Bipancyclic Properties of Faulty Hypercubes

Abstract

A bipartite graph 𝐺=(𝑉,𝐸) is bipancyclic if it contains cycles of every even length from 4 to |𝑉| and edge bipancyclic if every edge lies on a cycle of every even length from 4 to |𝑉|. Let 𝑄_𝑛 denote the 𝑛-dimensional hypercube. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒, where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑉(𝑄_𝑛), and 𝐹_𝑒 consists of 𝑓_𝑒 edges. We prove that 𝑄_𝑛−𝐹 is bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2. Moreover, 𝑄_𝑛−𝐹 is edge bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2 with 𝑓_𝑎𝑣<𝑛−2.

1. Introduction

Interconnection networks play an important role in parallel computing/communication systems. The graph embedding problem, which is a central issue in evaluating a network, asks if the guest graph is a subgraph of a host graph. A benefit of graph embedding is that we can apply an existing algorithm for guest graphs to host graphs. This problem has attracted a burst of studies in recent years. Note that cycle networks are useful for designing simple algorithms with low communication costs. Thus, there are many studies on the cycle embedding problem. The cycle embedding problem deals with identifying all possible lengths of the cycles in a given graph. For the graph definition and notation, we follow [1].

Let 𝐮=𝑢_𝑛𝑢_𝑛−1⋯𝑢₂𝑢₁ be 𝑛-bit binary strings. The Hamming weight of 𝐮, denoted by 𝑤(𝐮), is the number of 𝑖 such that 𝑢_𝑖=1. Let 𝐮=𝑢_𝑛𝑢_𝑛−1⋯𝑢₂𝑢₁ and 𝐯=𝑣_𝑛𝑣_𝑛−1⋯𝑣₂𝑣₁ be two 𝑛-bit binary strings. The Hamming distance ℎ(𝐮,𝐯) between two vertices 𝐮 and 𝐯 is the number of different bits in the corresponding strings of both vertices. The 𝑛-dimensional hypercube, denoted by 𝑄_𝑛, has all 𝑛-bit binary strings as its vertices; two vertices 𝐮 and 𝐯 are adjacent if and only if ℎ(𝐮,𝐯)=1. Obviously, 𝑄_𝑛 is a bipartite graph with bipartition 𝐴={𝐮∣𝑤(𝐮)iseven} and 𝐵={𝐮∣𝑤(𝐮)isodd}. A vertex 𝐮 of 𝑄_𝑛 is white if 𝑤(𝐮) is odd, otherwise 𝐮 is black. It is known that the distance between 𝐮 and 𝐯 is 𝑑_𝑄𝑛(𝐮,𝐯)=ℎ(𝐮,𝐯). For 𝑖=0, 1, let 𝑄^𝑖_𝑛 denote the subgraph of 𝑄_𝑛 induced by {𝐮=𝑢_𝑛𝑢_𝑛−1⋯𝑢₂𝑢₁∣𝑢_𝑗=𝑖forsome1≤𝑗≤𝑛}. Obviously, 𝑄^𝑖_𝑛 is isomorphic to 𝑄_𝑛−1. For any vertex 𝐮=𝑢_𝑛𝑢_𝑛−1…𝑢₂𝑢₁, we use (𝐮)_𝑗 to denote the bit 𝑢_𝑗. Moreover, we use (𝐮)^𝑘 to denote the vertex 𝐯=𝑣_𝑛𝑣_𝑛−1⋯𝑣₂𝑣₁ with 𝑣_𝑖=𝑢_𝑖 for 1≤𝑖≠𝑘≤𝑛 and 𝑣_𝑘=1−𝑢_𝑘. An edge (𝐮,𝐯)∈𝐸(𝑄_𝑛) is of dimension 𝑘 if 𝐯=(𝐮)^𝑘.

The hypercube 𝑄_𝑛 is one of the most popular interconnection networks for parallel computers/communication systems [2]. This is partly due to its attractive properties, such as regularity, recursive structure, vertex and edge symmetry, maximum connectivity as well as effective routing and broadcasting algorithms.

Note that the hypercube 𝑄_𝑛 is a bipartite graph for every integer 𝑛. The corresponding cycle embedding problem on bipartite graphs is called the bipancyclic property. A bipartite graph 𝐺 is bipancyclic if it contains cycles of every even length from 4 to |𝑉(𝐺)|, inclusive.

There are some variations of the bipancyclic property. A bipartite graph 𝐺 is edge bipancyclic if every edge lies on a cycle of every even length from 4 to |𝑉(𝐺)|, inclusive. A bipartite graph is 𝑘-edge fault tolerant bipancyclic if 𝐺−𝐹 is bipancyclic for any 𝐹⊂𝐸(𝐺) with |𝐹|≤𝑘. Moreover, a bipartite graph is 𝑘-edge fault tolerant edge bipancyclic if 𝐺−𝐹 is edge bipancyclic for any 𝐹⊂𝐸(𝐺) with |𝐹|≤𝑘. The following theorem is proved.

Theorem 1.1 (see [3]). 𝑄_𝑛 is (𝑛−2)-edge fault tolerant edge bipancyclic if 𝑛≥2.

In this paper, we improve Theorem 1.1 by considering both edge faults and vertex faults. However, we restrict the faults on the vertex set to those occurring only on disjoint adjacent pairs. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒 where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑄_𝑛, and 𝐹_𝑒 consists of 𝑓_𝑒 edges. More precisely, 𝐹_𝑎𝑣=∪^𝑓𝑎𝑣_𝑖=1{𝐛_𝑖,𝐰_𝑖}, where (𝐛_𝑖,𝐰_𝑖)∈𝐸(𝑄_𝑛) and {𝐛_𝑖,𝐰_𝑖}∩{𝐛_𝑗,𝐰_𝑗}=∅ for 𝑖≠𝑗. Without loss of generality, we assume that {𝐛_𝑖∣1≤𝑖≤𝑓_𝑎𝑣} is a set of 𝑓_𝑎𝑣 black vertices, and {𝐰_𝑖∣1≤𝑖≤𝑓_𝑎𝑣} is a set of 𝑓_𝑎𝑣 white vertices. We will prove that 𝑄_𝑛−𝐹 is bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2. Moreover, 𝑄_𝑛−𝐹 is edge bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2 with 𝑓_𝑎𝑣<𝑛−2.

2. Preliminary

We need the following lemmas.

Lemma 2.1 (see [4]). Let 𝑒 be any edge of 𝑄_𝑛 for 𝑛≥2. There are 𝑛−1 cycles of length four that contain 𝑒 in common.

Lemma 2.2 (see [5]). Assume that 𝑛 is any positive integer with 𝑛≥2, and 𝐹 is a subset of 𝐸(𝑄_𝑛) with |𝐹|≤𝑛−2. Then there exists a Hamiltonian path of 𝑄_𝑛−𝐹 joining any two vertices from different bipartite sets. Moreover, there exists a Hamiltonian path joining 𝐲 to 𝐳 of 𝑄_𝑛−𝐹−{𝐱} for 𝐱 in some partite set and 𝐲,𝐳 in the other partite set for |𝐹|≤𝑛−3.

Lemma 2.3 (see [6]). Assume that 𝑛 is any positive integer with 𝑛≥2. Let 𝐮 and 𝐱 be two distinct white vertices of 𝑄_𝑛 and 𝐯 and 𝐲 be two distinct black vertices of 𝑄_𝑛. There are two disjoint paths 𝑃₁ and 𝑃₂ such that (1) 𝑃₁ joins 𝐮 to 𝐯, (2) 𝑃₂ joins 𝐱 to 𝐲, and (3) 𝑃₁∪𝑃₂ spans 𝑄_𝑛.

We extend the above lemma by considering the occurrence of edge faults.

Lemma 2.4. Assume that 𝑛 is any positive integer with 𝑛≥3, and 𝐹 is a subset of 𝐸(𝑄_𝑛) with |𝐹|≤𝑛−3. Let 𝐮 and 𝐱 be two distinct white vertices of 𝑄_𝑛 and 𝐯 and 𝐲 be two distinct black vertices of 𝑄_𝑛. There are two disjoint paths 𝑃₁ and 𝑃₂ of 𝑄_𝑛−𝐹 such that (1) 𝑃₁ joins 𝐮 to 𝐯, (2) 𝑃₂ joins 𝐱 to 𝐲, and (3) 𝑃₁∪𝑃₂ spans 𝑄_𝑛−𝐹.

Proof. We prove this lemma by induction on 𝑛. By Lemma 2.3, this lemma is true for 𝑛=3. Thus, we assume 𝑛≥4 and |𝐹|≥1. For 1≤𝑖≤𝑛, let 𝐹_𝑖 denote the set of 𝑖-dimensional edges in 𝐹. Thus, ∑^𝑛_𝑖=1|𝐹_𝑖|=|𝐹|. Without loss of generality, we assume that |𝐹_𝑛|≥1. For 𝑖=0, 1, we use 𝐹^𝑖 to denote the set 𝐸(𝑄^𝑖_𝑛)∩𝐹. Obviously, |𝐹^𝑖|≤𝑛−4. Let 𝑆^𝑖={𝐮,𝐯,𝐱,𝐲}∩𝑉(𝑄^𝑖_𝑛) for 𝑖=0, 1. Without loss of generality, we can assume that |𝑆⁰|≥|𝑆¹|. We have the following cases.
Case  1  (|𝑆⁰|=4). By induction, there exist two spanning disjoint paths 𝑅₁ and 𝑅₂ of 𝑄⁰_𝑛−𝐹⁰ such that 𝑅₁ joins 𝐮 to 𝐯, and 𝑅₂ joins 𝐱 to 𝐲. Since 2^𝑛−1−2>2(𝑛−3), there exists an edge (𝐬,𝐭) of 𝑅₁ or 𝑅₂ such that {(𝐬,(𝐬)^𝑛),(𝐭,(𝐭)^𝑛)}∩𝐹=∅. Without loss of generality, we can assume that (𝐬,𝐭) is in 𝑅₁. Thus, 𝑅₁ can be written as ⟨𝐮,𝐻₁,𝐬,𝐭,𝐻₂,𝐯⟩. By Lemma 2.2, there exists a Hamiltonian path 𝑄 joining (𝐬)^𝑛 to (𝐭)^𝑛 of 𝑄¹_𝑛−𝐹¹. We set 𝑃₁ as ⟨𝐮,𝐻₁,𝐬,(𝐬)^𝑛,𝑄,(𝐭)^𝑛,𝐭,𝐻₂,𝐯⟩ and set 𝑃₂ as 𝑅₂.
Case  2  (|𝑆⁰|=3). Without loss of generality, we can assume that 𝐲∈𝑄¹_𝑛. Since there are 2^𝑛−2 black vertices in 𝑄⁰_𝑛 and 2^𝑛−2−1>𝑛−3 for 𝑛≥4, there exists a black vertex 𝐳 of 𝑄⁰_𝑛 such that 𝐳≠𝐯 and (𝐳,(𝐳)^𝑛)∉𝐹. By induction, there exist two spanning disjoint paths 𝑃₁ and 𝑅₁ of 𝑄⁰_𝑛−𝐹⁰ such that 𝑃₁ joins 𝐮 to 𝐯, and 𝑅₁ joins 𝐱 to 𝐳. By Lemma 2.2, there exists a Hamiltonian path 𝑅₂ joining (𝐳)^𝑛 to 𝐲 of 𝑄¹_𝑛−𝐹¹. We set 𝑃₂ as ⟨𝐱,𝑅₁,𝐳,(𝐳)^𝑛,𝑅₂,𝐲⟩. Thus, 𝑃₁ and 𝑃₂ form the required paths.
Case  3  (|𝑆⁰|=2and𝑆^𝑖={𝐮,𝐯}forsome𝑖=0,1). Without loss of generality, we can assume that 𝑆⁰={𝐮,𝐯}. By Lemma 2.2, there exists a Hamiltonian path 𝑃₁ of 𝑄⁰_𝑛−𝐹⁰ joining 𝐮 to 𝐯, and there exists a Hamiltonian path 𝑃₂ of 𝑄¹_𝑛−𝐹¹ joining 𝐱 to 𝐲. Obviously, 𝑃₁ and 𝑃₂ form the required paths.
Case  4  (|𝑆⁰|=2and𝑆^𝑖≠{𝐮,𝐯}foreach𝑖=0,1). Without loss of generality, we can assume that 𝑆⁰={𝐮,𝐱}. Since there are 2^𝑛−2 black vertices in 𝑄⁰_𝑛 and 2^𝑛−2>𝑛−2 for 𝑛≥4, there exist two black vertices 𝐬 and 𝐭 of 𝑄⁰_𝑛 such that {(𝐬,(𝐬)^𝑛),(𝐭,(𝐭)^𝑛)}∩𝐹=∅. By induction, there exist two spanning disjoint paths 𝑄₁ and 𝑄₂ of 𝑄⁰_𝑛−𝐹⁰ such that 𝑄₁ joins 𝐮 to 𝐬 and 𝑄₂ joins 𝐱 to 𝐭. Similarly, there exist two spanning disjoint paths 𝑅₁ and 𝑅₂ of 𝑄¹_𝑛−𝐹¹ such that 𝑅₁ joins (𝐬)^𝑛 to 𝐯, and 𝑅₂ joins (𝐭)^𝑛 to 𝐲. We set 𝑃₁ as ⟨𝐮,𝑄₁,𝐬,(𝐬)^𝑛,𝑅₁,𝐯⟩ and set 𝑃₂ as ⟨𝐱,𝑄₂,𝐭,(𝐭)^𝑛,𝑅₂,𝐲⟩.

Lemma 2.5 (see [7]). Assume that 𝑛≥3. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒 where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑉(𝑄_𝑛) and 𝐹_𝑒 consists of 𝑓_𝑒 edges. Then there exists a Hamiltonian cycle of 𝑄_𝑛−𝐹 if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2.

Lemma 2.6 (see [7]). Assume that 𝑛≥3. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒, where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑉(𝑄_𝑛), and 𝐹_𝑒 consists of 𝑓_𝑒 edges. Then there exists a Hamiltonian path of 𝑄_𝑛−𝐹 between any two vertices from different partite sets of 𝑄_𝑛 if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−3.

We improve Lemma 2.6 into the following lemma.

Lemma 2.7. Assume that 𝑛≥3. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒, where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑉(𝑄_𝑛), and 𝐹_𝑒 consists of 𝑓_𝑒 edges. Then there exists a Hamiltonian path of 𝑄_𝑛−𝐹 between any two vertices from different partite sets of 𝑄_𝑛 if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2 with 𝑓_𝑎𝑣≤𝑛−3.

Proof. Let 𝐰 be any fault-free white vertex and 𝐛 be any fault-free black vertex. We need to construct a Hamiltonian path of 𝑄_𝑛−𝐹 joining 𝐰 to 𝐛 by induction on 𝑛. Let 𝐻=𝐹_𝑒∪{(𝐛_𝑖,𝐰_𝑖)∣{𝐛_𝑖,𝐰_𝑖}⊂𝐹_𝑎𝑣}. For 1≤𝑖≤𝑛, let 𝐻_𝑖 denote the set of 𝑖-dimensional edges in 𝐻. Thus, ∑^𝑛_𝑖=1|𝐻_𝑖|=𝑓_𝑒+𝑓_𝑎𝑣. Without loss of generality, we assume that |𝐻_𝑛|=0.
By brute force, we can check that the required paths exist for 𝑛=3,4. By Lemmas 2.2 and 2.6, the required paths exist when 𝑓_𝑎𝑣=0 or 𝑓_𝑒+𝑓_𝑎𝑣≤𝑛−3. Therefore, we only need to consider the case 𝑓_𝑎𝑣≥1 and 𝑓_𝑎𝑣+𝑓_𝑒=𝑛−2 for 𝑛≥5. Thus, 𝐹_𝑒≠∅. Let 𝐹^𝑖_𝑒=𝐹_𝑒∩𝐸(𝑄^𝑖_𝑛), 𝐹^𝑖_𝑎𝑣=𝐹_𝑎𝑣∩𝑉(𝑄^𝑖_𝑛), and 𝐹^𝑖=𝐹^𝑖_𝑎𝑣∪𝐹^𝑖_𝑒 for 𝑖=0, 1. Let 𝐛₁ and 𝐰₁ be a pair of 𝐹⁰_𝑎𝑣 where 𝐛₁ is a black vertex.
Case  1   (𝐹⊄𝑄^𝑖_𝑛and|𝐹^𝑖_𝑎𝑣|≤𝑛−4,for𝑖=0,1).We first consider the case that 𝐰 and 𝐛 are in the same subcube. Without loss of generality, we can assume that both 𝐰 and 𝐛 are in 𝑄⁰_𝑛. By induction, there exists a Hamiltonian path 𝑃₁ of 𝑄⁰_𝑛−𝐹⁰ joining 𝐰 to 𝐛. Note that 𝑙(𝑃₁)−4𝑓¹_𝑎𝑣=2^𝑛−1−2𝑓⁰_𝑎𝑣−1−4𝑓¹_𝑎𝑣≥2^𝑛−1−4𝑛+15>0. We can write 𝑃₁ as ⟨𝐰,𝑅₁,𝐮,𝐯,𝑅₂,𝐛⟩ for some 𝐮 and 𝐯 such that {(𝐮)^𝑛,(𝐯)^𝑛}∩𝐹¹_𝑎𝑣=∅. By induction, there exists a Hamiltonian path 𝑅₃ of 𝑄¹_𝑛−𝐹¹ joining (𝐮)^𝑛 to (𝐯)^𝑛. Thus, ⟨𝐰,𝑅₁,𝐮,(𝐮)^𝑛,𝑅₃,(𝐯)^𝑛,𝐯,𝑅₂,𝐛⟩ is a desired path.
Thus we consider the case that 𝐰 and 𝐛 are in different subcubes. Without loss of generality, we can assume that 𝐰∈𝑄⁰_𝑛 and 𝐛∈𝑄¹_𝑛. Since there are 2^𝑛−2 black vertices in 𝑄⁰_𝑛 and 2^𝑛−2≥𝑛−3 for 𝑛≥5, there exists a black vertex 𝐱 in 𝑄⁰_𝑛 such that {𝐱,(𝐱)^𝑛}∩𝐹_𝑎𝑣=∅. By induction, there exists a Hamiltonian path 𝑃₁ of 𝑄⁰_𝑛−𝐹⁰ joining 𝐰 to 𝐱 and there exists a Hamiltonian path 𝑃₂ of 𝑄¹_𝑛−𝐹¹ joining (𝐱)^𝑛 to 𝐛. Thus, ⟨𝐰,𝑃₁,𝐱,(𝐱)^𝑛,𝑃₂,𝐛⟩ is a desired path.
Case  2 (𝐹⊂𝑄^𝑖_𝑛 or |𝐹^𝑖_𝑎𝑣|=𝑛−3 for some 𝑖=0,1). Without loss of generality, we can assume that 𝐹⊂𝑄⁰_𝑛 or |𝐹⁰_𝑎𝑣|=𝑛−3. Thus, |𝐹¹_𝑎𝑣|=0.
Assume that both 𝐰 and 𝐛 are in 𝑄⁰_𝑛. By induction, there exists a Hamiltonian path 𝑃₁ of 𝑄⁰_𝑛−(𝐹−{𝐛₁,𝐰₁}) joining 𝐰 to 𝐛. Suppose that the edge (𝐛₁,𝐰₁) is in 𝑃₁. Without loss of generality, we can write 𝑃₁ as ⟨𝐰,𝑍₁,𝐱,𝐛₁,𝐰₁,𝐲,𝑍₂,𝐛⟩. By Lemma 2.2, there exists a Hamiltonian path 𝑍₃ of 𝑄¹_𝑛−𝐹¹_𝑒 joining (𝐱)^𝑛 to (𝐲)^𝑛. Obviously, ⟨𝐰,𝑍₁,𝐱,(𝐱)^𝑛,𝑍₃,(𝐲)^𝑛,𝐲,𝑍₂,𝐛⟩ is a desired Hamiltonian path. Thus, we consider the case that (𝐛₁,𝐰₁) is not in 𝑃₁. Without loss of generality, 𝑃₁ can be written as ⟨𝐰,𝑅₁,𝐮,𝐛₁,𝐯,𝑅₂,𝐱,𝐰₁,𝐲,𝑅₃,𝐛⟩. By Lemma 2.4, there are two disjoint spanning paths 𝑅₄ and 𝑅₅ of 𝑄¹_𝑛−𝐹¹_𝑒 such that 𝑅₄ joins (𝐮)^𝑛 to (𝐱)^𝑛 and 𝑅₅ joins (𝐯)^𝑛 to (𝐲)^𝑛 of 𝑄¹_𝑛−𝐹¹_𝑒. Obviously, ⟨𝐰,𝑅₁,𝐮,(𝐮)^𝑛,𝑅₄,(𝐱)^𝑛,𝐱,𝑅₂⁻¹,𝐯,(𝐯)^𝑛,𝑅₅,(𝐲)^𝑛,𝐲,𝑅₃,𝐛⟩ is a desired path.
Now, we consider the case that 𝐰 and 𝐛 are in different subcubes. Without loss of generality, we can assume that 𝐰∈𝑄⁰_𝑛 and 𝐛∈𝑄¹_𝑛. By induction, there exists a Hamiltonian path 𝑃₁ of 𝑄⁰_𝑛−(𝐹−{𝐛₁,𝐰₁}) joining 𝐰 and 𝐛₁. Suppose that the edge (𝐛₁,𝐰₁) is in 𝑃₁. We can write 𝑃₁ as ⟨𝐰,𝑍₁,𝐲,𝐰₁,𝐛₁⟩. By Lemma 2.2, there exists a Hamiltonian path 𝑍₂ of 𝑄¹_𝑛−𝐹¹_𝑒 joining (𝐲)^𝑛 to 𝐛. Obviously, ⟨𝐰,𝑍₁,𝐲,(𝐲)^𝑛,𝑍₂,𝐛⟩ is a desired path. Thus, we consider the case that (𝐛₁,𝐰₁) is not in 𝑃₁. We can write 𝑃₁ as ⟨𝐰,𝑅₁,𝐮,𝐰₁,𝐯,𝑅₂,𝐱,𝐛₁⟩. Assume that (𝐱)^𝑛≠𝐛. By Lemma 2.4, there are two disjoint spanning paths 𝑅₃ and 𝑅₄ of 𝑄¹_𝑛−𝐹¹_𝑒 such that 𝑅₃ joins (𝐮)^𝑛 to (𝐱)^𝑛 and 𝑅₄ joins (𝐯)^𝑛 to 𝐛. Obviously, ⟨𝐰,𝑅₁,𝐮,(𝐮)^𝑛,𝑅₃,(𝐱)^𝑛,𝐱,𝑅₂⁻¹,𝐯,(𝐯)^𝑛,𝑅₄,𝐛⟩ is a desired path. Assume that (𝐱)^𝑛=𝐛. By Lemma 2.2, there exists a Hamiltonian path 𝑅₅ of 𝑄¹_𝑛−𝐹¹_𝑒−{𝐛} joining (𝐮)^𝑛 to (𝐯)^𝑛. Obviously, ⟨𝐰,𝑅₁,𝐮,(𝐮)^𝑛,𝑅₅,(𝐯)^𝑛,𝐯,𝑅₂,𝐱,(𝐱)^𝑛=𝐛⟩ is a desired path.
Finally, we consider the case that 𝐰 and 𝐛 are in 𝑄¹_𝑛. Let 𝑒=(𝐱,𝐲) be any faulty edge of 𝐹, where 𝐱 is a white vertex. By Lemma 2.5, there exists a Hamiltonian cycle 𝐶₁ of 𝑄⁰_𝑛−(𝐹−{𝑒}).
Suppose that 𝑒 is not an edge of 𝐶₁. Since the length of 𝐶 is at least 2^𝑛−1−2(𝑛−3), we can write 𝐶₁ as ⟨𝐮,𝑍₁,𝐯,𝐮⟩ such that {(𝐮)^𝑛,(𝐯)^𝑛}∩{𝐰,𝐛}=∅ and 𝐮 is a white vertex. By Lemma 2.4, there exist two disjoint spanning paths 𝑍₂ and 𝑍₃ of 𝑄¹_𝑛−𝐹¹_𝑒 such that 𝑍₂ joins 𝐰 to (𝐮)^𝑛 and 𝑍₃ joins (𝐯)^𝑛 to 𝐛. Obviously, ⟨𝐰,𝑍₂,(𝐮)^𝑛,𝐮,𝑍₁,𝐯,(𝐯)^𝑛,𝑍₃,𝐛⟩ is a desired path.
Thus, we consider the case that 𝑒 is in 𝐶₁. We can write 𝐶₁ as ⟨𝐱,𝑃₁,𝐲,𝐱⟩. Suppose that {(𝐱)^𝑛,(𝐲)^𝑛}∩{𝐰,𝐛}=∅. By Lemma 2.4, there exist two disjoint spanning paths 𝑃₂ and 𝑃₃ of 𝑄¹_𝑛−𝐹¹_𝑒 such that 𝑃₂ joins 𝐰 to (𝐱)^𝑛 and 𝑃₃ joins (𝐲)^𝑛 to 𝐛. Obviously, ⟨𝐰,𝑃₂,(𝐱)^𝑛,𝐱,𝑃₁,𝐲,(𝐲)^𝑛,𝑃₃,𝐛⟩ is a desired path. Suppose that |{(𝐱)^𝑛,(𝐲)^𝑛}∩{𝐰,𝐛}|=1. Without loss of generality, we assume that (𝐱)^𝑛=𝐛. By Lemma 2.2, there exists a Hamiltonian path 𝑃₄ of 𝑄¹_𝑛−𝐹¹_𝑒−{𝐛} joining 𝐰 to (𝐲)^𝑛. Thus, ⟨𝐰,𝑃₄,(𝐲)^𝑛,𝐲,𝑃₁⁻¹,𝐱,(𝐱)^𝑛=𝐛⟩ is a Hamiltonian path of 𝑄_𝑛−𝐹. Suppose that {(𝐱)^𝑛,(𝐲)^𝑛}={𝐰,𝐛}. Obviously, 𝐶₁ can be written as ⟨𝐲,𝑃₅,𝐮,𝐯,𝑃₆,𝐱,𝐲⟩ for some black vertex 𝐮. By induction, there exists a Hamiltonian path 𝑃₇ of 𝑄¹_𝑛−𝐹¹_𝑒−{𝐰,𝐛} joining (𝐮)^𝑛 to (𝐯)^𝑛. Obviously, ⟨𝐰=(𝐲)^𝑛,𝐲,𝑃₅,𝐮,(𝐮)^𝑛,𝑃₇,(𝐯)^𝑛,𝐯,𝑃₆,𝐱,(𝐱)^𝑛=𝐛⟩ is a desired path.

3. Bipancyclic Properties

Theorem 3.1. Assume that 𝑛≥3. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒, where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑉(𝑄_𝑛), and 𝐹_𝑒 consists of 𝑓_𝑒 edges. Then 𝑄_𝑛−𝐹 is bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2.

Proof. To prove this theorem, we will construct a cycle in 𝑄_𝑛−𝐹 of length 𝑡 for every even integer with 4≤𝑡≤2^𝑛−2𝑓_𝑎𝑣 by induction. Let 𝐻=𝐹_𝑒∪{(𝐛_𝑖,𝐰_𝑖)|{𝐛_𝑖,𝐰_𝑖}⊂𝐹_𝑎𝑣}. For 1≤𝑖≤𝑛, let 𝐻_𝑖 denote the set of 𝑖-dimensional edges in 𝐻. Thus, ∑^𝑛_𝑖=1|𝐻_𝑖|=𝑓_𝑒+𝑓_𝑎𝑣. Without loss of generality, we assume that |𝐻_𝑛|=0. Let 𝐹^𝑖_𝑎𝑣=𝐹_𝑎𝑣∩𝑉(𝑄^𝑖_𝑛),  𝐹^𝑖_𝑒=𝐹_𝑒∩𝐸(𝑄^𝑖_𝑛), 𝐹^𝑖=𝐹^𝑖_𝑎𝑣∪𝐹^𝑖_𝑒 and 𝑓^𝑖_𝑎𝑣=|𝐹^𝑖_𝑎𝑣| for 𝑖=0, 1.
By brute force, we can check that the required cycles exist for 𝑛=3, 4. By Theorem 1.1, the required cycles exist if 𝑓_𝑎𝑣=0. Therefore, we consider the case 𝑓_𝑎𝑣≥1 and 𝑛≥5. Assume 𝑡 is an even integer from 4 to 2^𝑛−2𝑓_𝑎𝑣. Let 𝐻^𝑖=𝐻∩𝐸(𝑄^𝑖_𝑛) for 𝑖=0, 1. Without loss of generality, we can assume |𝐻⁰|≥|𝐻¹|.
Suppose that |𝐻⁰|≤𝑛−3. We first consider the case that 𝑡≤2^𝑛−1−2𝑓¹_𝑎𝑣. By induction, the desired cycle exists in 𝑄¹_𝑛−𝐹¹. Suppose that 𝑡>2^𝑛−1−2𝑓¹_𝑎𝑣. Let 𝐶 be a cycle of length 2^𝑛−1−2𝑓⁰_𝑎𝑣 in 𝑄⁰_𝑛−𝐹⁰. Let 𝑡^′=𝑡−2^𝑛−1+2𝑓⁰_𝑎𝑣−1. Note that there are 2^𝑛−1−2𝑓⁰_𝑎𝑣+1 sections in 𝐶 of length 𝑡^′ depending on choice of the beginning and terminating vertices. Since 2^𝑛−1>4(𝑛−2)≥2𝑓⁰_𝑎𝑣+4𝑓¹_𝑎𝑣, one such section, say 𝑃, joins vertex 𝐩 to vertex 𝐪 such that {(𝐩)^𝑛,(𝐪)^𝑛}∩𝐹¹_𝑎𝑣=∅. By Lemma 2.6, there exists a Hamiltonian path 𝑅 of 𝑄¹_𝑛−𝐹¹ joining (𝐪)^𝑛 to (𝐩)^𝑛. Obviously, ⟨𝐩,𝑃,𝐪,(𝐪)^𝑛,𝑅,(𝐩)^𝑛,𝐩⟩ is a desired cycle.
Thus, we consider |𝐻⁰|=𝑛−2. Therefore, |𝐻¹|=0. Assume that 4≤𝑡≤2^𝑛−1. By Theorem 1.1, the desired cycle exists in 𝑄¹_𝑛. Thus, we consider 𝑡>2^𝑛−1. Let {𝐛₁,𝐰₁} be a pair of adjacent vertices of 𝐹_𝑎𝑣 with 𝐛₁ being a black vertex. By Lemma 2.5, there exists a Hamiltonian cycle 𝐶 in 𝑄⁰_𝑛−(𝐻−{𝐛₁,𝐰₁}). Obviously, 𝐶 can be written as ⟨𝐛₁,𝑃₁,𝐰₁,𝑃₂,𝐛₁⟩. Without loss of generality, we assume that 𝑙(𝑃₁)≤𝑙(𝑃₂). Thus, 2^𝑛−2−𝑓⁰_𝑎𝑣≤𝑙(𝑃₂)≤2^𝑛−1−2𝑓⁰_𝑎𝑣−1. We can write 𝑃₂ as ⟨𝐰₁,𝐱,𝑃^′₂,𝐲,𝐛₁⟩.
Suppose that 2^𝑛−1<𝑡≤2^𝑛−1+𝑙(𝑃₂)−1. Let 𝑡^′=𝑡−2^𝑛−1−1. Let 𝑅 be the section of 𝑃^′₂ joining vertex 𝐱 to 𝐳 of length 𝑡^′. By Lemma 2.2, there exists a Hamiltonian path 𝑆 of 𝑄¹_𝑛 joining (𝐳)^𝑛 to (𝐱)^𝑛. Obviously, ⟨𝐱,𝑅,𝐳,(𝐳)^𝑛,𝑆,(𝐱)^𝑛,𝐱⟩ is a desired cycle.
Suppose that 2^𝑛−1+𝑙(𝑃₂)<𝑡≤2^𝑛−2𝑓_𝑎𝑣. Since 𝑙(𝑃₁)+𝑙(𝑃₂)=2^𝑛−1−2𝑓_𝑎𝑣, 𝑙(𝑃₁)≥3. We can write 𝑃₁ as ⟨𝐰₁,𝐮,𝑃^′₁,𝐯,𝐛₁⟩. Let 𝑡^′=𝑡−2^𝑛−1−𝑙(𝑃₂). Let 𝑇 be the section of 𝑃^′₁ joining vertex 𝐮 to 𝐜 of length 𝑡^′. By Lemma 2.4, there exist two spanning disjoint paths 𝑅₁ and 𝑅₂ of 𝑄¹_𝑛 such that 𝑅₁ joins (𝐲)^𝑛 to (𝐮)^𝑛 and 𝑅₂ joins (𝐜)^𝑛 to (𝐱)^𝑛. Thus, ⟨𝐱,𝑃^′₂,𝐲,(𝐲)^𝑛,𝑅₁,(𝐮)^𝑛,𝐮,𝑇,𝐜,(𝐜)^𝑛,𝑅₂,(𝐱)^𝑛,𝐱⟩ forms a desired cycle.
The theorem is proved.

Theorem 3.2. Assume that 𝑛≥3. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒 where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑉(𝑄_𝑛) and 𝐹_𝑒 consists of 𝑓_𝑒 edges. Then 𝑄_𝑛−𝐹 is edge bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2 and 𝑓_𝑎𝑣≤𝑛−3.

Proof. Let 𝑒=(𝐮,𝐯) be any fault-free edge of 𝑄_𝑛 where 𝐮 is a black vertex. For 4≤𝑡≤2^𝑛−2𝑓_𝑎𝑣, we will construct a cycle containing (𝐮,𝐯) of length 𝑡 in 𝑄_𝑛−𝐹 by induction to prove this theorem. Suppose 𝑡=4, the desired cycle exists by Lemma 2.1. Thus, we consider 𝑡≥6.
Let 𝐻=𝐹_𝑒∪{(𝐛_𝑖,𝐰_𝑖)∣{𝐛_𝑖,𝐰_𝑖}⊂𝐹_𝑎𝑣}∪{(𝐮,𝐯)}. For 1≤𝑖≤𝑛, let 𝐻_𝑖 denote the set of 𝑖-dimensional edges in 𝐻. Since 𝑓_𝑒+𝑓_𝑎𝑣≤𝑛−2, we can assume without loss of generality that |𝐻_𝑛|=0. Thus, 𝐮,𝐯 are vertices in 𝑄^𝑖_𝑛 for some 𝑖∈{0,1}. Let 𝐹^𝑖_𝑎𝑣=𝑉(𝑄^𝑖_𝑛)∩𝐹_𝑎𝑣 and 𝑓^𝑖_𝑎𝑣=|𝐹^𝑖_𝑎𝑣| for 𝑖=0,1.
By Theorem 1.1, the desired cycles exist if 𝑓_𝑎𝑣=0. Thus, the desired cycles exist for 𝑛=3. By brute force, we can check that the desired cycles exist for 𝑛=4. Thus, we only consider the case 𝑓_𝑎𝑣≥1 with 𝑛≥5. Without loss of generality, we can assume that 𝑓⁰_𝑎𝑣+𝑓⁰_𝑒≥𝑓¹_𝑎𝑣+𝑓¹_𝑒.
Suppose that 𝑓⁰_𝑎𝑣+𝑓⁰_𝑒≤𝑛−3 and 𝑓⁰_𝑎𝑣≤𝑛−4. Without loss of generality, we can assume that both 𝐮 and 𝐯 are in 𝑄⁰_𝑛. Suppose that 6≤𝑡≤2^𝑛−1−2𝑓⁰_𝑎𝑣. By induction, the desired cycle exists in 𝑄⁰_𝑛−𝐹⁰. Thus, we consider that 2^𝑛−1−2𝑓⁰_𝑎𝑣+2≤𝑡≤2^𝑛−2𝑓_𝑎𝑣. Let 𝑡^′=𝑡−2^𝑛−1−2𝑓⁰_𝑎𝑣−1. By induction, there exists a Hamiltonian cycle 𝐶 of 𝑄⁰_𝑛−𝐹⁰ containing the edge (𝐮,𝐯). Since 2^𝑛−1>4(𝑛−3)+1≥2𝑓⁰_𝑎𝑣+4𝑓¹_𝑎𝑣+1, there exists an edge (𝐱,𝐲) in 𝐶 such that {(𝐱)^𝑛,(𝐲)^𝑛}}∩𝐹¹_𝑎𝑣=∅. Thus, 𝐶 can be written as ⟨𝐮,𝐯,𝑃₁,𝐱,𝐲,𝑃₂,𝐮⟩. By induction, there exist cycles ⟨(𝐱)^𝑛,𝑄,(𝐲)^𝑛,(𝐱)^𝑛⟩ of length 𝑡^′ of 𝑄¹_𝑛−𝐹¹. Obviously, ⟨𝐮,𝐯,𝑃₁,𝐱,(𝐱)^𝑛,𝑄,(𝐲)^𝑛,𝐲,𝑃₂,𝐮⟩ forms the desired cycle.
Thus, we consider 𝑓⁰_𝑎𝑣+𝑓⁰_𝑒=𝑛−2 or 𝑓⁰_𝑎𝑣=𝑛−3. We have the following two cases.
Case  1 ((𝐮,𝐯)∈𝑄¹_𝑛). Suppose that 6≤𝑡≤2^𝑛−1. By Theorem 1.1, the desired cycle exists. Thus, we consider 𝑡≥2^𝑛−1+2. Let 𝐛₁ and 𝐰₁ be a pair of adjacent vertices in 𝐹⁰_𝑎𝑣. By induction, there exists a Hamiltonian cycle 𝐶 in 𝑄⁰_𝑛−𝐹⁰_𝑒−(𝐹⁰_𝑎𝑣−{𝐛₁,𝐰₁}) containing the edge (𝐛₁,𝐰₁). Thus, we can write 𝐶 as ⟨𝐛₁,𝐰₁,𝐱,𝑃,𝐲,𝐛₁⟩.
Suppose that 2^𝑛−1+2≤𝑡≤2^𝑛−2𝑓_𝑎𝑣. Let 𝑡^′=𝑡−2^𝑛−1−1. Obviously, there exists a section 𝑃^′ of 𝑃 joining 𝐬 and 𝐭 of length 𝑡^′ such that 𝐬 is a black vertex. Suppose that {(𝐬)^𝑛,(𝐭)^𝑛}∩{𝐮,𝐯}=∅. By Lemma 2.4, there exist two spanning disjoint paths 𝑄₁ and 𝑄₂ of 𝑄¹_𝑛−𝐹¹_𝑒 such that 𝑄₁ joins 𝐮 to (𝐬)^𝑛 and 𝑄₂ joins (𝐭)^𝑛 to 𝐯. Obviously, ⟨𝐮,𝑄₁,(𝐬)^𝑛,𝐬,𝑃^′,𝐭,(𝐭)^𝑛,𝑄₂,𝐯,𝐮⟩ forms the desired cycle. Suppose that |{(𝐬)^𝑛,(𝐭)^𝑛}∩{𝐮,𝐯}|=1. Without loss of generality, we assume that 𝐮=(𝐭)^𝑛. By Lemma 2.2, there exists a Hamiltonian path 𝑄 of 𝑄¹_𝑛−𝐹¹_𝑒−{𝐮} joining (𝐬)^𝑛 to 𝐯. Obviously, ⟨𝐮=(𝐭)^𝑛,𝐭,𝑃^′−1,𝐬,(𝐬)^𝑛,𝑄,𝐯,𝐮⟩ forms the desired cycle. Suppose that {(𝐬)^𝑛,(𝐭)^𝑛}={𝐮,𝐯}. Thus, 𝐮=(𝐭)^𝑛 and 𝐯=(𝐬)^𝑛. We can write 𝑃^′−1 as ⟨𝐭,𝑅₁,𝐜,𝐝,𝑅₂,𝐬⟩ for some 𝐜 and 𝐝. By induction, there exists a Hamiltonian cycle 𝐶^′ of 𝑄¹_𝑛−𝐹¹_𝑒−{𝐮,𝐯} containing the edge ((𝐜)^𝑛,(𝐝)^𝑛). Thus, 𝐶^′ can be written as ⟨(𝐜)^𝑛,𝑄^′,(𝐝)^𝑛,(𝐜)^𝑛⟩. Obviously, ⟨𝐮=(𝐭)^𝑛,𝑡,𝑅₁,𝐜,(𝐜)^𝑛,𝑄^′,(𝐝)^𝑛,𝐝,𝑅₂,𝐬,(𝐬)^𝑛=𝐯,𝐮⟩ forms the desired cycle.
Case  2 ((𝐮,𝐯)in𝑄⁰_𝑛). Suppose that 6≤𝑡≤2^𝑛−1+2. Let 𝑡^′=𝑡−2. By Theorem 1.1, there exists a cycle 𝐶 of length 𝑡^′ in 𝑄¹_𝑛−𝐹¹_𝑒 containing the edge ((𝐮)^𝑛,(𝐯)^𝑛). We can write 𝐶 as ⟨(𝐮)^𝑛,(𝐯)^𝑛,𝑅,(𝐮)^𝑛⟩. Obviously, ⟨𝐮,𝐯,(𝐯)^𝑛,𝑅,(𝐮)^𝑛,𝐮⟩ forms the desired cycle.
Now, we consider 2^𝑛−1+4≤𝑡≤2^𝑛−2𝑓_𝑎𝑣. Let 𝐛₁ and 𝐰₁ be a pair of adjacent vertices in 𝐹⁰_𝑎𝑣. By induction, there exists a Hamiltonian cycle 𝐶 in 𝑄⁰_𝑛−𝐹_𝑒−(𝐹_𝑎𝑣−{𝐛₁,𝐰₁}) containing (𝐮,𝐯).
Suppose that (𝐛₁,𝐰₁)∈𝐶. Obviously, 𝐶 can be written as ⟨𝐰₁,𝐛₁,𝐲,𝑃,𝐱⟩. Let 𝑡^′=𝑡−2^𝑛−1−1. Obviously, there exists a section 𝑄 of 𝑃 joining 𝐬 to 𝐭 of length 𝑡^′ containing (𝐮,𝐯). By Lemma 2.2, there exists a Hamiltonian path 𝑅 joining (𝐭)^𝑛 to (𝐬)^𝑛 of 𝑄¹_𝑛−𝐹¹_𝑒. Obviously, ⟨𝐬,𝑄,𝐭,(𝐭)^𝑛,𝑅,(𝐬)^𝑛,𝐬⟩ forms the desired cycle.
Suppose that (𝐛₁,𝐰₁)∉𝐶. The cycle 𝐶 can be written as ⟨𝐰₁,𝐱,𝑃₁,𝐲,𝑏₁,𝐬,𝑃₂,𝐭,𝐰₁⟩. Without loss of generality, we can assume that (𝐮,𝐯) is on the path 𝑃₁. Suppose that 2^𝑛−1+4≤𝑡≤2^𝑛−1+1+𝑙(𝑃₁). Let 𝑡^′=𝑡−2^𝑛−1−1. There exists a section 𝑃^′₁ of 𝑃₁ such that (1) 𝑃^′₁ joins 𝐜 to 𝐝, (2) 𝑃^′₁ contains (𝐮,𝐯), and (3) 𝑃^′₁ is of length 𝑡^′. By Lemma 2.2, there exists a Hamiltonian path 𝑅 of 𝑄¹_𝑛−𝐹¹_𝑒 joining (𝐝)^𝑛 to (𝐜)^𝑛. Obviously, ⟨𝐜,𝑃^′₁,𝐝,(𝐝)^𝑛,𝑅,(𝐜)^𝑛,𝐜⟩ forms the desired cycle. Suppose that 2^𝑛−1+3+𝑙(𝑃₁)≤𝑡≤2^𝑛−2𝑓_𝑎𝑣. Let 𝑡^′=𝑡−2^𝑛−1−2−𝑙(𝑃₁). There exists a section 𝑃^′₂ of 𝑃₂ joining 𝐫 to 𝐭 of length 𝑡^′. By Lemma 2.4, there exist two spanning disjoint paths  𝑅₁ and 𝑅₂ of 𝑄¹_𝑛−𝐹¹_𝑒 such that 𝑅₁ joins (𝐲)^𝑛 to (𝐫)^𝑛 and 𝑅₂ joins (𝐭)^𝑛 to (𝐱)^𝑛. Obviously, ⟨𝐱,𝑃₁,𝐲,(𝐲)^𝑛,𝑅₁,(𝐫)^𝑛,𝐫,𝑃^′₂,𝐭,(𝐭)^𝑛,𝑅₂,(𝐱)^𝑛,𝐱⟩ forms the desired cycle.

4. Conclusion

In this paper, we study the bipancyclic property of faulty hypercubes. We improve previous results by considering both edge faults and vertex faults. Let 𝐹 be a subset of 𝑉(𝑄_𝑛)∪𝐸(𝑄_𝑛) such that 𝐹 can be decomposed into two parts 𝐹_𝑎𝑣 and 𝐹_𝑒 where 𝐹_𝑎𝑣 is a union of 𝑓_𝑎𝑣 disjoint adjacent pairs of 𝑉(𝑄_𝑛) and 𝐹_𝑒 consists of 𝑓_𝑒 edges. We prove that 𝑄_𝑛−𝐹 is bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2. This result is optimal. Let 𝐱 be any vertex of 𝑄_𝑛. Assume that 𝐹={(𝐱,(𝐱)^𝑖)∣1≤𝑖≤𝑛−1} forms a set of (𝑛−1) faulty edges. Obviously, deg_{𝑄𝑛−𝐹}(𝐱)=1. 𝑄_𝑛−𝐹 is not bipancyclic.

We also prove that 𝑄_𝑛−𝐹 is edge bipancyclic if 𝑓_𝑎𝑣+𝑓_𝑒≤𝑛−2 with 𝑓_𝑎𝑣<𝑛−2. Again, this result is optimal. Assume that 𝐹=∪^𝑛−1_𝑖=2{(𝐱)^𝑖,((𝐱)¹)^𝑖} forms a set of (𝑛−2) adjacent faulty vertices. Obviously, deg_{𝑄𝑛−𝐹}(𝐱)=deg_{𝑄𝑛−𝐹}((𝐱)¹)=2. Thus, any Hamiltonian cycle of 𝑄_𝑛−𝐹 contains the path ⟨(𝐱)^𝑛,𝐱,(𝐱)¹,((𝐱)¹)^𝑛⟩. Therefore, there is no Hamiltonian cycle containing the edge ((𝐱)^𝑛,((𝐱)¹)^𝑛) for 𝑛≥3. Thus, 𝑄_𝑛−𝐹 is not edge bipancyclic.

Two interesting observations, Lemmas 2.4 and 2.7, are used in this paper. In the following, we claim that these two lemmas are also optimal.

Let 𝐳, 𝐮, and 𝐱 be three distinct white vertices of 𝑄_𝑛. Assume that 𝐹={(𝐳,(𝐳)^𝑖)∣ for 2≤𝑖≤𝑛−1} forms a set of (𝑛−2) faulty edges. It is observed that {(𝐳)¹,(𝐳)^𝑛} are the neighbors of 𝐳 in 𝑄_𝑛−𝐹. Moreover, (𝐳)¹ and (𝐳)^𝑛 are black vertices. Obviously, 𝐳 cannot be any vertex in 𝑃₁∪𝑃₂ for any two disjoint paths 𝑃₁ and 𝑃₂ of 𝑄_𝑛−𝐹 such that 𝑃₁ joins 𝐮 to (𝐳)¹ and 𝑃₂ joins 𝐱 to (𝐳)^𝑛. Therefore, there do not exist two disjoint paths 𝑃₁ and 𝑃₂ of 𝑄_𝑛−𝐹 such that 𝑃₁ joins 𝐮 to (𝐳)¹ and 𝑃₂ joins 𝐱 to (𝐳)^𝑛. Thus, the number of faulty edges of Lemma 2.4 is optimal.

Assume that 𝑛≥3. Let 𝐱 be any vertex of 𝑄_𝑛 and 𝐹=∪^𝑛−1_𝑖=2{(𝐱)^𝑖,((𝐱)¹)^𝑖} be a set of (𝑛−2) adjacent faulty vertices. Obviously, (𝐱)^𝑛 and ((𝐱)¹)^𝑛 are vertices in different partite sets. Assume that there is a Hamiltonian path 𝑃 joining (𝐱)^𝑛 to ((𝐱)¹)^𝑛. Since deg_{𝑄𝑛−𝐹}(𝐱)=deg_{𝑄𝑛−𝐹}((𝐱)¹)=2, 𝑃 must include the section ⟨(𝐱)¹,𝐱,(𝐱)^𝑛,((𝐱)^𝑛)¹⟩. Obviously, 𝑃 is not a Hamiltonian path and we get a contradiction. Thus, the number of adjacent faulty vertices of Lemma 2.7 is optimal.

Acknowledgment

This work was supported in part by the National Science Council of Republic of China under NSC 100-2221-E-212-014.