The Study for Adjacent Vertices Fault-Tolerance
Bifanability of Hypercube
Chun-Nan Hung
Department of Computer Science and Information Engineering
Da-Yeh University Email:[email protected]
Po-Tsun Lin
Department of Computer Science and Information Engineering
Da-Yeh University
Email:[email protected]
Abstract—Let Qn = (Vb ∪ Vw, E) be the
n-dimensional hypercube. An Let Fa be the set of fa
pairs of adjacently faulty vertices. Let s1, t1
2, t21,· · · , t k1 1 ∈ Vb, s2, t1 1, t22,· · · , t k2
2 ∈ Vw be arbitrary fault-free vertices
of Qn. In this paper, we construct the spanning internally
disjoint paths P(s1, ti1) and P (s2, t j
2) of Qn − Fa for fa+ k1+ k2≤ n − 1 and 1 ≤ i ≤ k1,1 ≤ j ≤ k2.
Index Terms—hypercube, vertices fault-tolerance,
fan-ability, bifanfan-ability, Hamiltonian-laceable
I. INTRODUCTION
The hypercube network is one of the most popular interconnection networks. It has many attractive properties, such as regularity, symmetry, small de-gree and diameter, maximum fault tolerance, easy routing algorithms.
Network topology is usually represented by a graph where vertices represent processors and edges represent links between processors. A bipartite graph G = (Vb ∪ Vw, E) is a graph such that
Vb ∩ Vw = ∅ and every e = (u, v) ∈ E, u ∈ Vb
and v ∈ Vw. Let Vb(Vw) be the set of black(white)
vertices. Ann-dimensional hypercube Qnis a
bipar-tite graph with 2n vertices. Each vertex is denoted
as an n-bit binary string. Two vertices of
hyper-cube are adjacent if and only if their binary string representations differ exactly in one bit position. A
path is a sequence of adjacent vertices, denoted as hv1, v2, · · · , vni, where all the vertices except v1 and
vn are distinct. A cycle, written ashv1, v2, ..., vni, is
a path for v1 = vn. A Hamiltonian cycle(path) is a
cycle(path) that visits every vertex exactly once. A graphG is Hamiltonian if G contains a Hamiltonian
cycle. A bipartite graphG = (Vb∪ Vw, E) is
Hamil-tonian laceable if there exists a HamilHamil-tonian path
between each pair of verticesu and v for u ∈ Vb and
v ∈ Vw. A bipartite graphG = (Vb∪Vw, E) is
hyper-Hamiltonian laceable if there exists a hyper-Hamiltonian
path between each pair of vertices u and v of G − {y} for u, v ∈ Vi, z ∈ Vj, {i, j} = {b, w}. A
graph G is k edges hyper-Hamiltonian laceable if G − Fe is hyper-Hamiltonian laceable∀Fe ⊂ E and
|Fe| = k. A graph G is k edges hyper-Hamiltonian
laceable if G − Fe is hyper-Hamiltonian laceable.
∀Fe ⊂ E and |Fe| = k.
The following concepts are introduced in [2]. A k-container C(s, t) of a graph G is a set of k internal vertex disjoint paths between s and t.
Let V (C(s, t)) be the set of vertices incident with
some paths in C(s, t). A k∗-container C(s, t) is a
k-container that V (C(s, t)) = V (G). A k∗-laceable
graph is a bipartite graph that there exists a k∗
-container between every two vertices with different color. Thus, a graph is1∗-laceable and2∗-laceable if
and only if it is Hamiltonian laceable. In [2], Chang et al. showed that Qn is k∗-laceable for1 ≤ k ≤ n.
In [3], Chen et al. proposed some more general concepts. A k-fan A(s → T ) of a bipartite graph G = (Vb∪Vw, E) is a set of k paths from s ∈ Vb and
T = {ti|t1 ∈ Vw, and tj ∈ Vb, for 2 ≤ j ≤ k}, such
that every two paths of them share only the vertex
s. Let V (A(s → T )) be the set of vertices incident
with some paths inA(s → T ). A k∗-fan(s, T ) of G
is ak-fan with V (A(s → T )) = V (G). A graph G is k∗-fanable, if ∀s ∈ V
i and∀T = {t1, t2, · · · , tk|t1 ∈
Vj and t2, t3, · · · , tk∈ Vi} for {i, j} = {b, w}, there
exists a k∗-fan. A graph G is f
e edgesk∗-fanable if
G − Fe is k∗-fanable ∀Fe ⊂ E(G) with |Fe| = fe.
for n ≥ 3, 0 ≤ fe ≤ n − 2, 1 ≤ k + fe ≤ n. Let Fa
be the set of fa pairs of adjacently faulty vertices.
In this paper, we will prove that Qn− Fa is (n-fa)∗
-fanable.
In this paper, we furthermore investigate the fanability with two source vertices, named bifan-ability. A k-bifan B(s1 → T1, s2 → T2) of a
bipartite graph G = (Vb ∪ Vw, E) is a set of k
paths constructed by |T1|-fan A(s1 → T1) and
|T2|-fan A(s2 → T2) for |T1| + |T2| = k. Let
V (B(s1 → T1, s2 → T2)) be the set of vertices
incident with some paths in B(s1 → T1, s2 → T2).
A k∗-bifan B(s 1 → T1, s2 → T2) of G is a k-fan with V (B(s1 → T1, s2 → T2)) = V (G). A graph G is k∗-bifanable, if ∀s 1 ∈ Vx, s2 ∈ Vy and ∀T1 = {t1 1, t21, · · · , ti1|t11 ∈ Vy and t21, t31, · · · , ti1 ∈ Vx} and T2 = {t12, t22, · · · , t j 2|t12 ∈ Vx and t22, t32, · · · , t j 2 ∈ Vy}
for {x, y} = {b, w} and i + j = k, there exists a k∗-bifan. In this paper, we will prove that Q
n− Fa
is (n-1-fa)∗-bifanable of hypercube.
II. THE ADJACENT VERTICES FAULT-TOLERANCE OF FANABILITY
In this section, we will prove the adjacent vertices fault-tolerance of fanability of hypercube.
We first need to define some notations. Let Fa
be the set of fa pairs of adjacently faulty vertices.
Let Vi be the vertex set of Qi
n−1, for i = 0, 1. Let
Fi
a = Fa ∩ Vi and fai be the number of pairs of
adjacently vertices of Fi
a for i = 0, 1.
The following lemma is introduced in [1].
Lemma 1: The graphQn isf -adjacency (n − 2 −
f ) edges Hamiltonian for 0 ≤ f ≤ (n − 2), f
-adjacency (n−2−f ) edges Hamiltonian laceable for 0 ≤ f ≤ (n − 3), and f -adjacency (n − 3 − f ) edges
hyper-Hamiltonian laceable for 0 ≤ f ≤ (n − 3).
The following lemma is proposed in [3].
Lemma 2: TheQnisfeedgesk∗-fanable forn ≥
3, 0 ≤ fe ≤ n − 2 and 1 ≤ k ≤ n − fe.
The following lemma is proved in [5].
Lemma 3: The graphQn isf -adjacency (n−3 −
f ) edges property 2H, for 0 ≤ f ≤ n − 3, n ≥ 3. Theorem 1: The graph Qn is fa-adjacency (n −
fa)∗-fanable, for 0 ≤ fa ≤ n − 3, n ≥ 3.
Proof: We will prove this theorem by induction
on n. Applying Lemma 2, we can obtain that this
theorem is true for fa = 0. The graph Q4 can
be verified to be 1-adjacency 3∗-fanable by brute
force. Thus, this theorem is true forn ≤ 4. Without
loss of generality, we can assume that s ∈ V0. In
the follows, we will assume that fa ≥ 1 and n ≥ 5.
Case 1 t1 ∈ V0. Case 1.1 |T1| = 0 and f1 a = 0. Case 1.1.1 f0 a ≤ n − 4. Since Q0
n−1 is fa-adjacency (k − 1)∗-fanable, there
exists a (k − 1)∗-fan A(s → {t
1, · · · , tk−1})
of Q0
n−1 − Fa0. Thus, tk is on some path
of A(s → {t1, · · · , tk−1}). Without loss
of generality, we can assume tk is on
P (s, tk−1). Thus, the path P (s, tk−1) can
be written as hs P(s,tk)
−→ tk, a
P(a,tk−1)
−→ tk−1i.
Applying Lemma 1, we can construct a
Hamiltonian path P (φ(s), φ(a)) of Q1
n−1. Hence, A(s → {t1, · · · , tk−1}) ∪ {hs, φ(s) P(φ(s),φ(a)) −→ φ(a), a P(a,t−→k−1)tk−1i, P (s, tk)} − {P (s, tk−1)} is a k∗-fan of Q
n− Fa, as illustrated in Figure 1 (a).
Case 1.1.2 f0
a = n − 3.
Applying Lemma 1, we can construct a Hamiltonian cycle in Q0
n−1 − Fa. Without loss of generality,
we can assume that this cycle is denoted as
ht1, a1 P(a1,t2) −→ t2, a2 P(a2,t3) −→ t3, a3 P(a3,s) −→ s P−→ t(s,t1) 1i.
Applying Lemma 3, we can construct two spanning disjoint paths P (φ(s), φ(a2)) and P (φ(a3), φ(a1)).
Thus, the three disjoint paths hs P(s,a3)
−→ a3, φ(a3) P(φ(a3),φ(a1)) −→ φ(a1), a1 P(a1,t2) −→ t2i, P (s, t1)
and hs, φ(s) P(φ(s),φ(a−→ 2)) φ(a2), a2
P(a2,t3)
−→ t3, i form
the 3∗-fan A(s → {t
1, t2, t3}) of Qn − Fa, as
illustrated in Figure 1 (b).
Fig. 1. Illustration of Case 1.1
Case 1.2 |T1| = 0 and 1 ≤ f1
Since f1
a ≥ 1, Q0n−1 − Fa0 is k∗-fanable. There
exists a k∗-fan A0(s → T ) of Q0
n−1 − Fa0.
Without loss of generality, we can assume that the path P (s, tk) = hs P(s,a) −→ a, b P(b,tk) −→ tki for φ(a), φ(b) ∈/ F1 a. Since fa1 ≤ n − 4, Q1
n−1 − Fa1 is Hamiltonian laceable. There exists
a Hamiltonian path P (φ(a), φ(b)) of Q1
n−1 − Fa1. Thus, A0(s → T ) − {P (s, tk)} ∪ {hs P(s,a) −→ a, φ(a) P(φ(a),φ(b))−→ φ(b), b P(b,tk) −→ tki} is a k∗-fan of
Qn− Fa, as illustrated in Figure 2 (a).
Fig. 2. Illustration of Case 1.2 and Case 1.3
Case 1.3 |T1| = 0 and f1
a = n − 3.
Applying Lemma 1, we can construct a Hamiltonian cycle C of Q1
n−1 − Fa1. We can choose a
neighbor of t3, denoted a, such that the cycle
C = hφ(a), φ(b) P(φ(b),φ(a))−→ φ(a)i for b /∈ {t2, t3}.
By induction hypothesis, there exists a 3∗-fan
A(s → {t1, t2, b}) of Q0n−1 − {t3, a}. Thus,
A(s → {t1, t2, b}) − {P (s, b)} ∪ {hs
P(s,b)
−→ b, φ(b) P(φ(b),φ(a))−→ φ(a), a, t3i} is a 3∗-fan of
Qn− Fa, as illustrated in Figure 2 (b).
Case 1.4 f0
a = n − 3 and |T1| = 1.
Without loss of generality, we can assume that
t3 ∈ V1. Applying Lemma 1, we can obtain a
Hamiltonian cycle C of Q0
n−1 − Fa0. Suppose
that (t1, t2) is an edge of C. Thus, there exist
two spanning disjoint paths P (s, t1) and P (s, t2)
of Q0
n−1 − Fa0. There also exists a Hamiltonian
path P (φ(s), t3) of Q1n−1. These three paths form
the 3∗-fan A(s → T ) of Q
n − Fa. Suppose that
(t1, t2) is not an edge of C. We can denote C as
hs P(s,t1)
−→ t1, a P(a,b)
−→ b, t2
P(t2,s)
−→ si. Suppose that φ(b) 6= t3. Applying Lemma 3, we can construct
two spanning disjoint paths P (φ(s), φ(b)) and
P (φ(a), t3) of Q1n−1. Thus, P (s, t1), P (t2, s) and
hs, φ(s) P(φ(s),φ(b))−→ φ(b), b P−→ a, φ(a)(b,a) P(φ(a),t3)
−→ t3i
form a 3∗-fan A(s → T ) of Q
n − Fa. Suppose
that φ(b) = t3. Applying Lemma 1, we can
obtain a Hamiltonian path P (φ(s), φ(a)) of Q1
n−1 − {t3}. Thus, P (s, t1), P (t2, s) and
hs, φ(s) P(φ(s),φ(a))−→ φ(a), a P−→(a,b) b, t3i form a
3∗-fan A(s → T ) of Q
n− Fa.
Case 1.5 f0
a = n − 3 and |T1| = 2.
Applying Lemma 1, we can construct a Hamiltonian cycle hs P(s,t1)
−→ t1, a
P(a,s)
−→ si of Q0
n−1 − Fa0.
Applying Lemma 3, we can obtain two spanning disjoint paths P (φ(s), t2) and P (φ(a), t3) of
Q1
n−1. Thus, P (s, t1), hs, φ(s)
P(φ(s),t2)
−→ t2i and
hs P−→(s,a) a, φ(a) P(φ(a),t3)
−→ t3i form a 3∗-fan
A(s → T ) of Qn− Fa.
Case 1.6 f0
a ≤ n − 4 and |T1| = 1.
Without loss of generality, we can assume that
tk ∈ V1. By induction hypothesis, Q0n−1 is fa0
-adjacency (n − 1 − f0
a)-bifanable for fa0 ≤ n − 4.
There exists a (k − 1)∗-fan A(s → {t
1, · · · , tk−1})
of Q0
n−1− Fa0. Applying Lemma 1, we can obtain
a Hamiltonian path P (φ(s), tk) of Q1n−1. Thus,
A(s → {t1, · · · , tk−1}) ∪ {hs, φ(s) P(φ(s),tk) −→ tki} is a k∗-fan A(s → T ) of Q n− Fa. Case 1.7 f0 a ≤ n − 4 and |T1| ≥ 2.
Without loss of generality, we can assume that
tj+1, · · · , tk ∈ V1. Let (φ(ai), ti) be edges
of Q1
n−1 such that ai ∈ (F/ a0 ∪ {t2, · · · , tj})
for j + 1 ≤ i ≤ k − 2. Let φ(bk−1) be a
white node of Q1
n−1 for bk−1 ∈ F/ a0. Applying
Lemma 3, we can construct two spanning
disjoint paths P (φ(s), tk) and P (φ(bk−1), tk−1)
of Q1
n−1 − {tm, φ(am)| for 2 ≤ m ≤ k − 2}. By
induction hypothesis, there exists a (k − 1)∗-fan
A(s → {t1, t2, · · · , tj, aj+1, · · · , ak−2, bk−1})
of Q0
n−1 − Fa0. Thus, the following
paths hs, φ(s) P(φ(s),tk) −→ tki, hs P(s,bk−1) −→ bk−1, φ(bk−1) P(φ(bk−1),tk−1) −→ tk−1i, P (s, tm) and hs P(s,ai)
−→ ai, φ(ai), tii form a k∗-fan A(s → T ) of
Qn− Fa for 1 ≤ m ≤ j and j + 1 ≤ i ≤ k − 2.
Case 1.8 |T1| = 1 and f1
a ≤ n − 4.
tk ∈ V1. By induction hypothesis, there exists a
k∗-fan A(s → {t
1, · · · , tk−1, ak}) of Q0n−1− Fa0 for
ak, φ(ak) /∈ Fa. Applying Lemma 1, we can obtain a
Hamiltonian path P (φ(ak), tk) of Q1n−1− Fa1. Thus,
A(s → {t1, · · · , tk−1, ak}) − {P (s, ak)} ∪ {hs P(s,ak) −→ ak, φ(ak) P(φ(ak),tk) −→ tki} is a k∗-fan A(s → T ) of Qn− Fa. Case 1.9 |T1| ≥ 2 and f1 a + |T1| ≤ n − 2.
Without loss of generality, we can assume that
tj+1, · · · , tk ∈ V1. Let (φ(ai), ti) be edges of Q1n−1
such that ai ∈ (F/ a0∪ {t2, · · · , tj}) for j + 1 ≤ i ≤ k.
By induction hypothesis, there exists a k∗-fan
A(s → {t1, t2, · · · , tj, aj+1, · · · , ak}) of Q0n−1− Fa0.
Applying Lemma 3, we can construct two spanning disjoint paths P (φ(ak), tk) and P (φ(ak−1, tk−1) of
Q1
n−1− {tm, φ(am)| for j + 1 ≤ m ≤ k − 2}. Thus,
the following paths P (s, tm), hs P(s,ai) −→ ai, φ(ai), tii and hs P(s,ar) −→ ar, φ(ar) P(φ(ar),tr) −→ tri form a k∗-fan A(s → T ) of Qn− Fa for 1 ≤ m ≤ j, r = k − 1, k and j + 1 ≤ i ≤ k − 2. Case 1.10 |T1| = 1 and f1 a = n − 3.
Without loss of generality, we can assume that
t3 ∈ V1. Applying Lemma 1, we can obtain
a Hamiltonian cycle ht3, φ(a3)
P(φ(a3),t3)
−→ t3i of
Q1
n−1 − Fa1 for a3 6= t2. By induction hypothesis,
there exists a 3∗-fan A(s → {t
1, t2, a3}) of Q0n−1.
Thus, the following paths P (s, t1), P (s, t2), and
hs P(s,a3) −→ a3, φ(a3) P(φ(a3),t3) −→ t3i form a 3∗-fan A(s → T ) of Qn− Fa. Case 1.11 |T1| ≥ 2 and f1 a + |T1| = n − 1.
Let (φ(ai), ti) be edges of Q1n−1 for 4 ≤ i ≤ k.
Applying Lemma 1, we can obtain a Hamiltonian cycle ht3, φ(a2) P(φ(a2),t2) −→ t2, φ(a3) P(φ(a3),t3) −→ t3i of Q1 n−1 − Fa1 − {ti, φ(ai)| for 4 ≤ i ≤ k.
By induction hypothesis, there exists a k∗-fan
A(s → {t1, a2, a3, · · · , ak}) of Q0n−1. Thus, the
following paths P (s, t1), hs
P(s,ai)
−→ ai, φ(ai), tii,
and hs P−→ a(s,aj) j, φ(aj)
P(φ(aj),tj)
−→ tji form a k∗-fan
A(s → T ) of Qn− Fa for i = 2, 3 and 4 ≤ j ≤ k.
Case 2 t1 ∈ V1 and fa1 = 0.
Without loss of generality, we can assume that
t1 6= φ(s).
Case 2.1 f0
a = n − 3 and |T1| = 1.
Applying Lemma 1, we can construct a Hamiltonian cycle hs P(s,t2) −→ t2, a2 P(a2,t3) −→ t3, a3 P(a3,s) −→ si of Q0 n−1 − Fa0. Applying Lemma 3, we
can construct two spanning disjoint paths
P (φ(s), φ(a2)) and P (φ(a3), t1). Thus, these
three paths hs P(s,a3)
−→ a3, φ(a3) P(φ(a3),t1) −→ t1i, P (s, t2), hs, φ(s) P(φ(s),φ(a2)) −→ φ(a2), a2 P(a2,t3)
−→ t3i form a 3∗-fan A(s → T ) of
Qn− Fa.
Case 2.2 f0
a = n − 3 and |T1| = 2.
Without loss of generality, we can assume that
t2 ∈ V1. Applying Lemma 1, we can construct
a Hamiltonian cycle hs P(s,t3)
−→ t3, a P(a,s)
−→ si of Q0
n−1 − Fa0 for φ(a) 6= t2. Applying Lemma 3,
we can construct two spanning disjoint paths
P (φ(s), t2) and P (φ(a), t1). Thus, these three
paths hs P−→ a, φ(a)(s,a) P(φ(a),t1)
−→ t1i, hs, φ(s)
P(φ(s),t2)
−→ t2i, P (s, t3) form a 3∗-fan A(s → T ) of Qn− Fa.
Case 2.3 fa0 = n − 3 and |T1| = 3.
Let (φ(a2), t2) be an edge of Q1n−1 for
a2 ∈/ (Fa ∪ {s}). Applying Lemma 1,
we can construct a Hamiltonian cycle
hs P(s,a2)
−→ a2, a1
P(a1,s)
−→ si of Q0
n−1 − Fa0 for
φ(a1) 6= t3. Applying Lemma 3, we can construct
two spanning disjoint paths P (φ(s), t3) and
P (φ(a1), t1). Thus, these three paths hs
P(s,a1) −→ a1, φ(a1) P(φ(a1),t1) −→ t1i, hs P(s,a2) −→ a2, φ(a2), t2i, and hs, φ(s) P(φ(s),t3)
−→ t3i form a 3∗-fan A(s → T ) of
Qn− Fa.
Case 2.4 f0
a ≤ n − 4 and |T1| = 1.
Let a be a white node in Q0
n−1. By induction
hypothesis, there exists a (k − 1)∗-fan
A(s → {a, t2, · · · , tk−1}) of Q0n−1 − Fa0. Without
loss of generality, we can assume that tk is on
the path P (s, tk−1). We can denote P (s, tk−1) as
hs P(s,tk)
−→ tk, b
P(b,tk−1)
−→ tk−1i. Applying Lemma
3, we can obtain two spanning disjoint paths
P (φ(s), φ(b)) and P (φ(a), t1) of Q1n−1. Thus,
the following paths hs P−→(s,a) a, φ(a) P(φ(a),t1)
−→ t1i, P (s, ti), hs, φ(s)
P(φ(s),φ(b))
−→ φ(b), b P(b,tk−1)
−→ tk−1i, and P (s, tk) form a k∗-fan A(s → T ) of
Case 2.5 f0
a ≤ n − 4 and 2 ≤ |T1| ≤ n − 2.
Without loss of generality, we can assume that
t2, t3, · · · , tj ∈ V1. Let (φ(ai), ti) be edges
of Q1
n−1 such that ai ∈ (F/ a0 ∪ {tj+1, · · · , tk})
for 1 ≤ i ≤ j − 1. By induction
hypothesis, there exists a (k − 1)∗-fan
A(s → {a1, a2, · · · , aj−1, tj+1, · · · , tk}) of
Q0
n−1 − Fa0. Applying Lemma 3, we can construct
two spanning disjoint paths P (φ(aj−1), tj−1)
and P (φ(s), tj) of Q1n−1 − {ti, φ(ai)| for
1 ≤ i ≤ j − 2}. Thus, the following
paths hs P(s,ai) −→ ai, φ(ai), tii, hs P(s,aj−1) −→ aj−1, φ(aj−1) P(φ(aj−1),tj−1) −→ tj−1i, hs, φ(s) P(φ(s),tj) −→ tji
and P (s, tm) form a k∗-fan A(s → T ) of Qn− Fa
for 1 ≤ i ≤ j − 2 and j + 1 ≤ m ≤ k. Case 2.6 f0
a = 1 and |T1| = n − 1.
Let x be a black node of V1 such that the distance
of x and the black faulty node is more than 2. By
induction hypothesis, there exists a (n − 1)∗-fan
A(x → T ) of Q1
n−1. Without loss of generality,
we can assume that φ(s) is on the path P (x, t1).
Without loss of generality, we can assume that
P (x, t1) = hx
P(x,φ(an−1))
−→ φ(an−1), φ(s)
P(φ(s),t1)
−→ t1i
for an−1 ∈ F/ a0 . We can denote P (x, ti) as
hx, φ(ai)
P(φ(ai),ti)
−→ tii for 2 ≤ i ≤ n − 2.
By induction hypothesis, there exists a
(n − 2)∗-fan A(s → {a
2, a3, · · · , an−1})
of Q0
n−1 − Fa0. Thus, the following paths
hs, φ(s) P(φ(s),t1) −→ t1i, hs P(s,ai) −→ ai, φ(ai) P(φ(ai),ti) −→ tii and hs P(s,an−1) −→ an−1, φ(an−1) P(φ(an−1),x) −→ xP(x,tn−1) −→ tn−1i form a k∗-fan A(s → T ) of Qn − Fa for
1 ≤ i ≤ j − 2.
Case 3 t1 ∈ V1 and fa1 ≥ 1.
Case 3.1 f1
a + |T1| = n.
Let (φ(ai), ti) be edges of Q1n−1 for 4 ≤ i ≤ k.
Ap-plying Lemma 1, we can obtain a Hamiltonian cycle
ht1, φ(a3) P(φ(a3),t3) −→ t3, φ(a2) P(φ(a2),t2) −→ t2, φ(a1) P(φ(a1),t1) −→ t1i of Q1n−1 − Fa1 − {φ(ai), ti| for
4 ≤ i ≤ k}. By induction hypothesis, there exists
a k∗-fan A(s → {a
1, a2, · · · , ak}) of Q0n−1. Thus,
the following paths hs P(s,ai)
−→ ai, φ(ai)
P(φ(ai),ti)
−→ tii, hs
P(s,am)
−→ am, φ(am), tmi form a k∗-fan
A(s → T ) of Qn − Fa for 1 ≤ i ≤ 3 and
4 ≤ m ≤ k.
Case 3.2 f1
a + |T1| ≤ n − 1 and fa1 ≤ n − 4.
Without loss of generality, we can assume that
t2, t3, · · · , tj ∈ V1. Let b be a white node of
Q0
n−1 such that |N(b) ∩ (Fa∪ {tj+1, · · · , tk})| ≤ 1
where N(b) is the vertex set of the neighbor
of b. By induction hypothesis, there exists a j∗-fan A(φ(b) → {t 1, t2, · · · , tj}) of Q1n−1 − Fa1. We can denote (φ(b) → {t1, t2, · · · , tj}) as hφ(b), φ(ai) P(φ(ai),ti) −→ tii for 1 ≤ i ≤ j.
Without loss of generality, we can assume that a2, · · · , aj ∈/ (Fa ∪ {tj+1, · · · , tk}). By
induction hypothesis, there exists a k∗-fan
A(s → {b, a2, · · · , aj, tj+1, · · · , tk}) of Q0n−1− Fa0.
Thus, the following paths hs P−→ b, φ(b)(s,b) P(φ(b),t1)
−→ t1i, hs P(s,ai) −→ ai, φ(ai) P(φ(ai),ti) −→ tii and P (s, tm)
form a k∗-fan A(s → T ) of Q
n− Fa for 2 ≤ i ≤ j
and j + 1 ≤ m ≤ k. Case 3.3 f1
a = n − 3 and |T1| = 1.
Applying Lemma 1, we can obtain a Hamiltonian cycle ht1, φ(a)
P(φ(a),t1)
−→ t1i of Q1n−1 − Fa1.
By induction hypothesis, there exists a 3∗-fan
A(s → {a, t2, t3} of Q0n−1. Thus, the following
paths hs P−→ a, φ(a)(s,a) P(φ(a),t1)
−→ t1i, P (s, t2), P (s, t3)
form a 3∗-fan A(s → T ) of Q
n− Fa.
Case 3.4 f1
a = n − 3 and |T1| = 2.
Without loss of generality, we can assume that t2 ∈ V1. Applying Lemma 1, we can
obtain a Hamiltonian cycle ht1, φ(a2)
P(φ(a2),t2)
−→ t2, φ(a1)
P(φ(a1),t1)
−→ t1i of Q1n−1− Fa1 for φ(a2) 6= t3.
By induction hypothesis, there exists a 3∗-fan
A(s → {a1, a2, t3} of Q0n−1. Thus, the following
paths hs P(s,ai) −→ ai, φ(ai) P(φ(ai),ti) −→ tii, P (s, t3) form a 3∗-fan A(s → T ) of Q n− Fa for i = 1, 2.
III. THE ADJACENT VERTICES FAULT-TOLERANCE OF BIFANABILITY The following lemma is proved in [5].
Lemma 4: Let Qn = {Vb ∪ Vw, E}. For a ∈
Vb, b ∈ Vw, the graph Qn− {a, b} is fa-adjacency
(n − fa− 3) edges Hamiltonian laceable for fa ≤
n − 4, n ≥ 4.
Lemma 5: Lets1, t1 ∈ Vw and s2, t2 ∈ Vb be two
pairs of fault-free vertices andFabe the set withfa
two spanning disjoint paths P (s1, t1) and P (s2, t2)
of Qn− Fa for fa≤ n − 4, n ≥ 4.
Proof. By symmetry of hypercube, we can arrange
every adjacently faulty nodes either in Q0 n−1 or
Q1
n−1. Let T = {s1, t1, s2, t2} and Ti be the subset
of vertices of T in Qi
n−1 for i = 0, 1. Without loss
of generality, we can assume that |T0| ≥ |T1|. We
will prove this lemma in the following cases.
Case 1 |T0| = 4 and f1 a = 0.
Applying Lemma 3, we can construct two span-ning disjoint paths hs1
P(s1,b1) −→ b1, w1 P(w1,t2) −→ t2i and hs2 P(s2,b2) −→ b2, w2 P(w2,t1) −→ t1i of Q0n−1 − F0
a. Applying Lemma 3, we can further
con-struct two spanning disjoint paths P (φ(b1), φ(w2))
and P (φ(b2), φ(w1)) of Q1n−1. Thus, hs1 P(s1,b1) −→ b1, φ(b1) P(φ(b1),φ(w2)) −→ φ(w2), w2 P(w2,t1) −→ t1i and hs2 P(s2,b2) −→ b2, φ(b2) P(φ(b2),φ(w1)) −→ φ(w1), w1 P(w1,t2) −→ t2i are two spanning disjoint paths of Qn− Fa.
Case 2 |T0| = 4 and f1 a ≥ 1.
Let (x1, t1) be an edge of Q0n−1 for φ(x1) /∈ Fa1.
Let E(t2) = {(v, t2)|φ(v) ∈ Fa1}. Applying Lemma
1, we can construct a Hamiltonian path hs2
P(s2,t2) −→ t2, x2 P(x2,s1) −→ s1i of Q0n−1 − Fa0 − {x1, t1} − E(t2). Thus, hs1 P(s1,x2) −→ x2, φ(x2) P(φ(x2),φ(x1)) −→ φ(x1), x1, t1i and P (s2, t2) are two spanning disjoint
paths of Qn− Fa.
Case 3 |T0| = 3.
Without loss of generality, we can assume that
t2 ∈ V1. Let E(t1) = {(v, t1)|φ(v) ∈ Fa1}.
Applying Lemma 1, we can construct a Hamiltonian path hs1
P(s1,t1)
−→ t1, x
P(x,s2)
−→ s2i of Q0n−1 − Fa0 −
E(t1). Applying Lemma 1, we can also construct a
Hamiltonian path P (φ(x), t2) of Q1n−1− Fa1. Thus,
P (s1, t1) and hs2 P(s2,x)
−→ x, φ(x)P(φ(x),t2)
−→ t2i are two
spanning disjoint paths of Qn− Fa.
Case 4 |T0| = 2.
Without loss of generality, we can assume that s1 ∈
V0 andt
2 ∈ V1. Supposet1 ∈ V0 and s2 ∈ V1. Let
x1 be a black node ofQ0n−1forx1, φ(x1) /∈ (Fa∪T ).
Let E(t1) = {(v, t1)|φ(v) ∈ Fa1}. Applying Lemma
1, we can construct a Hamiltonian path hs1
P(s1,t1)
−→ t1, x2
P(x2,x1)
−→ x1i of Q0n−1− Fa0 − E(t1). Applying
Lemma 3, we can construct two spanning disjoint paths P (φ(x1), t2) and P (s2, φ(x2)) of Q1n−1− Fa1. Thus, P (s1, t1) and hs2 P(s2,φ(x2)) −→ φ(x2), x2 P(x2,x1) −→ x1, φ(x1) P(φ(x1),t2)
−→ t2i are two spanning disjoint
paths of Qn− Fa.
Suppose s2 ∈ V0 and t1 ∈ V1. Applying Lemma
1, we can construct a Hamiltonian path hs1
P(s1,w)
−→ w, b P−→ s(b,s2) 2i of Qn−10 − Fa0 for w ∈ Vw, b ∈ Vb
and φ(b), φ(w) /∈ (F1
a ∪ T1). Applying Lemma
3, we can construct two spanning disjoint paths
P (φ(b), t2) and P (φ(w), t1) of Q1n−1 − Fa1. Thus, hs1 P(s1,w) −→ w, φ(w) P(φ(w),t−→1) t1i and hs2 P(s2,b) −→ b, φ(b) P(φ(b),t−→2) t2i are two spanning disjoint paths
of Qn− Fa.
In the following, we will prove the adjacent ver-tices fault-tolerance for bifanability of hypercube.
Theorem 2: The graphQn−Fais(n − |Fa| −
1)-bifanable graph if |Fa| ≤ n − 3 for n ≥ 3.
Proof: We will prove this theorem by induction on n. Since Qn has property 2H, this theorem is true
forn = 3. We can verify this theorem for n = 4 by
brute force. Applying Lemma 3, we can obtain that this theorem holds if |Fa| = n − 3. In the follows,
we will assume that |Fa| ≤ n − 4 and n ≥ 5. By
symmetry of hypercube, we can assume that every pair of adjacently faulty vertices is either inQ0
n−1 or
Q1
n−1. Let s1, s2 be the source vertices for s1 ∈ Vb
and s2 ∈ Vw. Let T1 = {t11, t21, · · · , tk 1 1 } and T2 = {t1 2, t22, · · · , tk 2
2 } be the sets of end vertices of s1 and
s2, respectively. And let the vertices t11, ti2 ∈ Vw and
t1 2, t
j
1 ∈ Vw for 2 ≤ i ≤ k1, 2 ≤ j ≤ k2. Let Tkr be
the set of end vertices of sk in Qr for r = 0, 1 and
k = 1, 2. Let Tr= Tr
1∪T2rforr = 0, 1. Without loss
of generality, we can assume that k1 ≥ k2. Thus,
k1 ≥ 2. We will prove the induction step with the
following cases.
Case 1 s1, s2 ∈ V0 or s1, s2 ∈ V1.
Without loss of generality, we can assume that
s1, s2 ∈ V0.
Case 1.1 |T1| = 0 and |F1 a| = 0.
Let T0
1 = T1 − {tk11}. By induction hypothesis, we
can construct (n-2-fa)∗-bifan B(s1 → T10, s2 → T2)
ofQ0n−1− F0
a. Suppose the vertext k1
1 is on the path
P (s1, ti1) for some 1 ≤ i ≤ k1− 1. We can denote
the path P (s1, ti1) as hs1 P(s1,tk11 ) −→ tk1 1 , x P(x,ti 1) −→ ti 1i.
Applying Lemma 1, we can construct a Hamiltonian path P (φ(s1), φ(x)) of Q1n−1. We can construct the
B(s1 → T1, s2 → T2) from B(s1 → T10, s2 → T2) by replacing P (s1, ti1) with hs1, φ(s1) P(φ(s1),φ(x)) −→ φ(x), xP(x,t i 1) −→ ti 1i and P (s1, tk11) of Qn− Fa.
Suppose the vertex tk1
1 is on the path P (s2, ti2) for
some 1 ≤ i ≤ k2. We can denote the path P (s2, ti2)
as hs2 P(s2,x1) −→ x1, tk11, x2 P(x2,ti2) −→ ti 2i. Applying
Lemma 5, we can construct two Hamiltonian paths
P (φ(s1), φ(tk11)) and P (φ(x1), φ(x2)) of Q1n−1. We
can construct the B(s1 → T1, s2 → T2) from
B(s1 → T10, s2 → T2) by replacing P (s2, ti2) with hs1, φ(s1) P(φ(s1),φ(tk11 )) −→ φ(tk1 1 ), t k1 1 i and hs2 P(s2,x1) −→ x1, φ(x1) P(φ(x1),φ(x2)) −→ φ(x2), x2 P(x2,ti2) −→ ti 2i of Qn− Fa.
Fig. 3. Illustration of Case 2.1
Case 1.2 |T1| = 0 and |F1 a| ≥ 1.
By induction hypothesis, we can construct the
B(s1 → T1, s2 → T2) of Q0n−1 − Fa0. Without loss
of generality, we can assume that the path P (s1, t11)
can be denoted as hs1 P(s1,x1) −→ x1, x2 P(x2,t11) −→ t1 1i
for φ(x1), φ(x2) /∈ Fa1. Applying Lemma 1, we
can construct a Hamiltonian path P (φ(x1), φ(x2))
of Q1
n−1 − Fa1. Thus, we can construct B(s1 →
T1, s2 → T2) by replacing P (s1, t11) with hs1 P(s1,x1) −→ x1, φ(x1) P(φ(x1),φ(x2)) −→ φ(x2), x2 P(x2,t11) −→ t1 1i. Case 1.3 |T1| ≥ 1 and |F1 a| ≥ 1. Case 1.3.1 |T1| = 1.
Without loss of generality, we can assume that
t1
1 ∈ Q1n−1. Let x1 ∈ Q0n−1 be a white vertex such
that x1, φ(x1) /∈ (Fa∪ T1 ∪ T2). Applying Lemma
1, we can construct a Hamiltonian pathP (φ(x1), t11)
ofQ1
n−1−Fa1. By induction hypothesis, we can
con-structB(s1 → {x1, t21, · · · , t k1
1 }, s2 → T2) of Q0n−1−
F0
a. Thus, we can construct B(s1 → T1, s2 → T2)
of Qn − Fa from B(s1 → {x1, t21, · · · , t k1
1 }, s2 →
T2) with replacing the path P (s1, x1) by the path
hs1 P(s1,x1) −→ x1, φ(x1) P(φ(x1),t11) −→ t1 1i. Case 1.3.2 |T1| ≥ 2 and |T1| + |F1 a| ≤ n − 2.
Without loss of generality, we can assume that
t1 1, t21, · · · , t i1 1, t12, t22, · · · , t i2 2 ∈ V1 and i1 ≥ 2. Let (φ(xj1 1 ), t j1 1 ) and (φ(x j2 2 ), t j2 2 ) be edges of Q1n−1 such that xj1 1 , x j2 2 , φ(x j1 1 ), φ(x j2 2 ) /∈ (Fa ∪ T1 ∪ T2 ∪ {s1, s2}) for 1 ≤ j1 ≤ i1, 1 ≤
j2 ≤ i2. Applying Lemma 3, we can
con-struct two spanning disjoint paths P (φ(x1
1), t11) and
P (φ(x2
1), t21). By induction hypothesis, we can
con-struct B(s1 → {x11, · · · , xi 1 1 , ti 1+1 1 , · · · , tk 1 1 }, s2 → {x1 2, · · · , xi 2 2 , ti 2+1 2 , · · · , tk 2 2 }) of Q0n−1 − Fa0. Thus, hs1 P(s1,xm1 ) −→ xm 1 , φ(xm1 ) P(φ(xm 1 ),tm1) −→ tm 1 i, hs1 P(s1,xj11 ) −→ xj1 1 , φ(x j1 1 ), t j1 1 i, P (s1, tr 1 1 ), hs2 P(s2,xj22 ) −→ xj2 2 , φ(x j2 2 ), tj2
2 i, P (s2, tr22) form the (n-1-fa)∗-bifan of Qn− Fa
for 1 ≤ m ≤ 2, 3 ≤ j1 ≤ i1, i1+ 1 ≤ r1 ≤ k1, 1 ≤ j2 ≤ i2, i2+ 1 ≤ r2 ≤ k2. Case 1.3.3 |T1| ≥ 2 and |T1| + |F1 a| = n − 1. Let (φ(xj1 1 ), t j1 1 ) and (φ(x j2 2 ), t j2 2 ) be edges of Q1n−1 such that xj1 1 , x j2 2 , φ(x j1 1 ), φ(x j2 2 ) /∈ (Fa∪ T1∪ T2 ∪ {s1, s2}) for 1 ≤ j1 ≤ k1, 1 ≤ j2 ≤ k2. Applying
Lemma 3, we can construct two spanning disjoint paths ht2 1 P(t2 1,φ(x 2 1)) −→ φ(x2 1), φ(x11) P(φ(x1 1),t 1 1) −→ t1 1i and P (φ(x3 1), t31) of Q1n−1−Fa1−{φ(xj 1), tj1 1 , φ(x j2 2 ), t j2 2 | for 4 ≤ j1 ≤ k1 and 1 ≤ j2 ≤
k2}. By induction hypothesis, we can construct
B(s1 → {x11, x21, · · · , x k1 1 }, s2 → {x12, x22, · · · , x k2 2 }) of Q0 n−1. Thus, hs1 P(s1,xm1) −→ xm 1 , φ(xm1 ) P(φ(xm 1 ),tm1 ) −→ tm 1 i, hs1 P(s1,xj11 ) −→ xj1 1 , φ(x j1 1 ), t j1 1 i, hs2 P(s2,xj22 ) −→ xj2 2 , φ(x j2 2 ), t j2
2 i form the (n-1-fa)∗-bifan ofQn− Fa
for 1 ≤ m ≤ 3, 4 ≤ j1 ≤ k1, 1 ≤ j2 ≤ k2.
Case 1.4 |T1| ≥ 1 and |F1 a| = 0.
Case 1.4.1 t1
1 ∈ V1 or t12 ∈ V1.
Without loss of generality, we can assume that
t1
1 ∈ V1. Suppose that k2 ≥ 2. By
in-duction hypothesis, we can construct B(s1 →
{t2 2, t21, · · · , tk 1 1 }, s2 → {t12, t32, · · · , xk 2 2 }) of Q0n−1 − F0 a. We can denote P (s1, t22) as hs1, x1 P(x1,t22) −→ t2 2i.
span-ning disjoint paths P (φ(s1), t11), P (φ(s2), φ(x1))
of Q1
n−1. Thus, we can construct B(s1 →
T1, s2 → T2) of Qn − Fa from B(s1 → {t2 2, t21, · · · , t k1 1 }, s2 → {t12, t32, · · · , x k2 2 }) with
replac-ing P (s1, t22) by the paths hs1, φ(s1)
P(φ(s1),t11) −→ t1 1i and hs2, φ(s2) P(φ(s2),φ(x1)) −→ φ(x1), x1 P(x1,t22) −→ t2 2i.
Suppose that k2 = 1. Applying Theorem 1, we
can construct a fan A(s1 → {t21, · · · , tk
1
1 , s2, t12}) of
Q0
n−1− Fa. We can denote the paths P (s2, s1) and
P (s1, t12) as hs2 P(s2,x1) −→ x1, s1i and hs1, x2 P(x2,t12) −→ t1
2i, respectively. Applying Lemma 5, we can
con-struct two spanning disjoint paths P (φ(s1), t11)
and P (φ(x1), φ(x2)) of Q1n−1. Thus, we can
con-struct the B(s1 → T1, s2 → T2) of Qn − Fa
from A(s1 → {t21, · · · , t k1
1 , s2, t12}) by removing
the paths P (s1, s2) and P (s1, t12) and adding the
paths hs1, φ(s1) P(φ(s1),t11) −→ t1 1i and hs2 P(s2,x1) −→ x1, φ(x1) P(φ(x1),φ(x2)) −→ φ(x2), x2 P(x2,t12) −→ t1 2i. Case 1.4.2 t1 1, t12 ∈ Q1n−1 and |Fa| ≥ 1.
Let b1 and w1 be a pair of adjacently faulty
vertices of Q0 n−1. By induction hypothesis, we can construct B(s1 → {w1, t21, · · · , t k1 1 }, s2 → {b1, t22, · · · , t k2 2 }) of Q0n−1 − (Fa0 − {b1, w1}). We
can denote the paths P (s1, w1) and P (s2, b1) as
hs1
P(s1,x1)
−→ x1, w1i and hs2
P(s2,x2)
−→ x2, b2i,
respec-tively. Suppose that {φ(x1), φ(x2)} ∩ {t11, t12} = ∅.
Applying Lemma 5, we can construct two span-ning disjoint paths P (φ(x1), t11) and P (φ(x2), t12)
of Q1
n−1. Thus, we can construct the B(s1 →
T1, s2 → T2) of Qn − Fa from B(s1 → {w1, t21, · · · , tk 1 1 }, s2 → {b1, t22, · · · , tk 2 2 }) by
remov-ing the paths P (s1, w1) and P (s2, b1) and adding
the paths hs1 P(s1,x1) −→ x1, φ(x1) P(φ(x1),t11) −→ t1 1i and hs2 P(s2,x2) −→ x2, φ(x2) P(φ(x2),t12) −→ t1 2i.
Suppose that |{φ(x1), φ(x2)} ∩ {t11, t12}| = 1.
With-out loss of generality, we can assume that φ(x2) =
t1
2. Applying Lemma 1, we can construct a
Hamilto-nian pathP (φ(x1), t11) of Q1n−1−{t12}. Thus, we can
construct theB(s1 → T1, s2 → T2) of Qn−Fafrom
B(s1 → {w1, t21, · · · , t k1
1 }, s2 → {b1, t22, · · · , t k2
2 })
by removing the paths P (s1, w1) and P (s2, b1) and
adding the paths hs1
P(s1,x1) −→ x1, φ(x1) P(φ(x1),t11) −→ t1 1i and hs2 P(s2,x2) −→ x2, t12i.
Suppose that {φ(x1), φ(x2)} = {t11, t12}. That is,
φ(x1) = t11 and φ(x2) = t12. We denote P (s1, t21)
as hs1, y1
P(y1,t21)
−→ t2
1i. Applying Lemma 4, we
can construct a Hamiltonian path P (φ(s1), φ(y1))
of Q1
n−1 − {t11, t21}. Thus, we can construct the
B(s1 → T1, s2 → T2) of Qn− Fa from B(s1 → {w1, t21, · · · , t k1 1 }, s2 → {b1, t22, · · · , t k2 2 }) by
remov-ing the paths P (s1, w1), P (s2, b1), P (s1, t21) and
adding the paths hs1
P(s1,x1) −→ x1, t11i, hs2 P(s2,x2) −→ x2, t12i, hs1, φ(s1) P(φ(s1),φ(y1)) −→ φ(y1), y1 P(y1,t21) −→ t2 1i. Case 1.4.3 t1 1, t12 ∈ Q1n−1 and |Fa| = 0.
Suppose that n = 5 and k1 = k2 = 2.
Ap-plying Lemma 5, we can construct two spanning disjoint paths P (s1, t21), P (s2, t22) of Q0n−1 and two
spanning disjoint pathsP (φ(s1), t11), P (φ(s2), t12) of
Q1
n−1. Thus, the following pathshs1, φ(s1)
P(φ(s1),t11) −→ t1 1i, P (s1, t21), and hs2, φ(s2) P(φ(s2),t12) −→ t1 2i, P (s2, t22)
form the 4∗-bifan of Q
n. Suppose that n = 5
and k1 = 3, k2 = 1. Applying Lemma 1,
we can construct a Hamiltonian path ht2 1 P(t2 1,s1) −→ s1 P(s1,t31) −→ t3 1i of Q0n−1 − {s2}. Applying
Lemma 5, we can construct two spanning dis-joint paths P (φ(s1), t11), P (φ(s2), t12) of Q1n−1.
Thus, the following paths hs1, φ(s1)
P(φ(s1),t11) −→ t1 1i, P (s1, t21), P (s1, t31) and hs2, φ(s2) P(φ(s2),t12) −→ t1 2i
form the 4∗-bifan of Q n.
Suppose that n ≥ 6. Let (φ(x1
1), t11) be an edge of
Q1
n−1forx11 ∈ (T/ 2∪{s2}). By induction hypothesis,
we can construct B(s1 → {x11, t31, · · · , t k1 1 }, s2 → {t2 1, t22, · · · , t k2
2 }) of Q0n−1. We can denote the
path P (s2, t21) as hs2, x2
P(x2,t21)
−→ t2
1i. Applying
Lemma 5, we can construct two spanning dis-joint paths P (φ(s1), φ(x2)) and P (φ(s2), t12) of
Q1
n−1 − {t11, φ(x1)}. Thus, we can construct the
B(s1 → T1, s2 → T2) of Qn− Fa from B(s1 → {x1 1, t31, · · · , t k1 1 }, s2 → {t21, t22, · · · , t k2 2 }) by
remov-ing the paths P (s1, x11) and P (s2, t21) and adding
the paths hs1 P(s1,x1) −→ x1, t11i, hs1, φ(s1) P(φ(s1),φ(x2)) −→ φ(x2), x2 P(x2,t21) −→ t2 1i and hs2, φ(s2) P(φ(s2),t12) −→ t1 2i. Case 1.4.4 |(T1 ∪ T2 − {t11, t12}) ∩ T1| ≥ 1 and |T1| ≤ n − 3.
Without loss of generality, we can assume that
t1 1, · · · , t j1 1 , t12, · · · , t j2 2 ∈ V1 for j1 ≥ 2. Let (φ(xi1 1), t i1 1 ) and (φ(x i2 2), t i2 2) be edges of Q1n−1
such that φ(xi1 1), xi 1 1, φ(xi 2 2), xi 2 2 ∈ (F/ a ∪ T1 ∪ T2 ∪ {s1, s2}) for 1 ≤ i1 ≤ j1 − 1, 1 ≤
i2 ≤ j2. By induction hypothesis, we can
con-struct B(s1 → {x11, · · · , x j1−1 1 , t j1+1 1 , · · · , t k1 1 }, s2 → {x1 2, · · · , x j2 2 , t j2+1 2 , · · · , t k2 2 }) of Q0n−1 − Fa0.
Ap-plying Lemma 1, we can construct a Hamiltonian pathP (φ(s1), tj11) of Q1n−1− {t i1 1, φ(xi 1 1), ti 2 2, φ(xi 2 2)|
for 1 ≤ i1 ≤ j1 − 1 and 1 ≤ i2 ≤ j2}. Thus,
the following paths hs1
P(s1,xi11 ) −→ xi1 1, φ(xi 1 1), ti 1 1i, hs1, φ(s1) P(φ(s1),tj11 ) −→ tj1 1 i, P (s1, ti12) and hs2 P(s2,xr12 ) −→ xr1 2 , φ(x r1 2 ), t r1
2 i, P (s2, trr2) form the (n-1-fa)∗-bifan
B(s1 → T1, s2 → T2) of Qn− Fa for1 ≤ i1 ≤ j1−
1, j1+ 1 ≤ i2 ≤ k1, 1 ≤ r1 ≤ j2, j2+ 1 ≤ r2 ≤ k2.
Case 1.4.5 |T1| = n − 2.
Suppose that |Fa| = 1. Let (φ(xi11), ti
1
1) and
(φ(xi2
2 ), ti
2
2) be edges of Q1n−1 such that x i1
1, xi
2
2 ∈/
(Fa ∪ {s1, s2}) for 1 ≤ i1 ≤ k1 − 1, 1 ≤
i2 ≤ k2. By induction hypothesis, we can construct
B(s1 → {x11, · · · , x k1−1 1 }, s2 → {x12, · · · , x k2 2 }) of Q0n−1 − F0
a. Applying Lemma 3, we can construct
two spanning disjoint paths P (φ(xk1−1
1 ), tk 1−1 1 ) and P (φ(s1), tk11) of Q1n−1− {t i1 1, φ(x i1 1 ), t i2 2, φ(x i2 2)| for 1 ≤ i1 ≤ k1 − 2 and 1 ≤ i2 ≤ k2}. Thus,
the following paths hs1
P(s1,xi11 ) −→ xi1 1, φ(xi 1 1), ti 1 1i, hs1 P(s1,xk1−11 ) −→ xk1−1 1 , φ(x k1−1 1 ) P(φ(xk1−11 ),t k1−1 1 ) −→ tk1−1 1 i, hs1, φ(s1) P(φ(s1),tk11 ) −→ tk1 1 i and hs2 P(s2,xi22 ) −→ xi2 2 , φ(xi 2 2), ti 2
2i form the (n-2)∗-bifan B(s1 →
T1, s2 → T2) of Qn− Fa for 1 ≤ i1 ≤ k1 − 2, 1 ≤
i2 ≤ k2.
The proof for |F1
a| = 0 is similar as |Fa1| = 1.
Case 1.4.6 |T1| = n − 1.
Suppose that |T1| ≥ 3. Let (φ(xi11), t i1
1) and
(φ(xi2
2 ), t i2
2) be edges of Q1n−1 such that
xi1
1 , xi
2
2 ∈ {s/ 1, s2} for 4 ≤ i1 ≤ k1, 1 ≤ i2 ≤ k2.
Applying Lemma 3, we can construct two
spanning disjoint paths P (φ(s1), t31) and
ht1 1 P(t1 1,φ(x 1 1)) −→ φ(x1 1), φ(x21) P(φ(x2 1),t 2 1) −→ t2 1i of Q1 n−1 − {t i1 1, φ(xi 1 1), ti 2 2, φ(xi 2 2)| for 4 ≤ i1 ≤ k1 and 1 ≤ i2 ≤ k2} with x11, x21 ∈ {s/ 1, s2}. By
induction hypothesis, we can construct (n-2)∗
-bifan B(s1 → {x11, x21, x41, · · · , x k1−1 1 }, s2 → {x1 2, · · · , xk 2
2 }) of Q0n−1. Thus, the following
paths hs1 P(s1,xi11) −→ xi1 1, φ(x i1 1 ) P(φ(xi11 ),t i1 1 ) −→ ti1 1i, hs1, φ(s1) P(φ(s1),t31) −→ t31i, hs1 P(s1,xi21 ) −→ xi2 1, φ(xi 2 1), ti 2 1 i, and hs2 P(s2,xi32 ) −→ xi3 2, φ(x i3 2), t i3 2 i form the (n-1)∗-bifan B(s 1 → T1, s2 → T2) of Qn for 1 ≤ i1 ≤ 2, 4 ≤ i2 ≤ k1, 1 ≤ i3 ≤ k2.
The proof for |T1| = 2 is similar as |T1| ≥ 3.
Case 2 s1 ∈ Vi and s2 ∈ Vj for i 6= j.
Without loss of generality, we can assume thats1 ∈
V0 and s
2 ∈ V1. We can also assume that |T11| +
|F1
a| ≥ |T20| + |Fa0|.
Case 2.1 |T0
2| + |Fa0| ≤ |T11| + |Fa1| ≤ n − 4.
Without loss of generality, we can assume that
t1 1, · · · , t j1 1 ∈ V1 and t12, · · · , t j2 2 ∈ V0. Let (φ(xi1 ), ti1
1 ) be edges of Q1n−1such thatxi
1 , φ(xi1 ) /∈ (Fa ∪ T1 ∪ T2 ∪ {s1, s2}) for 1 ≤ i1 ≤ j1. Let (yi2 , ti2
2) be edges of Q0n−1 such that yi
2
, φ(yi2
) /∈ (Fa ∪ T1 ∪ T2 ∪ {s1, s2} ∪ {xi1, φ(xi1)| for 1 ≤
i1 ≤ j1}) with 1 ≤ i2 ≤ j2. Applying
The-orem 1, we can construct a k∗
1-fan A(s1 → {x1, · · · , xj1 , tj1+1 1 , · · · , t k1 1 }) of Q0n−1−Fa0−{t i2 2, yi 2| for 1 ≤ i2 ≤ j2} and a k∗2-fan A(s2 →
{φ(y1), · · · , φ(yj2 ), tj2+1 2 , · · · , tk 2 2 }) of Q1n−1− Fa1− {ti1 1 , φ(xi 1
)| for 1 ≤ i1 ≤ j1}. Thus, the following
paths hs1 P(s1,xi1) −→ xi1 , φ(xi1 ), ti1 1i, P (s1, ti12) and hs2 P(s2,φ(yi3)) −→ φ(yi3 ), yi3 , ti3 2i, P (s2, ti24) form the (n-1-fa)∗-bifanB(s1 → T1, s2 → T2) of Qn−Fafor 1 ≤ i1 ≤ j1, j1+ 1 ≤ i2 ≤ k1, 1 ≤ i3 ≤ j2, j2+ 1 ≤ i4 ≤ k2. Case 2.2 |T1 1| + |Fa1| = n − 3. Case 2.2.1 |T2| = 2. Suppose that t1 2, t22 ∈ V0 and d(s2, t22) ≥ 3. Let (y1, t1 2), (y2, t22) be edges of Q0n−1
for φ(y1), φ(y2) ∈/ (T1 ∪ F1
a ∪ {s2}).
Let (φ(xi), ti
1) be an edge of Q1n−1 for
xi ∈/ {s
1, t12, t22, y1, y2}, 2 ≤ i ≤ k1.
Applying Theorem 1, we can construct
a 3∗-fan A(s
2 → {φ(y1), φ(y2), t11}) of
Q1
n−1 − Fa1 − {ti1, φ(xi)|2 ≤ i ≤ k1}.
We denote the three paths of A(s2 →
{φ(y1), φ(y2), t1 1}) as P (s2, φ(y1)), P (s2, φ(y2)) and hs2, φ(x1), P(p(φ(x1 )),t1 1) −→ t1 1i. Applying
Theorem 1, we can construct a k∗
1
-fan A(s1 → {xi|1 ≤ i ≤ k1}) of
Q0
n−1 − {t12, y1, t22, y2}. Thus, the paths
hs1, P(s1,x1) −→ x1, φ(x1),P(φ(x 1 ),t1 1) −→ t1 1i, hs1, P(s1,xi) −→ xi, φ(xi), ti 1i, hs2, P(s2,φ(yj)) −→ φ(yj), yj, tj 2i form the
(n − 1 − fa)∗-bifan B(s1 → T1, s2 → T2) of
Qn− Fa, for 2 ≤ i ≤ k1, j = 1, 2.
Suppose that d(s2, t22) = 2 and d(s2, t12) ≥ 2. Let
(y1, t1
2) be an edge of Q0n−1 for φ(y1) /∈ T1.
Let (φ(xi), ti
1) be an edge of Q1n−1 for
xi ∈ {s/
1, t12, t22, y1}, for 1 ≤ i ≤ k1. Applying
Lemma 3, we can construct two spanning
disjoint paths P (s2, φ(y1)) and P (φ(x1), t11)
of Q1
n−1 − Fa1 − {ti1, φ(xi)|2 ≤ i ≤ k1}.
Applying Theorem 1, we can also construct a k∗
1-fan A(s1 → {xi|1 ≤ x ≤ k1}) of
Q0
n−1 − {t12, y1, t22, φ(s2)}. Thus, the paths
hs1 P(s1,x1) −→ x1, φ(x1),P(φ(x 1 ),t1 1) −→ t1 1i, hs1 P(s1,xi) −→ xi, φ(xi), ti 1i, hs2, P(s2,φ(y1)) −→ φ(y1), y1, t1 2i, hs2, φ(s2), t2
2i form the (n-1-fa)∗-bifan B(s1 → T1, s2 → T2)
of Qn− Fa for 2 ≤ i ≤ k1.
Suppose that d(s2, t22) = 2 and d(s2, t12) = 1.
Let (z, t2
2), (y, z) be edges of Q0n−1 for
{y, z, φ(y)} ∈/ (T1 ∪ T2 ∪ Fa ∪ {s1, s2}).
Let (φ(xi), ti
1) be edges of Q1n−1 for
φ(xi), xi ∈ (T/
1∪ T2∪ Fa∪ {s1, s2, y, z, φ(y), φ(z)})
and 1 ≤ i ≤ k1. Applying Lemma 3,
we can construct two spanning disjoint
paths P (s2, φ(y)) and P (φ(x1), t11) of
Q1
n−1 − Fa1 − {ti1, φ(xi)|2 ≤ i ≤ k1}.
Applying Theorem 1, we can construct
a k∗
1-fan A(s1 → {xi|1 ≤ i ≤ k1})
of Q0
n−1 − {t12, t22, y, z}. Thus, the paths
hs1 P(s1,x1) −→ x1, φ(x1) P(φ(x 1 ),t1 1) −→ t1 1i, hs1 P(s1,xi) −→ xi, φ(xi), ti 1i, hs2, t12i, hs2 P(s2,φ(y)) −→ φ(y), y, z, t2 2i
form the (n-1-fa)∗-bifan B(s1 → T1, s2 → T2) of
Qn− Fa.
The proof of the case {t1
2, t22} 6⊂ V0 is similar to the proof of t1 2, t22 ∈ V0. Case 2.2.2 |T2| = 1. Suppose that |T0 1| = 1 and |T20| = 1. Without
loss of generality, we can assume that t1
1 ∈ V0. Let (y, t1 2) ∈ Q0n−1 for y, φ(y) /∈ (Fa ∪ T1 ∪ T2 ∪ {s1, s2}). Let (φ(xi), ti1) be edges of Q1n−1 for xi, φ(xi) /∈ (F a ∪ T1 ∪ T2 ∪ {s1, s2, y, φ(y)})
for 2 ≤ i ≤ k1. Applying Lemma 3, we can
con-struct two spanning disjoint paths P (s2, φ(y)) and
P (φ(x2), t2
1)of Q1n−1− Fa1− {φ(xi), xi|3 ≤ i ≤ k1}.
Applying Theorem 1, we can construct a k∗ 1-fan A(s1 → {t11, x2, x3, · · · xk 1} of Q0 n−1−{t12, y}. Thus, the paths P (s1, t11), hs1 P(s1,xi) −→ xi, φ(xi), ti 1i, and hs2 P(s2,φ(y)) −→ φ(y), y, t1
2i form the (n-1-fa)∗-bifan
B(s1 → T1, s2 → T2) of Qn− Fa.
The proofs of the cases |T0
2| = 0 or |T10| = 0 are
similar to the proof of |T0
1| = 1 and |T20| = 1
Case 2.3 |T1
1| + |Fa1| = n − 2.
Suppose that |T1
2| = 0. Let (y, t12) be an edge
of Q0
n−1 for y, φ(y) /∈ (Fa ∪ T1 ∪ T2 ∪ {s1, s2}).
Let (φ(xi), ti
1) be edges of Q1n−1 for φ(xi), xi ∈/
(Fa ∪ T1 ∪ T2 ∪ {s1, s2, y, φ(y)}) for 3 ≤ i ≤ k1.
Applying Lemma 1, we can construct a Hamilto-nian path P (t2
1, t11) of Q1n−1− Fa− {ti1, φ(xi)|3 ≤
i ≤ k1} − {(φ(y), φ(t12))}. Without loss of
gen-erality, we can assume that the path P (t2
1, t11) can be denoted as ht2 1 P(t2 1,φ(x 2 )) −→ φ(x2), s 2 P(s2,φ(y)) −→ φ(y), φ(x1) P(φ(x 1 ),t1 1) −→ t1 1i. Applying Theorem 1, we can construct a k∗ 1-fan A(s1 → {xi|1 ≤ x ≤ k1} of Q0
n−1 − {t12, y}. Thus, the paths hs1
P(s1,xi) −→ xi, φ(xi) P(φ(xi),ti1) −→ ti 1i, hs1 P(s1,xj) −→ xj, φ(xj), tj 1i and hs2 P(s2,φ(y)) −→ φ(y), y, t1
2i form the (n-1-fa)∗-bifan
B(s1 → T1, s2 → T2) of Qn − Fa for 1 ≤ i ≤
2, 3 ≤ j ≤ k1.
The proof for |T1
2| = 1 is similar to the proof for
|T1
2| = 0 .
IV. CONCLUSIONS AND FUTURE WORKS In this paper, we have shown that Qn− Fa is (
n-fa)∗-fanable and (n-1-fa)∗-bifanable if fa ≤ n − 3
where Fa is the set of fa pairs of adjacently faulty
vertices. The vertices fault-tolerance and edges fault-tolerance for fanability and bifanability are worth studying in the future.
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[2] C. H. Chang, C. K. Lin, H. M. Huang, and L. H. Hsu, ”The super laceability of the hypercubes,” Information Processing Letters, Vol. 92, pp. 15-21, 2004.
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