Edge-bipancyclicity of bubble-sort star graphs

TL;DR

This paper proves that the n-dimensional bubble-sort star graph is edge-bipancyclic for all n≥3, meaning each edge lies on cycles of all even lengths, enhancing understanding of its cycle structure.

Contribution

The paper establishes that the bubble-sort star graph is edge-bipancyclic for n≥3, providing new insights into its cycle properties and network robustness.

Findings

Each edge lies on cycles of all even lengths from 4 to n!

Every edge is part of at least four cycles of each even length

The graph is bipartite and (2n-3)-regular for n≥3

Abstract

The interconnection network considered in this paper is the bubble-sort star graph. The $n$-dimensional bubble-sort star graph $BS_n$ is a bipartite and $(2n-3)$-regular graph of order $n!$. A bipartite graph $G$ is edge-bipancyclic if each edge of $G$ lies on a cycle of all even length $l$ with $4\leq l\leq |V(G)|$. In this paper, we show that the $n$-dimensional bubble-sort star graph $BS_n$ is edge-bipancyclic for $n\ge 3$ and for each even length $l$ with $4\leq l\leq n!$, every edge of $BS_n$ lies on at least four different cycles of length $l$.

Full text

AMS classification: 05C50

Keywords: bubble-sort star graph, edge-bipancyclicity

1. Introduction

Network topology is a crucial factor for the interconnection networks since it determines the performance of the networks. Many interconnection network topologies have been proposed in the literature for the purpose of connecting hundreds or thousands of processing elements. In these interconnection network topologies, linear arrays and rings are two of the most fundamental structures. One of the important issues in parallel processing is to embed linear arrays and rings into interconnection networks (see [12] and [19]). Paths and cycles are popular interconnection networks owing to their simple structures and low degree. Moreover, many parallel algorithms have been devised on them [16, 18]. Many literatures have addressed how to embed cycles and paths into various interconnection networks [1, 4, 6, 7, 17].

An interconnection network is usually represented by an undirected simple graph where vertices represent processors and edges represent links between processors. In the rest of the paper, we will use Bondy and Murty [2] for terminology and notation not defined here and only consider simple undirected graphs.

Let $G=(V,E)$ be a graph. $G$ is called $k$-regular if every vertex in $G$ has exactly $k$ neighbors. $P_{k}=v_{1}v_{2}\cdots v_{k}$ on $k$ distinct vertices $v_{1},v_{2},\cdots,v_{k}$ in $G$ is a $k$-path if $(v_{i},v_{i+1})$ is an edge in $G$ for every $i=1,2,\cdots,k-1$. For a $k$-path $P_{k}=v_{1}v_{2}\cdots v_{k}$, if $(v_{1},v_{k})$ is an edge, then $C_{k}=v_{1}v_{2}\cdots v_{k}v_{1}$ is a $k$-cycle in $G$ and the length of $C_{k}$ is $k$. $G$ is Hamiltonian if it contains a cycle passing through every vertex exactly once and the cycle is called a Hamiltonian cycle. We say that $G$ of order $n$ is pancyclic if it contains a cycle of every possible length between 3 and $n$. $G$ is vertex-pancyclic (resp. edge-pancyclic) if every vertex (resp. edge) of $G$ lies on a cycle of all length $l$ with $3\leq l\leq|V(G)|$. $G$ is bipartite if the vertex set of $G$ can be partitioned into two vertex subsets $V_{1}$ and $V_{2}$ with $V(G)=V_{1}\cup V_{2}$, $V_{1}\cap V_{2}=\emptyset$ such that each edge of $G$ joins one vertex in $V_{1}$ and the other in $V_{2}$. It is well known that any bipartite graph contains no odd cycles. A bipartite graph $G$ of order $n$ is bipancyclic if it contains a cycle of every possible even length between 4 and $n$. A bipartite graph $G$ is vertex-bipancyclic (resp. edge-bipancyclic) if every vertex (resp. edge) of $G$ lies on a cycle of all even length $l$ with $4\leq l\leq|V(G)|$. It is obvious that every edge-pancyclic (resp. edge-bipancyclic) graph is vertex-pancyclic (resp. vertex-bipancyclic) graph and every vertex-pancyclic (resp. vertex-bipancyclic) graph is pancyclic (resp. bipancyclic) graph. The edge-bipancyclicity and the bipancyclicity of different interconnection networks are widely studied. For example, see [10, 11, 13, 14, 15, 20].

The bubble-sort star graphs [5], which belong to the class of Cayley graphs, have been attractive alternative to the hypercubes. It gains many nice properties, such as high degree of regularity and symmetry. In particular, the $n$-dimensional bubble-sort star graph $BS_{n}$ has $n!$ vertices, and is $(2n-3)$-regular and vertex transitive. But it is not edge transitive. $BS_{n}$ has received considerable attention in recent years. For example, Cai et al. [3] investigated the fault-tolerant maximal local-connectivity of $BS_{n}$. Guo et al. [9] gave the conditional diagnosability of $BS_{n}$. Gu et al. [8] determined the pessimistic diagnosability of $BS_{n}$. Wang et al. [21, 22] studied the 2-extra connectivity (resp. 2-good-neighbor connectivity) and the 2-extra diagnosability (resp. 2-good-neighbor diagnosability) of $BS_{n}$.

This paper deals with the edge-bipancyclicity of bubble-sort star graphs. In [5], Chou et al. showed that the bubble-sort star graph is Hamiltonian. In this paper, we will show that $n$-dimensional bubble-sort star graph $BS_{n}$ is edge-bipancyclic, vertex-bipancyclic and bipancyclic for $n\geq 3$ and for each even length $l$ with $4\leq l\leq n!$, every edge of $BS_{n}$ lies on at least four different cycles of length $l$.

2. Preliminaries

In this section, we first review bubble-sort star graphs and give some lemmas which will be used in the following proof.

Definition 2.1 [5] The $n$-dimensional bubble-sort star graph $BS_{n}$ has vertex set that consists of all $n!$ permutations on $\{1,2,\cdots,n\}$. A permutation $x$ on $\{1,2,\cdots,n\}$ is denoted as $x=x_{1}x_{2}\cdots x_{n}$. A vertex $x=x_{1}x_{2}\cdots x_{n}\in V(BS_{n})$ is adjacent to a vertex $y=y_{1}y_{2}\cdots y_{n}\in V(BS_{n})$ if and only if there exists an integer $i$ with $2\leq i\leq n$ such that $y_{i}=x_{i-1}$, $y_{i-1}=x_{i}$ and $x_{j}=y_{j}$ for every $j\in\{1,2,\cdots,n\}-\{i-1,i\}$ or $y_{1}=x_{i}$, $y_{i}=x_{1}$ and $x_{j}=y_{j}$ for every $j\in\{2,\cdots,n\}-\{i\}$.

By Definition 2.1, $BS_{n}$ is a bipartite graph that has $n!$ vertices, each of which is a permutation on $\{1,2,\cdots,n\}$ and each vertex has degree $2n-3$. Fig. 1 shows $BS_{2}$, $BS_{3}$, and $BS_{4}$, respectively.

Let $x=x_{1}x_{2}\cdots x_{n},~{}y=y_{1}y_{2}\cdots y_{n}\in V(BS_{n})$, we use “$\circ$” to denote an operation such that $x=y\circ(i,j)$ if and only if $x_{i}=y_{j}$, $x_{j}=y_{i}$ and $x_{k}=y_{k}$ for every $k\in\{1,2,\cdots,n\}-\{i,j\}$. Then $(x,y)\in E(BS_{n})$ if and only if $y=x\circ(1,i)$ or $y=x\circ(i-1,i)$ for some $i\in\{2,3,\cdots,n\}$. Let $x^{+}=x\circ(1,n)$ and $x^{-}=x\circ(n-1,n)$ for simplicity. For an integer $i$ with $1\leq i\leq n$, we use $BS_{n}(i)$ to denote the induced subgraph of $BS_{n}$ by the set of vertices $\{x=x_{1}x_{2}\cdots x_{n}~{}|~{}x_{n}=i\}$. By Definition 2.1, $BS_{n}(i)\cong BS_{n-1}$ for every $i=1,2,\cdots,n$. Let $e=(x,y)\in E(BS_{n}(i))$ and $x^{\prime}\in\{x^{+},x^{-}\},y^{\prime}\in\{y^{+},y^{-}\}$. If $e^{\prime}=(x^{\prime},y^{\prime})\in E(BS_{n}(j))$ for some $j\neq i$, then $e^{\prime}$ is called a coupled pair-edge of $e$.

Now we give some properties about $BS_{n}$.

Lemma 2.2 [5] For $n\geq 3$, the bubble-sort star graph $BS_{n}$ is Hamiltonian.

**Lemma 2.3 ** Let $n,~{}k,~{}m$ be three integers with $n\geq 4$, $1\leq k,m\leq n$ and $m\neq k$. Let $H$ be an arbitrary Hamiltonian cycle in $BS_{n}(k)$. Then for any vertex $u=u_{1}u_{2}\cdots u_{n-2}mk$, there is a vertex $v=v_{1}v_{2}\cdots v_{n-1}k$ such that $e=(u,v)$ has a coupled pair-edge $e^{\prime}\in E(BS_{n}(m))$, where $(u,v)\in E(H)$.

Proof.  If $v_{n-1}=m$, then $e^{\prime}=(u^{-},v^{-})$ is a coupled pair-edge of $e$ with $e^{\prime}\in E(BS_{n}(m))$. Now we suppose that $v_{n-1}\neq m$. Thus the two neighbors of $u$ in $H$ is $u\circ(1,n-1)$ and $u\circ(n-2,n-1)$. In this case, set $v=u\circ(1,n-1)$. Then $e^{\prime}=(u^{-},v^{+})$ is a coupled pair-edge of $e$ with $e^{\prime}\in E(BS_{n}(m))$.

3. Edge-bipancyclicity of $BS_{n}$

We first consider the edge-bipancyclicity of $BS_{3}$.

Lemma 3.1 ** $BS_{3}$ is edge-bipancyclic. Furthermore, every edge of $BS_{3}$ lies on at least four different cycles of length 4 and 6, respectively.

Proof.  Let $e$ be an arbitrary edge in $BS_{3}$. Since $BS_{3}$ is vertex transitive, without loss of generality, let $123$ be an end vertex of $e$. Then we will consider the following two cases.

Case 1. $e=(123,132)$* or $e=(123,213)$.*

Let $e=(123,132)$. The four different 4-cycles containing $e$ are showed in Fig. 2 and the four different 6-cycles containing $e$ are showed in Fig. 4 (a), (b), (c) and (d).

If $e=(123,213)$, the cycles of length 4 and 6 containing $e$ can be constructed similarly to the above case.

Case 2. $e=(123,321)$.

The four different $4$-cycles containing $e$ are showed in Fig. 3 and the four different 6-cycles containing $e$ are showed in Fig.4 (b), (c), (e) and (f).

Thus we get that $BS_{3}$ is edge-bipancyclic and every edge of $BS_{3}$ lies on at least four different cycles of length 4 and 6, respectively.

Now we show that $BS_{4}$ is edge-bipancyclic. Lemmas 3.1 and 3.2 are the basis of induction for proving $BS_{n}$ to be edge-bipancyclic.

Lemma 3.2 $BS_{4}$* is edge-bipancyclic. Furthermore, every edge of $BS_{4}$ lies on at least four different cycles of every even length $l$ with $4\leq l\leq 24$.*

Proof.  Let $(u,v)$ be an arbitrary edge of $BS_{4}$. Since $BS_{4}$ is vertex transitive, we can assume that $u=1234$. Then $u\in V(BS_{4}(4))$. Now we will consider the following three cases.

Case 1. $v\in V(BS_{4}(4))$.

Note that $BS_{4}(4)\cong BS_{3}$. By Lemma 3.1, $(u,v)$ lies on at least four different cycles of length $4$ and $6$ in $BS_{4}(4)$, respectively. Now we will construct cycles of every even length between $8$ and $24$ containing $(u,v)$ by the following two subcases.

Subcase 1.1. $v=1324$* or $v=2134$.*

Let $v=1324$. The four different $8$-cycles containing $(u,v)$ in $BS_{4}$ are listed in Table 1.

[TABLE]

Table 1. $8$-cycles containing $(1234,1324)$ in $BS_{4}$.

Note that $(2143,1243)\in E(BS_{4}(3))$ is an edge of $C_{8}^{1}$ in Table 1. Since $BS_{4}(3)\cong BS_{3}$, by Lemma 3.1, (2143,1243) can be contained in four different $t$-cycles in $BS_{4}(3)$ for $t=4,6$. Let $C^{1}$ be any $t$-cycle in $BS_{4}(3)$ containing (2143,1243), then the edge set $E(C_{8}^{1})\cup E(C^{1})-\{(2143,1243)\}$ forms a $(6+t)$-cycle containing (1234,1324). Thus (1234,1324) lies on at least four different cycles of length 10 and 12, respectively.

Let $C_{12}=(1234)(1324)(3124)(3214)(2314)(2134)(2143)(2413)(4213)(4123)(1423)\\ (1243)(1234)$ be a $12$-cycle containing the edge (1234,1324). Let $C_{6}^{1}=(1234)(1324)\\ (3124)(3214)(2314)(2134)(1234)$, $C_{6}^{2}=(2143)(2413)(4213)(4123)(1423)(1243)(2143)$. Thus $C_{6}^{1}$ (resp. $C_{6}^{2}$) is a Hamiltonian cycle in $BS_{4}(4)$ (resp. $BS_{4}(3)$). By Lemma 2.3, there exists an edge $e=(x,y)\not\in\{(1234,2134),(1243,2143)\}$ in $C_{6}^{1}$ (resp. $C_{6}^{2}$) that has a coupled pair-edge $e^{\prime}$ such that $e^{\prime}=(x^{\prime},y^{\prime})\in E(BS_{n}(k))$, where $k=1,2$. Hence the edge set $E(C_{12})\cup\{e^{\prime},(x,x^{\prime}),(y,y^{\prime})\}-\{e\}$ forms a $14$-cycle containing (1234,1324). Thus (1234,1324) lies on at least four different 14-cycles.

Let $C_{14}=(1234)(1324)(3124)(3214)(2314)(2134)(2143)(2413)(4213)(4123)(4132)\\ (1432)(1423)(1243)(1234)$ be a $14$-cycle containing the edge (1234,1324). Note that $(4132,1432)\in E(BS_{4}(2))$ is an edge of $C_{14}$. Since $BS_{4}(2)\cong BS_{3}$, by Lemma 3.1, $(4132,1432)$ can be contained in four different $t$-cycles in $BS_{4}(2)$, where $t=4,6$. Let $C^{2}$ be any $t$-cycle in $BS_{4}(2)$ containing $(4132,1432)$, then the edge set $E(C_{14})\cup E(C^{2})-\{(4132,1432)\}$ forms a $(12+t)$-cycle containing (1234,1324). Thus (1234,1324) lies on at least four different cycles of length 16 and 18, respectively.

Let $C_{18}=(1234)(1324)(3124)(3214)(2314)(2134)(2143)(2413)(4213)(4123)(4132)\\ (4312)(3412)(3142)(1342)(1432)(1423)(1243)(1234)$ be a $18$-cycle containing the edge (1234,1324). By Lemma 2.3, there exists an edge $e=(x,y)\not\in\{(1234,2134),(1243,\\ 2143),(1423,4123)\}$ in $C_{6}^{1}$ (resp. $C_{6}^{2}$) that has a coupled pair-edge $e^{\prime}$ such that $e^{\prime}=(x^{\prime},y^{\prime})\in E(BS_{n}(1))$. Let $C_{6}^{3}=(1432)(4132)(4312)(3412)(3142)(1342)(1432)$. Thus $C_{6}^{3}$ is a Hamiltonian cycle in $BS_{4}(2)$. We have that $e=(x,y)=(1342,1432)$ (resp. (4312, 3412)) in $C_{6}^{3}$ has a coupled pair-edge $e^{\prime}=(x^{\prime},y^{\prime})=(2341,2431)$ (resp. (4321, 3421)) in $BS_{n}(1)$. Hence the edge set $E(C_{18})\cup\{e^{\prime},(x,x^{\prime}),(y,y^{\prime})\}-\{e\}$ forms a $20$-cycle containing (1234,1324). Thus (1234,1324) lies on at least four different 20-cycles.

Let $C_{20}=(1234)(1324)(3124)(3214)(2314)(2134)(2143)(2413)(4213)(4123)(4132)\\ (4312)(4321)(3421)(3412)(3142)(1342)(1432)(1423)(1243)(1234)$ be a $20$-cycle containing the edge (1234,1324). Note that $(4321,3421)\in E(BS_{4}(1))$ is an edge of $C_{20}$. Since $BS_{4}(1)\cong BS_{3}$, by Lemma 3.1, $(4321,3421)$ can be contained in four different $t$-cycles in $BS_{4}(1)$ for $t=4,6$. Let $C^{3}$ be any $t$-cycle in $BS_{4}(1)$ containing $(4321,3421)$, then the edge set $E(C_{20})\cup E(C^{3})-\{(4321,3421)\}$ forms a $(18+t)$-cycle containing (1234,1324). Thus (1234,1324) lies on at least four different cycles of length 22 and 24, respectively.

If $v=2134$, the cycles of every even length between $8$ and $24$ containing (1234,2134) can be constructed similarly to the case $v=1324$.

Subcase 1.2. $v=3214$.

The four different $8$-cycles containing $(u,v)$ in $BS_{4}$ are listed in Table 2.

[TABLE]

Table 2. $8$-cycles containing $(1234,3214)$ in $BS_{4}$.

Note that $(2143,1243)\in E(BS_{4}(3))$ is an edge of $C_{8}^{1*}$ in Table 2. Since $BS_{4}(3)\cong BS_{3}$, by Lemma 3.1, $(2143,1243)$ can be contained in four different $t$-cycles in $BS_{4}(3)$ for $t=4,6$. Let $C^{1*}$ be any $t$-cycle in $BS_{4}(3)$ containing $(2143,1243)$, then the edge set $E(C_{8}^{1*})\cup E(C^{1*})-\{(2143,1243)\}$ forms a $(6+t)$-cycle containing (1234,3214). Thus (1234,3214) lies on at least four different cycles of length 10 and 12, respectively.

The cycles of every even length between $14$ and $24$ containing (1234,3214) can be constructed by the same argument as that of Subcase 1.1.

Case 2. $v=u^{-}=1243$.

The four different $4$-cycles containing $(u,v)$ in $BS_{4}$ are listed in Table 3.

[TABLE]

Table 3. $4$-cycles containing $(1234,1243)$ in $BS_{4}$.

Note that $(2134,1234)\in E(BS_{4}(4))$ is an edge of $C_{4}^{1}$ in Table 3. Since $BS_{4}(4)\cong BS_{3}$, by Lemma 3.1, $(2134,1234)$ can be contained in four different $t$-cycles in $BS_{4}(4)$ for $t=4,6$. Let $C^{1^{\prime}}$ be any $t$-cycle in $BS_{4}(4)$ containing $(2134,1234)$, then the edge set $E(C_{4}^{1})\cup E(C^{1^{\prime}})-\{(2134,1234)\}$ forms a $(2+t)$-cycle containing (1234,1243). Thus (1234,1243) lies on at least four different cycles of length 6 and 8, respectively.

Let $C_{8}^{1}$ be the 8-cycle containing (1234,1243) in Table 1. Then the cycles of every even length between $10$ and $24$ containing (1234,1243) can be constructed by the same argument as that of Subcase 1.1.

Case 3. $v=u^{+}=4231$.

The four different $4$-cycles containing $(u,v)$ in $BS_{4}$ are listed in Table 4.

Note that $(3214,1234)\in E(BS_{4}(4))$ is an edge of $C_{4}^{1*}$ in Table 4. Since $BS_{4}(4)\cong BS_{3}$, by Lemma 3.1, $(3214,1234)$ can be contained in four different $t$-cycles in $BS_{4}(4)$, where $t=4,6$. Let $C^{1^{\prime\prime}}$ be any $t$-cycle in $BS_{4}(4)$ containing $(3214,1234)$, then the edge set $E(C_{4}^{1*})\cup E(C^{1^{\prime\prime}})-\{(3214,1234)\}$ forms a $(2+t)$-cycle containing (1234,4231). Thus (1234,4231) lies on at least four different cycles of length 6 and 8, respectively.

[TABLE]

Table 4. $4$-cycles containing $(1234,4231)$ in $BS_{4}$.

The cycles of every even length between $10$ and $24$ containing (1234,4231) can be constructed by the same argument as that of Subcase 1.1.

Theorem 3.3 For $n\geq 3$, the bubble-sort star graph $BS_{n}$ is edge-bipancyclic. Furthermore, every edge of $BS_{n}$ lies on at least four different cycles of every even length $l$ with $4\leq l\leq n!$.

Proof.  We prove this theorem by induction on $n$. For $n=3,4$, the result holds by Lemmas 3.1 and 3.2. Assume $n\geq 5$. Let $e=(u,v)\in E(BS_{n})$ be an arbitrary edge. Since $BS_{n}$ is vertex transitive, we can assume that $u=12\cdots n$. Then $u\in V(BS_{n}(n))$. Now we will consider the following three cases.

Case 1. $v\in V(BS_{n}(n))$.

Note that $BS_{n}(n)\cong BS_{n-1}$. Hence $e=(u,v)$ lies on at least four different cycles of every even length between $4$ and $(n-1)!$ in $BS_{n}(n)$ by induction hypothesis. Let $q$ be an integer with $1\leq q\leq n-1$ and $p$ be an even integer with $2\leq p\leq(n-1)!$. We first prove the following claim.

Claim 1. There are at least four different cycles of length $l$ containing $e$, where $l=q(n-1)!+p$.

Proof of Claim 1. We prove this claim by induction on $q$.

Suppose $q=1$. Let $H_{n}$ be a Hamiltonian cycle in $BS_{n}(n)$ containing $e$. By Lemma 2.3, there exists an edge $e_{1}=(s_{1},r_{1})\in E(H_{n})$ with $e_{1}\neq e$ that has a coupled pair-edge $e_{1}^{\prime}=(s_{1}^{\prime},r_{1}^{\prime})$ such that $e_{1}^{\prime}\in E(BS_{n}(m))$ for every $1\leq m\leq n-1$. Thus the edge set $E(H_{n})\cup\{e_{1}^{\prime},(s_{1},s_{1}^{\prime}),(r_{1},r_{1}^{\prime})\}-\{e\}$ forms a $((n-1)!+2)-$cycle containing $e$ in $BS_{n}$. Since $e=(u,v)$ lies on at least four different Hamiltonian cycles in $BS_{n}(n)$, there are at least four $((n-1)!+2)-$cycles containing $e$ in $BS_{n}$. Now we will consider the case that $4\leq p\leq(n-1)!$.

Let $e_{11}^{\prime}=(s_{11}^{\prime},r_{11}^{\prime})\in E(BS_{n}(1))$ be a coupled pair-edge of $e_{11}=(s_{11},r_{11})\in E(H_{n})$ with $e_{11}\neq e$. Since $BS_{n}(1)\cong BS_{n-1}$, $e_{11}^{\prime}$ can be contained in at least four different $p$-cycles in $BS_{n}(1)$, where $4\leq p\leq(n-1)!$. Let $C_{p}$ be any $p$-cycle in $BS_{n}(1)$ containing $e_{11}^{\prime}$, then the edge set $E(H_{n})\cup E(C_{p})\cup\{(s_{11},s_{11}^{\prime}),(r_{11},r_{11}^{\prime})\}-\{e_{11}^{\prime},e_{11}\}$ forms a $((n-1)!+p)$-cycle containing $e$. Thus $e$ lies on at least four different cycles of length $(n-1)!+p$. Hence the claim holds for $q=1$.

Suppose the claim holds for $q=k-1$, where $2\leq k\leq n-1$. Thus there are at least four different cycles of every even length between $4$ and $k(n-1)!$ containing $e$. Let $C_{k(n-1)!}$ be a cycle of length $k(n-1)!$ containing $e$. Let $H_{i}$ be a Hamiltonian cycle in $BS_{n}(i)$ with $E(H_{i})=(E(C_{k(n-1)!})\cap E(BS_{n}(i)))\cup\{e^{\prime}_{i1},e_{(i+1)1}\}$, where $e^{\prime}_{i1}$ is a coupled pair-edge of $e_{i1}$ for every $i=1,2,\cdots,k-1$ (See Fig. 5).

Now we will prove the claim holds for $q=k$. By Lemma 2.3, there exists an edge $e*=(x,y)\in E(H_{i})-\{e^{\prime}_{i1},e_{(i+1)1}\}$ that has a coupled pair-edge $e^{\prime}=(x^{\prime},y^{\prime})\in E(BS_{n}(j))$ for every $i\in\{1,2,\cdots,k-1,n\}$ and $j\in\{k,k+1,\cdots,n-1\}$. Thus the edge set $E(C_{k(n-1)!})\cup\{e^{\prime},(x,x^{\prime}),(y,y^{\prime})\}-\{e*\}$ forms a $(k(n-1)!+2)$-cycle containing $e$. Since $k(n-k)\geq 4$ for $n\geq 5$ and $2\leq k\leq n-1$, there are at least four different $(k(n-1)!+2)-$cycles containing $e$ in $BS_{n}$. Now we will consider the case that $4\leq p\leq(n-1)!$.

Let $e_{k1}^{\prime}=(s_{k1}^{\prime},r_{k1}^{\prime})\in E(BS_{n}(k))$ be a coupled pair-edge of $e_{k1}=(s_{k1},r_{k1})\in E(H_{k-1})$, where $e_{k1}\neq e^{\prime}_{(k-1)1}$. Since $BS_{n}(k)\cong BS_{n-1}$, by induction hypothesis, $e_{k1}^{\prime}$ can be contained in at least four different $p$-cycles in $BS_{n}(k)$, where $4\leq p\leq(n-1)!$. Let $C_{p}$ be any $p$-cycle in $BS_{n}(k)$ containing $e_{k1}^{\prime}$, then the edge set $E(C_{k(n-1)!})\cup E(C_{p})\cup\{(s_{k1},s_{k1}^{\prime}),(r_{k1},r_{k1}^{\prime})\}-\{e_{k1}^{\prime},e_{k1}\}$ forms a $(k(n-1)!+p)$-cycle containing $e$. Thus $e$ lies on at least four different cycles of length $k(n-1)!+p$. Hence the claim holds.

By Claim 1, we have that $e$ lies on at least four different cycles of every even length $l$ with $4\leq l\leq n!$.

Case 2. $v=u^{-}$.

The four different $4$-cycles containing $e$ in $BS_{n}$ are listed in Table 5.

[TABLE]

Table 5. $4$-cycles containing $(u,u^{-})$ in $BS_{n}$.

Note that $(u,u\circ(1,2))\in E(BS_{n}(n))$ is an edge of $C_{4}^{1}$ in Table 5. Since $BS_{n}(n)\cong BS_{n-1}$, by induction hypothesis, $(u,u\circ(1,2))$ can be contained in four different $p$-cycles in $BS_{n}(n)$ for $p=4,6,\cdots,(n-1)!$. Let $C^{p^{\prime}}$ be any $p$-cycle in $BS_{n}(n)$ containing $(u,u\circ(1,2))$, then the edge set $E(C_{4}^{1})\cup E(C^{p^{\prime}})-\{(u,u\circ(1,2))\}$ forms a $(2+p)$-cycle containing $e$. Thus $e$ lies on at least four different cycles of every even length between 6 and $(n-1)!+2$.

The cycles of every even length between $(n-1)!+4$ and $n!$ containing $e$ can be constructed by the same argument as that of Case 1.

Case 3. $v=u^{+}$.

The four different $4$-cycles containing $e$ in $BS_{n}$ are listed in Table 6.

[TABLE]

Table 6. $4$-cycles containing $(u,u^{+})$ in $BS_{n}$.

Note that $(u,u\circ(2,3))\in E(BS_{n}(n))$ is an edge of $C_{4}^{1*}$ in Table 6. Since $BS_{n}(n)\cong BS_{n-1}$, by induction hypothesis, $(u,u\circ(2,3))$ can be contained in four different $p$-cycles in $BS_{n}(n)$ for $p=4,6,\cdots,(n-1)!$. Let $C^{p^{\prime\prime}}$ be any $p$-cycle in $BS_{n}(n)$ containing $(u,u\circ(2,3))$, then the edge set $E(C_{4}^{1*})\cup E(C^{p^{\prime\prime}})-\{(u,u\circ(2,3))\}$ forms a $(2+p)$-cycle containing $e$. Thus $e$ lies on at least four different cycles of every even length between 6 and $(n-1)!+2$.

The cycles of every even length between $(n-1)!+4$ and $n!$ containing $e$ can be constructed by the same argument as that of Case 1.

Thus we complete the proof of our main theorem.

By Theorem 3.3, we immediately obtain the following corollaries.

Corollary 3.4 For $n\geq 3$, the bubble-sort star graph $BS_{n}$ is vertex-bipancyclic.

Corollary 3.5 For $n\geq 3$, the bubble-sort star graph $BS_{n}$ is bipancyclic.

Acknowledgements This research is supported by National Natural Science Foundation of China (No. 11801450, 11771247), Natural Science Foundation of Shaanxi Province, China (No. 2019JQ-506).