯
x ∈ V00. Then hu, W1, y, ¯y, Q, ¯x, x, W2−1, vi is the hamiltonian path of G − {z} joining u to v. SeeFig. 1(c) for
an illustration.
3. Every B0
n-graph is super spanning laceable
Let n be any positive integer. To prove that every graph in Bn0 is w∗-laceable for every w, 1 ≤ w ≤ n, we need the concept of spanning fan. We note that there is another Menger-type Theorem. Let u be a node of G and S = {v1, v2, . . . , vk}be a subset of V(G) not including u. An (u, S)-fan is a set of disjoint paths {P1, P2, . . . , Pk}of
Gsuch that Pi joins u andvi [9]. It is proved that a graph G is k-connected if and only if there exists an(u, S)-fan
between any node u and any k-subset S of V(G) such that u /∈ S. With this observation, we define a spanning fan is a fan that spans G. Naturally, we can studyκ∗f an(G) as the largest integer k such that there exists a spanning (u, S)-fan between any node u and any k-node subset S with u /∈ S. However, we defer such a study for the following reasons:
First, let S be a cut set of a graph G. Let u be any node of V(G) − S. It is easy to see that there is no spanning (u, S)-fan in G. Thus, κ∗
f an(G) < κ(G) if G is not a complete graph.
Second, let G be a bipartite graph with bipartition V0 and V1 and |V0| = |V1|. Let u be a vertex in Vi,
S = {v1, v2, . . . , vk}be a subset of G not containing u, and k ≤ κ(G). Suppose that |S ∩ V1−i| = r. Without
loss of generality, we assume that {v1, v2, . . . , vr} ⊂ V1−i. Let {P1, P2, . . . , Pk}be any spanning(u, S)-fan of G.
Then l(Pj) is odd if j ≤ r, and l(Pj) is even if r < j ≤ k. Let l(Pj) = 2tj+1 if j ≤ r and l(Pj) = 2tj if j> r. For
j ≤ r, there are tj−1 nodes of Pj in Vi other than u and there are tjnodes of Pjin V1−i. For j > r, there are tjnodes
of Pj in Vi other than u and there are tj nodes of Pj in V1−i. Thus, |Vi| =1 − r +Pkj =1tj and |V1−i| =Pkj =1tj.
Since |Vi| = |V1−i|, r = 1. Thus, r = 1 is a fact requirement as we study the spanning fan of bipartite graphs with
equal size of bipartition.
Theorem 5. Let n and k be any two positive integer with k ≤ n. Let G be a graph in Bn0 with bipartition V0 and
V1. There exists a spanning(u, S)-fan in G for any node u in Vi and any node subset S with |S| ≤ n such that
Fig. 2. Illustration for Case 1 ofTheorem 5.
Proof. We prove this statement by induction on n. Let G = G0⊕G1in Bn0 such that V0i and V1i be the bipartition
of Gi for every i = 0, 1. Without loss of generality, we assume that V00∪V01and V10∪V11form the bipartition of
G. Let u be any node in V00∪V01and S = {v1, v2, . . . , vk}be any node subset in G − {u} withv1being the unique
node in(V10∪V11) ∩ S. Without loss of generality, we assume that u ∈ V00. ByTheorem 2, this statement holds for k = 1. Thus, we assume that k = 2 and n ≥ 2. ByTheorem 2, there is a hamiltonian path P of G joiningv1tov2.
Without loss of generality, we write P as hv1, P1, u, P2, v2i. Then {P1, P2}forms the spanning(u, S)-fan of G. Thus,
this statement holds for k = 2. Moreover, this statement holds for n = 2. We assume that 3 ≤ k ≤ n. Suppose that this statement holds for Bn−10 , and Gi ∈ Bn−10 for i = 0 and 1. Without loss of generality, we assume that u ∈ G0.
Let T = S − {v1}. We have the following cases:
Case 1: |T ∩ V00| = |T |. Thenvi ∈V00for every i , 2 ≤ i ≤ k.
Case 1.1:v1∈V10. Let H = S − {vk}. Obviously, H ⊂ G0, |H ∩ V10| =1, and |H | = k − 1. By induction, there is
a spanning(u, H)-fan {P1, P2, . . . , Pk−1}of G0. Without loss of generality, we assume that Pi is joining u tovi for
every i , 1 ≤ i ≤ k − 1.
Suppose thatvk∈V(P1). Without loss of generality, we write P1as hu, Q1, vk, x, Q2, v1i. Sincevk ∈V00, x ∈ V10.
(Note that x =v1if l(Q2) = 0.) ByTheorem 2, there is a hamiltonian path R of G1joining node ¯u ∈ V11to node
¯
x ∈ V01. We set W1 = hu, ¯u, R, ¯x, x, Q2, v1i, Wi = Pi for every i , 2 ≤ i ≤ k − 1, and Wk = hu, Q1, vki. Then
{W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 2(a) for an illustration where k = 6.
Suppose thatvk ∈ V(Pi) for some 2 ≤ i ≤ k − 1. Without loss of generality, we assume that vk ∈ V(Pk−1) and
we write Pk−1as hu, Q1, vk, x, Q2, vk−1i. Sincevk ∈ V00, x ∈ V10. ByTheorem 2, there is a hamiltonian path R of
G1joining node ¯u ∈ V11to node ¯x ∈ V01. We set Wi = Pi for every i ∈ hk − 2i, Wk−1= hu, ¯u, R, ¯x, x, Q2, vk−1i,
and Wk = hu, Q1, vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 2(b) for an illustration where
k =6.
Case 1.2:v1 ∈ V11. We choose a node x in V10. Let H = (T ∪ {x}) − {vk}. So H ⊂ G0, |H ∩ V10| = 1, and
|H | = k −1. By induction, there is a spanning(u, H)-fan {P1, P2, . . . , Pk−1}of G0. Without loss of generality, we
assume that P1is joining u to x and Piis joining u tovi for every 2 ≤ i ≤ k − 1. We have ¯u ∈ V11and ¯x ∈ V01.
Case 1.2.1:vk ∈ V(P1). Without loss of generality, we write P1as hu, Q1, y, vk, Q2, xi. Since vk ∈ V00, y ∈ V10
and ¯y ∈ V01.
Suppose thatv16= ¯u. ByTheorem 3, there are two disjoint paths R1and R2in G1such that(1) R1joins ¯y tov1,
(2) R2joins ¯uto ¯x, and(3) R1∪R2spans G1. We set W1= hu, Q1, y, ¯y, R1, v1i, Wi =Pi for every 2 ≤ i ≤ k − 1,
and Wk = hu, ¯u, R2, ¯x, x, Q−12 , vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 2(c) for an
illustration where k = 6.
Suppose that v1 = u. By¯ Theorem 4, there is a hamiltonian path R of G1− {v1} joining ¯y to ¯x. We set
W1 = hu, ¯u = v1i, Wi = Pi for every 2 ≤ i ≤ k − 1, and Wk = hu, Q1, y, ¯y, R, ¯x, x, Q−12 , vki. Then
{W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 2(d) for an illustration where k = 6.
Case 1.2.2:vk ∈V(Pi) for some 2 ≤ i ≤ k − 1. Without loss of generality, we assume that vk∈V(Pk−1) and we
Fig. 3. Illustration for Case 2 ofTheorem 5.
Fig. 4. Illustration for Case 3 ofTheorem 5.
Suppose thatv16= ¯u. ByTheorem 3, there are two disjoint paths R1and R2in G1such that(1) R1joins ¯xtov1,
(2) R2joins ¯uto ¯y, and(3) R1∪R2spans G1. We set W1= hu, P1, x, ¯x, R1, v1i, Wi = Pifor every 2 ≤ i ≤ k − 2,
Wk−1 = hu, ¯u, R2, ¯y, y, Q2, vk−1i, and Wk = hu, Q1, vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G.
SeeFig. 2(e) for an illustration where k = 6.
Suppose that v1 = u¯. By Theorem 4, there is a hamiltonian path R of G1 − {v1} joining ¯x to ¯y. We set
W1= hu, ¯u = v1i, Wi =Pifor every 2 ≤ i ≤ k − 2, Wk−1= hu, P1, x, ¯x, R, ¯y, y, Q2, vk−1i, and Wk = hu, Q1, vki.
Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 2(f) for an illustration where k = 6.
Case 2: |T ∩ V01| =1. Without loss of generality, we assume thatvk ∈V01. We have ¯u ∈ V11.
Case 2.1:v1∈V10. Let H = S − {vk}. Obviously, H ⊂ G0, |H ∩ V10| =1, and |H | = k − 1. By induction, there is
a spanning(u, H)-fan {P1, P2, . . . , Pk−1}of G0. Without loss of generality, we assume that Pi is joining u tovi for
every 1 ≤ i ≤ k − 1. ByTheorem 2, there is a hamiltonian path R of G1joining ¯utovk. We set Pk = hu, ¯u, R, vki.
Then {P1, P2, . . . , Pk}is the spanning(u, S)-fan of G. SeeFig. 3(a) for an illustration where k = 6.
Case 2.2:v1∈V11. ByTheorem 2, there is a hamiltonian path R of G1joiningv1tovk. Without loss of generality,
we write R as hv1, R1, ¯u, x, R2, vki. (Note thatv1= ¯uif l(R1) = 0 and x = vkif l(R2) = 0.) Since ¯u ∈ V11, x ∈ V01
and ¯x ∈ V10. Let H =(T ∪ { ¯x}) − {vk}. Obviously, H ⊂ G0, |H ∩ V10| =1, and |H | = k − 1. By induction, there
is a spanning(u, H)-fan {P1, P2, . . . , Pk−1}of G0. Without loss of generality, we assume that P1is joining u to ¯x
and Pi is joining u tovi for every 2 ≤ i ≤ k − 1. We set W1 = hu, ¯u, R−11 , v1i, Wi = Pi for every 2 ≤ i ≤ k − 1,
and Wk= hu, P1, ¯x, x, R2, vki. Then {W1, W2, . . . , Wk}is the(u, S)-fan of G. SeeFig. 3(b) for an illustration where
k =6. Case 3: |T ∩ V01| =2. Without loss of generality, we assume that {vk−1, vk} ⊂V01. We have |V00| ≥ n ≥ k.
We can choose a node x in V00− {u, v2, v3, . . . , vk−2}. Obviously, { ¯x, ¯u} ⊂ V11with ¯x 6= ¯u. ByTheorem 3, there are
two disjoint paths R1and R2in G1such that(1) R1joins ¯xtovk−1,(2) R2joins ¯utovk, and(3) R1∪R2spans G1.
Case 3.1:v1 ∈ V10. Let H = (S ∪ {x}) − {vk−1, vk}. Obviously, H ⊂ G0, |H ∩ V10| = 1, and |H | = k − 1.
By induction, there is a spanning(u, H)-fan {P1, P2, . . . , Pk−1}of G0. Without loss of generality, we assume that Pi
is joining u tovi for every 1 ≤ i ≤ k − 2 and Pk−1 is joining u to x. We set Wi = Pi for every 1 ≤ i ≤ k − 2,
Wk−1= hu, Pk−1, x, ¯x, R1, vk−1i, and Wk= hu, ¯u, R2, vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G.
SeeFig. 4(a) for an illustration where k = 6.
Case 3.2:v1 ∈ V11andv1 ∈ V(R1). Without loss of generality, we write R1as h ¯x, Q1, v1, y, Q2, vk−1i. Since
v1 ∈ V11, y ∈ V01 and ¯y ∈ V10. Let H = (T ∪ {x, ¯y}) − {vk−1, vk}. Obviously, H ⊂ G0, |H ∩ V10| = 1, and
|H | = k −1. By induction, there is a spanning(u, H)-fan {P1, P2, . . . , Pk−1}of G0. Without loss of generality,
we assume that P1 is joining u to x, Pi is joining u to vi for every i ∈ hk − 2i, and Pk−1 is joining u to ¯y.
We set W1 = hu, P1, x, ¯x, Q1, v1i, Wi = Pi for every 2 ≤ i ≤ k − 2, Wk−1 = hu, Pk−1, ¯y, y, Q2, vk−1i, and
Wk = hu, ¯u, R2, vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 4(b) for an illustration where
k =6.
Case 3.3:v1 ∈ V11 andv1 ∈ V(R2). Without loss of generality, we write R2 as h ¯u, Q1, v1, y, Q2, vki. Since
v1∈V11, y ∈ V01and ¯y ∈ V10. Let H =(T ∪{x, ¯y})−{vk−1, vk}. Obviously, H ⊂ G0, |H ∩ V10| =1, and |H | = k −1.
Fig. 5. Illustration for Case 4 ofTheorem 5.
is joining u to x, Piis joining u tovi for every 2 ≤ i ≤ k − 2, and Pk−1is joining u to ¯y. We set W1= hu, ¯u, Q1, v1i,
Wi = Pi for every 2 ≤ i ≤ k − 2, Wk−1 = hu, P1, x, ¯x, R1, vk−1i, and Wk = hu, Pk−1, ¯y, y, Q2, vki. Then
{W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 4(c) for an illustration where k = 6.
Case 4: |T ∩ V01| ≥3 and |T ∩ V00| ≥1. We have n ≥ k = |S| ≥ 5. Without loss of generality, we assume that A = T ∩ V00= {v2, v3, . . . , vt}and B = T ∩ V01= {vt +1, vt +2, . . . , vk}for some 2 ≤ t ≤ k − 3. Since t ≤ k − 3 and k ≤ n, | A| = t − 1 ≤ n − 4 and |B| ≤ n − 2. Since n ≥ 5,(n −1)|A|+|B| ≤ (n −1)(n −4)+(n −2) < 2n−2= |V01|. Thus, we can choose a node x in V01−B such that ¯vi /∈ NG1(x) for every 2 ≤ i ≤ t. Since 2 ≤ t ≤ k − 3 and
k ≤ n, k − t + 1 ≤ n − 1. Let H = B ∪ { ¯u}. Obviously, H ⊂ G1, |H ∩ V11| = 1, and |H | = k − t + 1. By
induction, there is a spanning(x, H)-fan {P1, P2, . . . , Pk−t +1}of G1. Without loss of generality, we assume that P1
is joining x to ¯uand Pi is joining x tovt +i −1for every 2 ≤ i ≤ k − t + 1. Moreover, we write P1= hx, x1, R1, ¯ui and
Pi = hx, xi, Ri, vt +i −1ifor every 2 ≤ i ≤ k − t + 1. Since x ∈ V01, xi ∈V11and ¯xi ∈V00for every 1 ≤ i ≤ k − t + 1.
We set C = { ¯x2, ¯x3, . . . , ¯xk−t}.
Case 4.1:v1 ∈ V10. Let H0 = A ∪ C ∪ {v1}. Obviously, H0 ⊂ G0, |H0 ∩V10| = 1, and |H0| = k −1. By
induction, there is a spanning (u, H0)-fan {Q1, Q2, . . . , Qk−1}of G0. Without loss of generality, we assume that
Qi is joining u tovi for every 1 ≤ i ≤ t and Qj is joining u to ¯xj −t +2 for every t + 1 ≤ j ≤ k − 1. We set
Wi = hu, Qi, vii for every 1 ≤ i ≤ t , Wj = hu, Qj, ¯xi −t +2, xi −t +2, Ri −t +2, vji for every t + 1 ≤ j ≤ k − 1,
and Wk = hu, ¯u, P1−1, x, Pk−t +1, vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 5(a) for an
illustration where k = 6 and t = 3.
Case 4.2:v1∈V11andv1∈V(P1). Without loss of generality, we write P1as hx, Z1, y, v1, Z2, ¯ui. Since v1∈V11,
y ∈ V01and ¯y ∈ V10. Let H0= A ∪ C ∩ { ¯y}. Obviously, H0⊂G0, |H0∩V10| =1, and |H0| =k −1. By induction,
there is a spanning(u, H0)-fan {Q1, Q2, . . . , Qk−1}of G0. Without loss of generality, we assume that Q1is joining
u to ¯y, Qi is joining u tovi for every 2 ≤ i ≤ t , and Qj is joining u to ¯xj −t +2for every t + 1 ≤ j ≤ k − 1. We
set W1= hu, ¯u, Z2−1, v1i, Wi = hu, Qi, viifor every 2 ≤ i ≤ t , Wj = hu, Qj, ¯xi −t +2, xi −t +2, Ri −t +2, vjifor every
t +1 ≤ j ≤ k − 1, and Wk = hu, Q1, ¯y, y, Z−11 , x, Pk−t +1, vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan
of G. SeeFig. 5(b) for an illustration where k = 6 and t = 3.
Case 4.3: v1 ∈ V11 and v1 ∈ V(Pi) for some 2 ≤ i ≤ k − t + 1. Without loss of generality, we assume
that v1 ∈ V(Pk−t +1) and we write Pk−t +1 as hx, Z1, v1, y, Z2, vki. Since v1 ∈ V11, y ∈ V01 and ¯y ∈ V10. Let
H0=A ∪ C ∪ { ¯y}. Obviously, H0⊂G0, |H0∩V10| =1, and |H
0| =k −1. By induction, there is a spanning(u, H0
)-fan {Q1, Q2, . . . , Qk−1}of G0. Without loss of generality, we assume that Q1is joining u to ¯y, Qi is joining u tovi
for every 2 ≤ i ≤ t , and Qj is joining u to ¯xj −t +2for every t + 1 ≤ j ≤ k − 1. We set W1= hu, ¯u, P1−1, x, Z1, v1i,
Wi = hu, Qi, vii for every 2 ≤ i ≤ t , Wj = hu, Qj, ¯xi −t +2, xi −t +2, Ri −t +2, vji for every t + 1 ≤ j ≤ k − 1,
and Wk = hu, Q1, ¯y, y, Z2, vki. Then {W1, W2, . . . , Wk}forms the spanning(u, S)-fan of G. SeeFig. 5(c) for an
illustration where k = 6 and t = 3.
Case 5: |T ∩ V01| = |T | ≥3. Let H =(T ∪ { ¯u}) − {vk}. Obviously, H ⊂ G1, |H ∩ V11| =1, and |H | = k − 1.
By induction, there is a spanning(vk, H)-fan {P1, P2, . . . , Pk−1}of G1. Without loss of generality, we assume that
P1 is joining vk to ¯u and Pi is joining vk tovi for every 2 ≤ i ≤ k − 1. Without loss of generality, we write
P1= hvk, x1, R1, ¯ui and write Pi = hvk, xi, Ri, viifor every 2 ≤ i ≤ k − 1. Sincevk ∈V01, xi ∈V11and ¯xi ∈V00for
every 1 ≤ i ≤ k − 1. We set C = { ¯x2, ¯x3, . . . , ¯xk−1}.
Case 5.1:v1∈ V10. Let H0 =C ∪ {v1}. Obviously, H0 ⊂ G0, |H0∩V10| = 1, and |H0| =k −1. By induction,
Fig. 6. Illustration for Case 5 ofTheorem 5.
utov1and Qi is joining u to ¯xi for every 2 ≤ i ≤ k − 1. We set W1 = Q1, Wi = hu, Qi, ¯xi, xi, Ri, viifor every
2 ≤ i ≤ k − 1, and Wk = hu, ¯u, P1−1, vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G. SeeFig. 6(a) for
an illustration where k = 6.
Case 5.2:v1 ∈ V11 andv1 ∈ V(P1). Without loss of generality, we write P1 = hvk, Z1, y, v1, Z2, ¯ui. Since
v1∈V11, y ∈ V01and ¯y ∈ V10. Let H0=C ∪{ ¯y}. Obviously, H0⊂G0, |H0∩V01| =1, and |H0| =k −1. By induction,
there is a spanning(u, H0)-fan {Q1, Q2, . . . , Qk−1}of G0. Without loss of generality, we assume that Q1is joining
uto ¯y and Qi is joining u to ¯xi for every 2 ≤ i ≤ k − 1. We set W1 = hu, ¯u, Z−12 , v1i, Wi = hu, Qi, ¯xi, xi, Ri, vii
for every 2 ≤ i ≤ k − 1, and Wk = hu, Q1, ¯y, y, Z−11 , vki. Then {W1, W2, . . . , Wk}is the spanning(u, S)-fan of G.
SeeFig. 6(b) for an illustration where k = 6.
Case 5.3: v1 ∈ V11 andv1 ∈ V(Pi) for some 2 ≤ i ≤ k − 1. Without loss of generality, we assume that
v1 ∈ V(Pk−1) and write Pk−1 = hvk, xk−1, Z1, v1, y, Z2, vk−1i. Since v1 ∈ V11, y ∈ V01 and ¯y ∈ V10. Let
H0 = C ∪ { ¯y}. Obviously, H0 ⊂ G
0, |H0∩V10| = 1, and |H0| = k −1. By induction, there is a (u, H0)-fan
{Q1, Q2, . . . , Qk−1}of G0. Without loss of generality, we assume that Q1is joining u to ¯y and Qi is joining u to
¯
xi for every 2 ≤ i ≤ k − 1. We set W1 = hu, Qk−1, ¯xk−1, xk−1, Z1, v1i, Wi = hu, Qi, ¯xi, xi, Ri, vii for every
2 ≤ i ≤ k − 2, Wk−1= hu, Q1, ¯y, y, Z2, vk−1i, and Wk = hu, ¯u, P1−1, vki. Then {W1, W2, . . . , Wk}is the spanning
(u, S)-fan of G. SeeFig. 6(c) for an illustration where k = 6. Theorem 6. Every graph in B0
nis super spanning laceable for n ≥1.
Proof. Suppose that G = G0⊕G1in Bn0 with bipartition V0and V1. Let u be any node in V0andv be any node in V1.
We need to show there is a k∗-container of G between u andv for every positive integer k with k ≤ n. ByTheorem 2, there is a 1∗-container of G joining u tov. Thus, we assume that k ≥ 2 and n ≥ 2. Since k ≤ n and |N
G(v)| = n,
we can choose(k − 1) distinct nodes x1, x2, . . . , xk−1in NG(v) − {u}. Since v is in V1, xi is in V0− {u}for i = 1
to k − 1. We set S = {v, x1, x2, . . . , xk−1}. ByTheorem 5, there is a spanning(u, S)-fan {R1, R2, . . . , Rk}of G.
Without loss of generality, we assume that R1is joining u tov and Ri is joining u to xi −1for every 2 ≤ i ≤ k. We
set P1=R1and Pi = hu, Ri, xi −1, vi for every 2 ≤ i ≤ k. Then {P1, P2, . . . , Pk}is the k∗-container of G between
uandv.
4. On thew∗-connectedness of Nn0-graphs
4.1. Every graph in Nn0 is3∗-connected
Lemma 2. According to isomorphism, there is only one graph in N30. Moreover, this graph is3∗-connected. Proof. By brute force, we can check the graph T inFig. 7is the only graph in N30.
Let x and y be two distinct nodes of T . By the symmetry of T , we can assume that x = 0 and y ∈ {1, 2, 3, 4}. The 3∗-containers {P1, P2, P3}of T between x and y are listed below:
y =1 {P1= h0, 1i, P2= h0, 4, 3, 2, 1i, P3= h0, 7, 6, 5, 1i}
y =2 {P1= h0, 1, 2i, P2= h0, 7, 3, 2i, P3= h0, 4, 5, 6, 2i}
y =3 {P1= h0, 4, 3i, P2= h0, 7, 3i, P3= h0, 1, 5, 6, 2, 3i}
Fig. 7. The only graph T in N30.
Thus, T is 3∗-connected.
Let n ≥ 3. Let G = G0⊕G1 ∈ Nn+10 with G0∈ Hn0 and G1∈ Hn0. Depending on G0and G1is bipartite or not,
we prove that G = G0⊕G1is 3∗-connected with the following lemmas.
Lemma 3. Let n ≥ 3. Assume that G = G0⊕G1in Nn+10 with both G0and G1in Nn0. Then G is3∗-connected.
Proof. Let u andv be any two distinct nodes of G. We need to construct a 3∗-container of G between u andv. Case 1: u, v ∈ G0. ByTheorem 1, there is a 2∗-container {P1, P2}of G0between u andv. ByTheorem 1again,
there is a hamiltonian path P of G1joining ¯uto ¯v. We set P3as hu, ¯u, P, ¯v, vi. Then {P1, P2, P3}is the 3∗-container
of G between u andv.
Case 2: u ∈ G0andv ∈ G1with ¯u =v. Since there are 2nnodes in G0and 2n> 3 for n ≥ 3, we can choose two
distinct nodes x and y in G0− {u}. ByTheorem 1, there is a hamiltonian path R of G0joining x to y. Again, there
is a hamiltonian path W of G1joining ¯x to ¯y. We write R = hx, R1, u, R2, yi and W = h ¯x, W1, v, W2, ¯yi. We set
P1= hu, R1−1, x, ¯x, W1, vi, P2= hu, R2, y, ¯y, W2−1, vi, and P3= hu, vi. Then {P1, P2, P3}is the 3∗-container of G
between u andv.
Case 3: u ∈ G0andv ∈ G1 with ¯u 6= v. Since there are 2n nodes in G0, we choose a node x in G0− {u, ¯v}.
By Theorem 1, there is a hamiltonian path R of G0 joining x to ¯v. Again, there is a hamiltonian path W of
G1 joining ¯x to ¯u. We write R = hx, R1, u, R2, ¯vi and W = h ¯x, W1, v, W2, ¯ui. We set P1 = hu, ¯u, W2−1, vi,
P2 = hu, R−11 , x, ¯x, W1, vi, and P3 = hu, R2, ¯v, vi. Then {P1, P2, P3}is the 3∗-container of G between u and
v.
Lemma 4. Let n ≥ 3. Assume that G = G0⊕G1in Nn+10 with G0in Bn0 and G1in Nn0. Then G is3
∗-connected.
Proof. Let V0and V1be the bipartition of G0. Let u andv be any two distinct nodes of G. We need to construct a
3∗-container of G between u andv.
Case 1: u, v ∈ G0. ByTheorem 2, there is a 2∗-container {P1, P2}of G0between u andv. ByTheorem 1, there
is a hamiltonian path P of G1joining ¯uto ¯v. We set P3= hu, ¯u, P, ¯v, vi. Then {P1, P2, P3}is the 3∗-container of G
between u andv.
Case 2: u, v ∈ G1. Without loss of generality, we assume that ¯u ∈ V0.
Case 2.1: ¯v ∈ V0. Since there are 2n−1nodes in V1and 2n−1 ≥ 4 for n ≥ 3, we can choose two distinct nodes
x and y in V1. By Theorem 1, there is a hamiltonian path R of G1 joining ¯x to ¯y. Without loss of generality, we
write R = h ¯x, R1, u, R2, v, R3, ¯yi. ByTheorem 3, there are two disjoint paths T1and T2of G0such that (1) T1joins
¯
u to y, (2) T2joins x to ¯v, and (3) T1∪T2 spans G1. We set P1 = hu, R2, vi, P2 = hu, R−11 , ¯x, x, T2, ¯v, vi, and
P3= hu, ¯u, T1, y, ¯y, R3−1, vi. Then {P1, P2, P3}is the 3∗-container of G between u andv.
Case 2.2: ¯v ∈ V1. ByTheorem 1, there is a 2∗-container {P1, P2}of G1between u andv. ByTheorem 2, there is
a hamiltonian path P of G0joining ¯u to ¯v. We set P3 = hu, ¯u, P, ¯v, vi. Then {P1, P2, P3}is the 3∗-container of G
between u andv.
Case 3: u ∈ G0andv ∈ G1with ¯u 6= v. By Theorem 2, there is a hamiltonian cycle C of G0. Without loss
of generality, we write C = hu, R1, ¯v, x, R2, ui. By Theorem 1, there is a hamiltonian path T of G1joining ¯u to
¯
x. Without loss of generality, we write T = h ¯u, T1, v, T2, ¯xi. We set P1 = hu, R1, ¯v, vi, P2 = hu, ¯u, T1, vi, and
P3= hu, R−12 , x, ¯x, T −1
2 , vi. Then {P1, P2, P3}is the 3∗-container of G between u andv.
Case 4: u ∈ G0andv ∈ G1with ¯u =v. Without loss of generality, we assume that u ∈ V0. We can choose a node
xin V0− {u}and a node y in V1. ByTheorem 2, there is a hamiltonian path R of G0joining x to y. ByTheorem 1,
there is a hamiltonian path T of G1 joining ¯x to ¯y. Without loss of generality, we write R = hx, R1, u, R2, yi
and T = h ¯x, T1, v, T2, ¯yi. We set P1 = hu, vi, P2 = hu, R1−1, x, ¯x, T1, vi, and P3 = hu, R2, y, ¯y, T2−1, vi. Then
0 1 0
Theorem 2, there is a hamiltonian path P of G1joining ¯uto ¯v. We set P3= hu, ¯u, P, ¯v, vi. Then {P1, P2, P3}is the
3∗-container of G between u andv.
Case 2:v ∈ V00 and ¯v ∈ V11. Since u ∈ V00, ¯u ∈ V11,v ∈ V00, and ¯v ∈ V11, we can choose a node x in V10 such that ¯x ∈ V01and choose a node y in V00such that ¯y ∈ V01. ByTheorem 2, there is a hamiltonian path R of G0
joining x to y. Without loss of generality, we write R = hx, R1, p, R2, q, R3, yi where {p, q} = {u, v}. Without loss
of generality, we assume that p = u and q =v. ByTheorem 3, there are two disjoint paths T1and T2of G1such that
(1) T1joins ¯xto ¯v, (2) T2joins ¯uto ¯y, and (3) T1∪T2spans G1. We set P1= hu, R2, vi, P2= hu, R−11 , x, ¯x, T1, ¯v, vi,
P3= hu, ¯u, T2, ¯y, y, R3−1, vi. Then {P1, P2, P3}is the 3∗-container of G between u andv.
Case 3:v ∈ V10and ¯v ∈ V11. Since u ∈ V00and ¯u ∈ V11,v ∈ V10, and ¯v ∈ V11, we can choose a node x in V10 such that ¯x ∈ V01and choose a node y in V00such that ¯y ∈ V01. ByTheorem 2, there is a hamiltonian path R of G0
joining x to y. Without loss of generality, we write R = hx, R1, p, R2, q, R3, yi where {p, q} = {u, v}. Without loss
of generality, we assume that p = u and q =v. ByTheorem 3, there are two disjoint paths T1and T2of G1such that
(1) T1joins ¯xto ¯v, (2) T2joins ¯uto ¯y, and (3) T1∪T2spans G1. We set P1= hu, R2, vi, P2= hu, R−11 , x, ¯x, T1, ¯v, vi,
P3= hu, ¯u, T2, ¯y, y, R3−1, vi. Then {P1, P2, P3}is the 3∗-container of G between u andv.
Case 4:v ∈ V1
0 ∪V11and ¯u 6=v.
Case 4.1: ¯v ∈ V00. Since u ∈ V00, ¯u ∈ V11, and ¯v ∈ V00, we can choose a node x ∈ V10 such that ¯x ∈ V01. By Theorem 2, there is a hamiltonian path R of G0joining x to ¯v. Again, byTheorem 2, there is a hamiltonian
path T of G1joining ¯x to ¯u. Write R = hx, R1, u, R2, ¯vi and T = h ¯x, T1, v, T2, ¯ui. We set P1 = hu, ¯u, T2−1, vi,
P2= hu, R2, ¯v, vi, and P3= hu, R1−1, x, ¯x, T1, vi. Then {P1, P2, P3}is the 3∗-container of G between u andv.
Case 4.2: ¯v ∈ V10andv ∈ V01. Since u ∈ V00, ¯u ∈ V11,v ∈ V01, and ¯v ∈ V10, we can choose a node x ∈ V00 such that ¯x ∈ V01. ByTheorem 2, there is a hamiltonian path R of G0 joining x to ¯v, and there is a hamiltonian
path T of G1joining ¯xto ¯u. We write R = hx, R1, u, R2, ¯vi and T = h ¯x, T1, v, T2, ¯ui. We set P1= hu, ¯u, T2−1, vi,
P2= hu, R2, ¯v, vi, and P3= hu, R1−1, x, ¯x, T1, vi. Then {P1, P2, P3}is the 3∗-container of G between u andv.
Case 4.3: ¯v ∈ V10andv ∈ V11. Since u ∈ V00, ¯u ∈ V11, andv ∈ V11, we can choose a node x ∈ V00such that ¯x ∈ V01. ByTheorem 2, there is a hamiltonian path R of G0joining x to ¯v, and there is a hamiltonian path T of G1joining ¯x
to ¯u. We write R = hx, R1, u, R2, ¯vi and T = h ¯x, T1, v, T2, ¯ui. We set P1= hu, ¯u, T2−1, vi, P2= hu, R2, ¯v, vi, and
P3= hu, R1−1, x, ¯x, T1, vi. Then {P1, P2, P3}is the 3∗-container of G between u andv.
Case 5:v = ¯u. Since u ∈ V00and ¯u ∈ V11, we can choose a node x ∈ V00such that ¯x ∈ V01and choose a node y ∈ V10such that ¯y ∈ V11. ByTheorem 2, there is a hamiltonian path R of G0joining x to y, and there is a hamiltonian
path T of G1joining ¯xto ¯y. Without loss of generality, we write that R = hx, R1, u, R2, yi and T = h ¯x, T1, v, T2, ¯yi.
We set P1 = hu, vi, P2 = hu, R1−1, x, ¯x, T1, vi, and Pis = hu, R2, y, ¯y, T2−1, vi. Then {P1, P2, P3}forms the 3∗
-container of G between u andv.
WithLemmas 2–5, we have the following theorem: Theorem 7. Every graph in Nn0 is3∗-connected. 4.2. An Nn0-graph H is not4∗-connected
We say that u = unun−1. . . u2u1is an n-bit binary string if ui ∈ {0, 1} for every 1 ≤ i ≤ n. For 1 ≤ i ≤ n, we
use(u)i to denote the binary string,vnvn−1. . . v2v1, such thatvi =1 − ui andvj =uj for every j 6= i . Moreover,
we use(u)i to denote ui. The Hamming weight of an n-bit binary strings u = unun−1. . . u2u1,w(u), is Pni =1ui.
The n-dimensional hypercube, Qn, consists of all n-bit binary strings as its nodes. Two nodes u = unun−1. . . u2u1
andv = vnvn−1. . . v2v1of Qn are adjacent if and only ifv = (u)i for some i ∈ {1, 2, . . . , n}. Note that Qn is a
{u ∈ V(Qn) | (u)n=i }for i ∈ {0, 1}. Then Qni is isomorphic to Qn−1. By the definition of Qn, Qn∈ Bn0. Let n ≥ 4
and let e = 00. . . 0 | {z }
n
be a node in Qn. We setv = (e)1, p =(e)n, and q =((e)1)n.
Let H be the graph with V(H) = V (Qn) and E(H) = (E(Qn) − {(e, p), (v, q)}) ∪ {(e, q), (v, p)}. Obviously,
H − {(e, q), (v, p)} is a bipartite graph with bipartition A = {x | w(x) is even} and B = {x | w(x) is odd}. Moreover, H is in Nn0 and H = Q0n⊕Q1nfor some 1–1 connectionφ. We will show that H is not k∗-connected for k ≥ 4.
Suppose that there is a k∗-container C = {P1, P2, . . . , Pk}of H between e and q for some k ≥ 4. We have the
following cases:
Case 1: (e, q) ∈ ∪ki =1Pi and(v, p) ∈ ∪ki =1Pi. Without loss of generality, we assume that(e, q) ∈ P1. Thus,
P1= he, qi. Again, we can assume without loss of generality that (v, p) ∈ P2. Obviously, the number of nodes in P2
is 2t2for some integer t2and the number of nodes in Pi is 2ti+1 for some integer ti for every 3 ≤ i ≤ k. Therefore,
there are t2nodes of V(P2)∩ B and (t2−2) nodes of V (P2)∩ A other than e and q, and there are ti nodes of V(Pi)∩ B
and(ti−1) nodes of V (Pi) ∩ A other than e and q for every 3 ≤ i ≤ k. As a consequence, |A| = Pki =2ti +2 − k
and |B| =Pk
i =2ti. Thus, | A| 6= |B|.
Case 2:(e, q) ∈ ∪ki =1Pi and(v, p) /∈ ∪ki =1Pi. Without loss of generality, we assume that(e, q) ∈ P1. Obviously,
the number of nodes in Piis(2ti+1) for some integer tifor every 2 ≤ i ≤ k. Moreover, there are tinodes of V(Pi)∩B,
and(ti−1) nodes of V (Pi) ∩ A other than e and q for every 2 ≤ i ≤ k. As a consequence, |A| = Pki =2ti +3 − k
and |B| =Pk
i =2ti. Thus, | A| 6= |B|.
Case 3:(e, q) /∈ ∪ki =1Pi and(v, p) ∈ ∪ki =1Pi. Without loss of generality, we assume that(v, p) ∈ P1. Obviously,
the number of nodes in P1is 2t1for some integer t1, and the number of nodes in Pi is(2ti+1) for some integer tifor
every 2 ≤ i ≤ k. Moreover, there are t1nodes of V(P1) ∩ B and (t1−2) nodes of V (P1) ∩ A other than e and q,
and there are ti nodes of V(Pi) ∩ B and (ti −1) nodes of V (Pi) ∩ A other than e and q for every 2 ≤ i ≤ k. As a
consequence, | A| =Pk
i =1ti +1 − k and |B| =Pki =1ti. Thus, | A| 6= |B|.
Case 4:(e, q) /∈ ∪i =1k Pi and(v, p) /∈ ∪ki =1Pi. Obviously, the number of nodes in Pi is(2ti +1) for some integer
ti for every 1 ≤ i ≤ k. Moreover, there are tinodes of V(Pi) ∩ B, and (ti−1) nodes of V (Pi) ∩ A other than e and q
for every 1 ≤ i ≤ k. As a consequence, | A| =Pk
i =1ti +2 − k and |B| =Pki =1ti. Thus, | A| 6= |B|.
With Case 1, Case 2, Case 3, and Case 4, C is not a k∗-container of H between e and q. Thus, H is not k∗-connected for any k, 4 ≤ k ≤ n.
5. Concluding remark
In this paper, we have shown that every Bn0 graph is super spanning laceable. With this result, we believe that there should exist more super spanning laceable graphs than we expected. Similarly, there are more superspanning connected graphs to be discussed. We have also shown that every Nn0-graph isw∗-connected for everyw, 1 ≤ w ≤ 3. It would be interesting to characterize those graphs being superspanning connected or superspanning laceable.
Finally, we prove that there exists a spanning(x, S)-fan in any Bn0 graph G with bipartition V0and V1, for any node
xin Vi with i ∈ {0, 1}, and any node subset S with |S| ≤ n such that |S ∩ V1−i| =1. We believe that there are other
bipartite graphs with such a nice property.
We also think that there exists a spanning(x, S)-fan in some incomplete graph G with κ(G) = k for any vertex x and any node subset S such that S is not a cut set with |S| ≤ k. We can easily prove that G is superspanning connected once the above property holds.
References
[1] S. Abraham, K. Padmanabhan, The twisted cube topology for multiprocessors: a study in network asymmetry, Journal of Parallel and Distributed Computing 13 (1991) 104–110.
[2] S.B. Akers, B. Krishnamurthy, A group-theoretic model for symmetric interconnection networks, IEEE Transactions on Computers 38 (1989) 555–566.
[3] M. Albert, R.E.L. Aldred, D. Holton, On 3∗-connected graphs, Australasian Journal of Combinatorics 24 (2001) 193–207. [4] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North Holland, New York, 1980.
[5] C.H. Chang, C.K. Lin, H.M. Huang, L.H. Hsu, The super laceability of the hypercube, Information Processing Letters 92 (2004) 15–21. [6] Y.C. Chen, J.M. Tan, L.H. Hsu, S.S. Kao, Super-connectivity and super-edge-connectivity for some interconnection networks, Applied
[12] D.F. Hsu, On container width and length in graphs, groups, and networks, IEICE Transactions Fundamentals E77-A (1994) 668–680. [13] H.C. Hsu, C.K. Lin, H.M. Huang, L.H. Hsu, The spanning connectivity of the(n, k)-star graphs, International Journal of Foundations of
Computer Science 17 (2006) 415–434.
[14] F.T. Leighton, Introduction to Parallel Algorithms and Architectures: Arrays · Trees · Hypercubes, Morgan Kaufmann, San Mateo, CA, 1992. [15] C.K. Lin, H.M. Huang, L.H. Hsu, The super connectivity of the pancake graphs and the super laceability of the star graphs, Theoretical
Computer Science 339 (2005) 257–271.
[16] C.K. Lin, H.M. Huang, L.H. Hsu, On the spanning connectivity of graphs, Discrete Mathematics 307 (2007) 285–289. [17] K. Menger, Zur allgemeinen Kurventheorie, Fundamentale Mathematik 10 (1927) 95–115.
[18] O. Ore, Hamiltonian connected graphs, Journal of Mathematic Pures Application 42 (1963) 21–27.
[19] C.D. Park, K.Y. Chwa, Hamiltonian properties on the class of hypercube-like networks, Information Processing Letters 91 (2004) 11–17. [20] G.J. Simmons, Almost all n-dimensional rectangular lattices are hamilton-laceable, in: Proceedings of the Ninth Southeastern Conference on
Combinatorics, Graph Theory, and Computing, 1978 pp. 649–661.
[21] C.H. Tsai, J.J.M. Tan, L.H. Hsu, The super connected property of recursive circulant graphs, Information Processing Letters 91 (2004) 293–298.
[22] A.S. Vaidya, P.S.N. Rao, S.R. Shankar, A class of hypercube-like networks, in: Proceedings of the Fifth IEEE Symposium on Parallel and Distributed Processing, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 800–803.
