Mutually independent hamiltonian cycles of binary wrapped butterfly graphs

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Contents lists available atScienceDirect

Mathematical and Computer Modelling

journal homepage:www.elsevier.com/locate/mcm

Mutually independent hamiltonian cycles of binary wrapped

butterfly graphs

Tz-Liang Kueng

a

, Tyne Liang

a,∗

, Lih-Hsing Hsu

b

a_{Department of Computer Science, National Chiao Tung University, 1001 University Road, Hsinchu, 30050, Taiwan, ROC}

b_{Department of Computer Science and Information Engineering, Providence University, 200 Chung Chi Road, Taichung, 43301, Taiwan, ROC}

a r t i c l e i n f o Article history:

Received 24 October 2006

Received in revised form 24 July 2008 Accepted 7 August 2008 Keywords: Interconnection network Graph Butterfly graph Hamiltonian cycle a b s t r a c t

Effective utilization of communication resources is crucial for improving performance in multiprocessor/communication systems. In this paper, the mutually independent hamiltonicity is addressed for its effective utilization of resources on the binary wrapped butterfly graph. Let G be a graph with N vertices. A hamiltonian cycle C of G is represented byhu1,u2, . . . ,uN,u1ito emphasize the order of vertices on C . Two hamiltonian cycles of G, namely C1 = hu1,u2, . . . ,uN,u1iand C2 = hv1, v2, . . . , vN, v1i, are said to be independent if u1 = v1and ui 6= vifor all 2 ≤ i ≤ N. A collection of m hamiltonian

cycles C1, . . . ,Cm, starting from the same vertex, are m-mutually independent if any two

different hamiltonian cycles are independent. The mutually independent hamiltonicity of a graph G, denoted byIH C(G), is defined to be the maximum integer m such that, for each vertex u of G, there exists a set of m-mutually independent hamiltonian cycles starting from

u. Let BF(n)denote the n-dimensional binary wrapped butterfly graph. Then we prove that IH C(BF(n)) =4 for all n≥3.

1. Introduction

A multiprocessor/communication interconnection network is usually modeled as a graph, in which the vertices correspond to processors/nodes, and the edges correspond to connections or communication links. In this paper, we use the terms, graphs and networks, interchangeably. Designing an interconnection network is multi-objected and complicated [1]. Hence, the topological properties of various interconnection networks have been widely addressed by many researchers [2–11]. Among various kinds of popular network topologies, butterfly networks are very suitable for VLSI implementation and parallel computing. In particular, the binary wrapped butterfly graph has gained many researchers’ efforts for its nice topological properties. For example, it belongs to the family of constant degree-four Cayley graphs [12,13]. Therefore, it is vertex-transitive. Moreover, the hamiltonian properties were addressed in research by [2,3,11]. Until recently it is believed that the presence of such a constant-degree network topology, with both logarithmic diameter and optimal fault tolerance is critical to improve the performance of peer-to-peer architectures [14,15]. In practice, Malkhi et al. [16] build a peer-to-peer lookup network on the basis of butterfly graphs.

Network embedding [1] is an interesting subject, because the portability of the guest network onto the host network would permit executing the guest specified algorithms on the host with as little modification as possible. In the research of [4,7,9–11], embedding of various topologies, such as rings, linear arrays, and binary trees, etc., onto the butterfly networks had been addressed. In particular, the ring is a popular network topology, since many efficient communication algorithms have been designed based on a ring structure. For instance, the token ring [17] often serves as the underlying connection

∗_{Corresponding author. Tel.: +886 3 5131365; fax: +886 3 5721490.} E-mail address:tliang@cs.nctu.edu.tw(T. Liang).

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T.-L. Kueng et al. / Mathematical and Computer Modelling 48 (2008) 1814–1825 1815

architecture of the local area network. In addition, the advantages of rings were discussed by Tel [18]. In this paper, we study the problem of embedding mutually independent hamiltonian cycles, as proposed by Sun et al. [19], onto a binary wrapped butterfly graph; that is, if some vertex is fixed as the start, any two of such hamiltonian cycles will traverse different vertices at every time step except the start-up and termination. Recently, Lin et al. [20] investigated how to embed the mutually independent hamiltonian cycles onto star networks and pancake networks. Moreover, Hsieh and Weng [21] further concerned the fault-tolerant embedding of pairwise independent hamiltonian paths on faulty hypercubes.

The concept of mutually independent hamiltonian cycles can be applied in many different areas. For example, communication applications on the interconnection network are often viewed as the interleaving of local computation and global communication stages. Such applications can be performed via a message routing protocol, by which information is transmitted along the communication links in packets of equal size. For the sake of simplification, the store-and-forward all-port communication model [8] has been widely adopted as one basic routing scheme, in which every processor is assumed to be capable of exchanging messages of fixed length, with all of its neighbors at each time step. Although routing messages over a spanning tree of the given network is intuitively the best strategy for message transmission, Baldi and Ofek [22] presented a systematic comparison between ring and tree embedding for group (many-to-many) multicast, and concluded that ring embedding remains a promising alternative. It is worth mentioning that there may be two potential shortcomings incurred by routing messages in a ring structured network [1]. One is that at least two message packets are likely to reside in the same processor, so as to provoke contention for the local computation resources. The other is that two or more message packets will contend for the use of some communication link (in the same direction). Clearly, mutually independent hamiltonian cycles can ease the effects of these two shortcomings.

As another example, a Latin square of order n is an n

×

n array containing the integers from 1 to n, arranged so that each integer appears exactly once in each row, and exactly once in each column. If we delete some rows from a Latin square, we will get a Latin rectangle. Obviously, a Latin square of order n can be thought of as the intermediate vertices of n mutually independent hamiltonian cycles on the complete graph with n

+

1 vertices. Thus, the concept behind mutually independent hamiltonian cycles can be interpreted as a Latin square/rectangle for graphs. Furthermore, we consider the following scenario. A tour agency will organize a 10-day tour to Japan in the Christmas vacation. Suppose that there will be many people joining this tour. However, the maximum number of people staying in each local area is limited, say 100 people, for the sake of a hotel contract. One trivial solution is based on the First-Come-First-Served intuition. So, only 100 people can join this tour. Note that we cannot schedule the tour in a pipelined manner, because the holiday period is fixed. Fortunately, we observe that scheduling a tour is like a hamiltonian cycle of a graph, in which a vertex denotes a hotel and an edge denotes the connection between two hotels if they can be traveled in a reasonable time. Therefore, we can organize all the attendants into a number of subgroups; each subgroup has its own tour in such a way, that no two subgroups will stay in the same area during the same time period. So any two different tours are indeed independent hamiltonian cycles. If there exist five mutually independent hamiltonian cycles, then we may allow up to 500 attendees to visit Japan on a Christmas vacation. Obviously, if we can find the maximum number of mutually independent hamiltonian cycles, the number of tour attendants would be maximized.

The rest of this paper is organized as follows. In Section2, the terminologies and notations are defined. In Section3, the nearly recursive construction of the n-dimensional binary wrapped butterfly network, denoted by BF

(

n

)

, is introduced. The basic properties of BF

(

n

)

are given in Section4. In Section5, we show that BF

(

n

)

has four mutually independent hamiltonian cycles starting from any vertex. Finally, the concluding remarks are given in Section6.

2. Definitions

In this paper, we concentrate on loopless undirected graphs. For the notations and graph-theoretic terminologies, we follow the ones given by Bondy and Murty [23]. A graph G is a two-tuple

(

V

,

E

)

, where V is a nonempty set, and E is a subset of

{

(

u

, v) | (

u

, v)

is an unordered pair of V

}

. We say that V

=

V

(

G

)

is the vertex set and E

=

E

(

G

)

is the edge set. Two vertices u and

v

are adjacent if

(

u

, v) ∈

E. The number of vertices in a graph G is denoted by

|

V

(

G

)|

. The degree of any vertex u in a graph G, denoted by degG

(

u

)

, is the number of edges incident with u. The maximum and minimum degrees of graph G are

denoted by∆

(

G

)

and

δ(

G

)

, respectively. A graph G is k-regular if∆

(

G

) = δ(

G

) =

k.

A graph H is a subgraph of a graph G if V

(

H

) ⊆

V

(

G

)

and E

(

H

) ⊆

E

(

G

)

. Let S be a nonempty subset of V

(

G

)

. The subgraph induced by S is the subgraph of G with the vertex set S and with the edge set consisting of those edges that join two vertices in S. Analogously, the subgraph generated by a nonempty set F

⊆

E

(

G

)

is the subgraph of G with the edge set F and with the vertex set consisting of those vertices incident to at least one edge of F . Two graphs G1and G2are isomorphic if there is a bijection

µ

from V

(

G1

)

onto V

(

G2

)

, such that

(

u

, v) ∈

E

(

G1

)

if and only if

(µ(

u

), µ(v)) ∈

E

(

G2

)

. The bijection

µ

is called an isomorphism.

A path P of length k from vertex x to vertex y in a graph G is a sequence of distinct vertices

h

v

₁

, v

₂

, . . . , v

_k₊₁

i

such that

v

1

=

x,

v

k+1

=

y, and

(v

i

, v

i+1

) ∈

E

(

G

)

for every 1

≤

i

≤

k. We also write P as

h

x

,

P

,

y

i

to emphasize its beginning and ending vertices. The i-th vertex of P is denoted by P

(

i

)

; i.e., P

(

i

) = v

i. Both P

(

1

)

and P

(

k

+

1

)

are terminal vertices of P. In

particular, let P−1

= h

v

_k₊₁

, v

_k

, . . . , v

₁

i

denote the reverse of P. For convenience, we use V

(

P

)

to denote the set of vertices traversed by P. A cycle is a path with at least three vertices, such that the first vertex is adjacent to the last one. To emphasize the vertex order on a cycle, a cycle of length k is represented by

h

v

₁

, v

₂

, . . . , v

_k

, v

₁

i

. A hamiltonian cycle (or hamiltonian path) of a graph G is a cycle (or path) that spans G. Two hamiltonian cycles starting from the same vertex s in a graph G, namely

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Fig. 1. (a) BF(3); (b) BF(3)with level-0 vertices replicated to ease visualization.

C1

= h

v

1

, v

2

, . . . , v

|V(G)|

, v

1

i

and C2

= h

u1

,

u2

, . . . ,

u|V(G)|

,

u1

i

, are independent if

v

1

=

u1

=

s and

v

i

6=

uifor 2

≤

i

≤ |

V

(

G

)|

.

A collection of m hamiltonian cycles C1

, . . . ,

Cm, starting from the same vertex, are m-mutually independent if Ciand Cjare

independent whenever i

6=

j. Moreover, the mutually independent hamiltonicity of a graph G, denoted byIH C

(

G

)

, is defined to be the maximum integer m, such that for any vertex u of G, there exists a set of m-mutually independent hamiltonian cycles starting from u. It is trivial thatIH C

(

G

) ≤ δ(

G

)

for any graph G.

Let Zn

= {

0

,

1

, . . . ,

n

−

1

}

denote the set of integers modulo n. The n-dimensional binary wrapped butterfly graph

(or butterfly graph for short) BF

(

n

)

is a graph with the vertex set Zn

×

Zn2. Each vertex is labeled by a two-tuple

h

`,

a0

. . .

a`

. . .

an−1

i

with a level

` ∈

Znand an n-bit binary string a0a1

. . .

an−1

∈

Zn2. A level-

`

vertex

h

`,

a0

. . .

a`

. . .

an−1

i

is adjacent to two vertices,

h

(` +

1

)

mod n

,

a0

. . .

a`

. . .

an−1

i

and

h

(` −

1

)

mod n

,

a0

. . .

a`−1

. . .

an−1

i

, by straight edges, and is adjacent to another two vertices,

h

(` +

1

)

mod n

,

a0

. . .

a`−1a`a`+1

. . .

an−1

i

and

h

(` −

1

)

mod n

,

a0

. . .

a`−2a`−1a`

. . .

an−1

i

, by cross edges. More formally, the edges of BF

(

n

)

can be defined in terms of four generators g, g−1_{, f , and f}−1_{as follows [}₁₃_]:

g

(h`,

a0

. . .

a`

. . .

an−1

i

) = h(` +

1

)

mod n

,

a0

. . .

a`

. . .

an−1

i

,

f

(h`,

a0

. . .

a`

. . .

an−1

i

) = h(` +

1

)

mod n

,

a0

. . .

a`−1a`a`+1

. . .

an−1

i

,

g−1

(h`,

a0

. . .

a`

. . .

an−1

i

) = h(` −

1

)

mod n

,

a0

. . .

a`

. . .

an−1

i

,

and f−1

(h`,

a0

. . .

a`−1

. . .

an−1

i

) = h(` −

1

)

mod n

,

a0a1

. . .

a`−2a`−1a`

. . .

an−1

i

,

where a_`

≡

a_`

+

1

(

mod 2

)

. Throughout this paper, a level-

`

edge of BF

(

n

)

is an edge that joins a level-

`

vertex and a

level-(`+

1

)

mod nvertex. To avoid the degenerate case, we only concern the case that n

≥

3. So, BF

(

n

)

is 4-regular.Fig. 1(a) depicts the structure of BF

(

3

)

andFig. 1(b) is another layout of BF

(

3

)

with the replication of level-0 vertices to ease visualization.

3. Nearly recursive construction of BF

(

n

)

For any

` ∈

_Znand i

∈

Z2, we use BF_`i

(

n

)

to denote the subgraph of BF

(

n

)

induced by

{h

h

,

a0

. . .

a`

. . .

an−1

i ∈

V

(

BF

(

n

)) |

a_`

=

i

}

. Obviously,

{

BF_`0

(

n

),

BF_`1

(

n

)}

forms a partition of BF

(

n

)

. Moreover, BF_`i

1

(

n

)

is isomorphic to BF

j

`2

(

n

)

for any i

,

j

∈

Z2 and any

`

1

, `

2

∈

Zn. With this observation, Wong [11] proposed a stretching operation to obtain BF_`i

(

n

)

from BF

(

n

−

1

)

.

More precisely, the stretching operation can be described as follows.

Let i

∈

_Z₂and

` ∈

_Znfor n

≥

3. Furthermore, let

=

ndenote the set of all subgraphs of BF

(

n

)

. Suppose that G

∈ =

n. We

define the following subsets of V

(

BF

(

n

+

1

))

and E

(

BF

(

n

+

1

))

:

V1

= {h

h

,

a0

. . .

a`−1ia`

. . .

an−1

i |

0

≤

h

< `, h

h

,

a0

. . .

a`−1a`

. . .

an−1

i ∈

V

(

G

)},

V2

= {h

h

+

1

,

a0

. . .

a`−1ia`

. . .

an−1

i |

` <

h

≤

n

−

1

, h

h

,

a0

. . .

a`−1a`

. . .

an−1

i ∈

V

(

G

)},

V3

= {h

`,

a0

. . .

a`−1ia`

. . .

an−1

i | h

`,

a0

. . .

a`−1a`

. . .

an−1

i

is incident to a level-

(` −

1

)

mod nedge in G

}

,

V4

= {h

` +

1

,

a0

. . .

a`−1ia`

. . .

an−1

i | h

`,

a0

. . .

a`−1a`

. . .

an−1

i

`

edge in G

}

,

E1

= {

(h

h

,

a0

. . .

a`−1ia`

. . .

an−1

i

, h

h

+

1

,

b0

. . .

b`−1ib`

. . .

bn−1

i

) |

0

≤

h

< `,

(h

h

,

a0

. . .

a`−1a`

. . .

an−1

i

, h

h

+

1

,

b0

. . .

b`−1b`

. . .

bn−1

i

) ∈

E

(

G

)},

E2

= {

(h

h

+

1

,

a0

. . .

a`−1ia`

. . .

an−1

i

, h(

h

+

2

)

mod(n+1)

,

b0

. . .

b`−1ib`

. . .

bn−1

i

) | ` ≤

h

≤

n

−

1

,

(h

h

,

a0

. . .

a`−1a`

. . .

an−1

i

, h(

h

+

1

)

mod n

,

b0

. . .

b`−1b`

. . .

bn−1

i

) ∈

E

(

G

)},

and E3

= {

(h`,

a0

. . .

a`−1ia`

. . .

an−1

i

, h` +

1

,

a0

. . .

a`−1ia`

. . .

an−1

i

) | h`,

a0

. . .

a`−1a`

. . .

an−1

i

is incident to at least one level-

(` −

1

)

_{mod n}edge and at least one level-

`

edge in G

}

.

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Fig. 2. (a) A subgraph G of BF(3); (b)γ0

0(G)inγ00(BF(3)); (c)γ10(G)inγ10(BF(3)). Fig. 3. (a) BF00,,10(4); (b) BF 0,0 0,2(4); (c) BF 0,0 0,3(4); (d) BF 0,0,0 0,2,3(4); (e) BF 0,0,0 0,1,3(4); (g) BF 0,0,0 0,1,2(4). Then, the stretching function

γ

_`i

:

S

n≥3

=

n

→

S

n≥4

=

nis defined by assigning

γ

_`i

(

G

)

as the graph with the vertex set

V1

∪

V2

∪

V3

∪

V4and the edge set E1

∪

E2

∪

E3. Clearly

γ

_`iis well-defined and one-to-one. We have

γ

_`i

(

G

) ∈ =

n+1if G

∈ =

n. In

particular,

γ

_`i

(

BF

(

n

)) =

BF_`i

(

n

+

1

)

. InFig. 2, we illustrate a subgraph G of BF

(

3

)

,

γ

₀0

(

G

)

in

γ

₀0

(

BF

(

3

))

, and

γ

₁0

(

G

)

in

γ

₁0

(

BF

(

3

))

. Obviously,

γ

i

`1

(

BF

(

n

))

is isomorphic to

γ

j

`2

(

BF

(

n

))

for any

`

1

, `

2

∈

Znand i

,

j

∈

Z2. Moreover,

γ

i

`

(

P

)

is a path in BF

(

n

+

1

)

if P is a path in BF

(

n

)

.

In fact, BF

(

n

)

can be further partitioned. Let m be an integer with 1

≤

m

≤

n. Assume that

`

1

, . . . , `

m

∈

Zn,

such that

`

1

< · · · < `

m. For any i1

, . . . ,

im

∈

Z2, we use BF_`i1₁,...,_,...,`im_m

(

n

)

to denote the subgraph of BF

(

n

)

induced by

{h

h

,

a0

. . .

an−1

i ∈

V

(

BF

(

n

)) |

a`j

=

ijfor 1

≤

j

≤

m

}

. InFig. 3, we illustrate BF 0,0

0,1

(

4

)

, BF0,20,0

(

4

)

, BF0,30,0

(

4

)

, BF0,2,30,0,0

(

4

)

, BF_0,1,30,0,0

(

4

)

, and BF_0,1,20,0,0

(

4

)

. Clearly BF_0,10,0

(

4

)

is isomorphic with BF_0,30,0

(

4

)

. Moreover, BF_0,2,30,0,0

(

4

)

, BF_0,1,30,0,0

(

4

)

, and BF_0,1,20,0,0

(

4

)

are also isomorphic. However, BF_0,10,0

(

4

)

is not isomorphic to BF_0,20,0

(

4

)

.

Lemma 1. Assume that n

≥

3 and i

,

j

,

k

∈

_Z₂. Then BF₀i,_,j₁

(

n

)

is isomorphic with BF₀i,_,j_n₋₁

(

n

)

; BF₀i,_,j₁,k_,₂

(

n

)

, BF₀i,_,j₁,k_,_n₋₁

(

n

)

, and BF_0,i,j_n,k_−2,_n₋₁

(

n

)

are isomorphic.

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Obviously,

{

BFi1,...,im

`1,...,`m

(

n

) |

i1

, . . . ,

im

∈

Z2

, `

1

, . . . , `

m

∈

Zn

, `

1

< · · · < `

m

}

forms a partition of BF

(

n

)

for any 1

≤

m

≤

n. To avoid the complication caused from modular arithmetic, we restrict our attention on the case that 1

≤

m

≤

n

−

1, 0

≤

`

₁

< · · · < `

_m, and

`

j

<

n

−

m

+

j

−

1 for each 1

≤

j

≤

m. The following two lemmas can be easily verified.

Lemma 2. Let 1

≤

m

≤

n

−

1. Suppose that i1

, . . . ,

im

∈

Z2and

`

1

, . . . , `

mare integers such that 0

≤

`

1

< · · · < `

mand

`

j

<

n

−

m

+

j

−

1 for each 1

≤

j

≤

m. Then

BFi1,...,im `1,...,`m

(

n

) =











γ

im `m

◦

γ

im−1 `m−1

◦ · · · ◦

γ

i3 `3

(

BF i1,i2 `1,`2

(

3

))

if m

=

n

−

1,

γ

im `m

◦

γ

im−1 `m−1

◦ · · · ◦

γ

i2 `2

(

BF i1 `1

(

3

))

if m

=

n

−

2,

γ

im `m

◦

γ

im−1 `m−1

◦ · · · ◦

γ

i1 `1

(

BF

(

n

−

m

))

otherwise.

Lemma 3. Let G be a connected spanning subgraph of BF₀i,_,j₁

(

n

)

, with i

,

j

∈

_Z₂and n

≥

3. Assume that 2

≤

` ≤

n

−

1. Let F0

= {h

`,

a0

. . .

an−1

i ∈

V

(

G

) | h`,

a0

. . .

an−1

i

is not incident to any level-

(` −

1

)

edge in G

}

,

F1

= {h

`,

a0

. . .

an−1

i ∈

V

(

G

) | h`,

a0

. . .

an−1

i

is not incident to any level-

`

edge in G

}

.

For any p

,

q

∈

_Z₂, let

F0

= {h

`,

a0

. . .

a`−1pqa`

. . .

an−1

i | h

`,

a0

. . .

a`−1a`

. . .

an−1

i ∈

F0

}

∪{h

` +

1

,

a0

. . .

a`−1pqa`

. . .

an−1

i | h

`,

a0

. . .

a`−1a`

. . .

an−1

i ∈

F0

}

,

F1

= {h

` +

1

,

a0

. . .

a`−1pqa`

. . .

an−1

i | h

`,

a0

. . .

a`−1a`

. . .

an−1

i ∈

F1

}

∪{h

` +

2

,

a0

. . .

a`−1pqa`

. . .

an−1

i | h

`,

a0

. . .

a`−1a`

. . .

an−1

i ∈

F1

}

,

M0

=

[

h`,a0...an−1i6∈F0∪F1

{

(h`,

a0

. . .

a`−1pqa`

. . .

an−1

i

, h` +

1

,

a0

. . .

a`−1pqa`

. . .

an−1

i

)} ,

and

M1

=

[

h`,a0...an−1i6∈F0∪F1

{

(h` +

1

,

a0

. . .

a`−1pqa`

. . .

an−1

i

, h` +

2

,

a0

. . .

a`−1pqa`

. . .

an−1

i

)}.

Then F0

∩

F1

= ∅

, F0

∩

F1

= ∅

, F0

∪

F1

=

V

(

BF0i,,j1,p,`,`+,q 1

(

n

+

2

)) −

V

(γ

q `+1

◦

γ

p `

(

G

))

, and M0

∪

M1

⊆

E

(γ

_`+q1

◦

γ

p `

(

G

))

. Let G be a subgraph of BF

(

n

)

. A cycle C in G is called an

`

-scheduled cycle of G if every level-

`

vertex of G is incident to a level-

(` −

1

)

mod nedge and a level-

`

edge on C [11]. Furthermore, a cycle C in G is a totally scheduled cycle of G if it is an

`

-scheduled cycle of G for all

` ∈

Zn[11]. Obviously,

γ

_`i

(

C

)

with i

∈ {

0

,

1

}

is a totally scheduled cycle of

γ

_`i

(

G

)

if C is a totally

scheduled cycle of G.

Lemma 4 ([11]). Let n

≥

3. Then BF

(

n

)

has a totally scheduled hamiltonian cycle. By stretching operation, we have the following two corollaries.

Corollary 1. Assume that n

≥

3 and i

,

j

,

k

∈

_Z₂. Then there exists a totally scheduled hamiltonian cycle of BF₀i,_,j₁,k_,₂

(

n

)

including all straight edges of level 0, level 1, and level 2.

≥

4 and i

,

j

,

p

,

q

∈

_Z₂. Then there exists a totally scheduled hamiltonian cycle of BF_0,1,2,3i,j,p,q

(

n

)

including all straight edges of level 0, level 1, level 2, and level 3 in BF_0,1,2,3i,j,p,q

(

n

)

.

4. Basic properties of BF

(

n

)

Suppose that e1

=

(

u1

, v

1

)

and e2

=

(

u2

, v

2

)

are either any two cross edges of BF

(

n

)

, or any two straight edges of BF

(

n

)

. Since BF

(

n

)

is vertex-transitive, there exists an isomorphism

µ

over V

(

BF

(

n

))

, such that u2

=

µ(

u1

)

and

v

2

=

µ(v

1

)

. Clearly, every hamiltonian cycle of BF

(

n

)

includes at least one cross edge and at least one straight edge.

Lemma 5. For any edge e of BF

(

n

)

with n

≥

3, there exists a totally scheduled hamiltonian cycle of BF

(

n

)

including e.

Lemma 6. Assume that i

,

j

,

k

∈

_Z₂. Let e be any edge of BF_0,1,2i,j,k

(

4

)

such that e

6∈ {

(h

3

,

ijk0

i

, h

0

,

ijk0

i

), (h

3

,

ijk1

i

, h

0

,

ijk1

i

)}

. Then there exists a totally scheduled hamiltonian cycle C of BF_0,1,2i,j,k

(

4

)

such that e

∈

E

(

C

)

.

Proof. Obviously,

hh

0

,

ijk0

i

, h

1

,

ijk0

i

, h

2

,

ijk0

i

, h

3

,

ijk0

i

, h

0

,

ijk1

i

, h

1

,

ijk1

i

, h

2

,

ijk1

i

, h

3

,

ijk1

i

, h

0

,

ijk0

ii

is the unique hamil-tonian cycle of BF_0,1,2i,j,k

(

4

)

. Thus, this lemma is proved.

(6)

Fig. 4. (a) A weakly 2-scheduled hamiltonian path P1of BF

i,j

0,1(4)joinsh1,ij00itoh2,ij10i; (b)γ30◦γ20(P1)in BF

i,j,0,0

0,1,2,3(6) = γ30◦γ20(BF

i,j

0,1(4)); (c) a weakly

2-scheduled hamiltonian path P2of BF

i,j

0,1(4)joinsh1,ij00itoh2,ij00i; (d)γ30◦γ20(P2)in BF

i,j,0,0 0,1,2,3(6). Table 1

Hamiltonian paths of BF0i,,j1(4)betweenh1,ij00iandh2,ijpqifor any p,q∈Z2

hh1,ij00i,h0,ij00i,h3,ij01i,h2,ij11i,h1,ij11i,h0,ij11i,h3,ij11i,h2,ij01i,h1,ij01i,h0,ij01i,h3,ij00i,h2,ij10i,h1,ij10i,h0,ij10i,h3,ij10i,h2,ij00ii hh1,ij00i,h0,ij00i,h3,ij00i,h2,ij00i,h3,ij10i,h0,ij11i,h1,ij11i,h2,ij11i,h3,ij01i,h0,ij01i,h1,ij01i,h2,ij01i,h3,ij11i,h0,ij10i,h1,ij10i,h2,ij10ii hh1,ij00i,h0,ij00i,h3,ij00i,h2,ij00i,h3,ij10i,h2,ij10i,h1,ij10i,h0,ij10i,h3,ij11i,h0,ij11i,h1,ij11i,h2,ij11i,h3,ij01i,h0,ij01i,h1,ij01i,h2,ij01ii hh1,ij00i,h0,ij00i,h3,ij01i,h2,ij01i,h1,ij01i,h0,ij01i,h3,ij00i,h2,ij00i,h3,ij10i,h2,ij10i,h1,ij10i,h0,ij10i,h3,ij11i,h0,ij11i,h1,ij11i,h2,ij11ii

By stretching operation andCorollary 1, we have the following corollary.

Corollary 3. Suppose that n

≥

5. Let e be any edge of BF_0,1,2i,j,k

(

n

)

with i

,

j

,

k

∈

_Z₂. Then there exists a totally scheduled hamiltonian cycle of BF_0,1,2i,j,k

(

n

)

including e.

A path P of BF

(

n

)

is weakly

`

-scheduled if there is at least one non-terminal level-

`

vertex

v

of P, such that

v

(` −

1

)

mod n edge and a level-

`

edge on P.Fig. 4illustrates two weakly 2-scheduled hamiltonian paths P1and P2of BF_0,1i,j

(

4

)

and their images

γ

30

◦

γ

20

(

P1

)

and

γ

30

◦

γ

20

(

P2

)

on

γ

30

◦

γ

20

(

BF

i,j

0,1

(

4

)) =

BF0,1,2,3i,j,0,0

(

6

)

, respectively.

Lemma 7. Let n

≥

4 and i

,

j

∈

_Z₂. Suppose that s is any level-1 vertex of BF_0,1i,j

(

n

)

and d is any level-2 vertex of BF_0,1i,j

(

n

)

. Then there exists a weakly 2-scheduled hamiltonian path of BF_0,1i,j

(

n

)

, joining s to d.

Proof. Without loss of generality, we assume that s

= h

1

,

ij0n−2

_i

_{and d}

_{= h}

₂

_,

_ijpqx

_i

_{with p}

_,

_q

_∈

Z2and x

∈

Zn2−4. We prove this lemma by induction on n. The induction bases are listed inTables 1and2.

As the inductive hypothesis, we assume that the statement holds for BF_0,1i,j

(

n

−

2

)

with n

≥

6. Now, we partition BF_0,1i,j

(

n

)

into

{

BF_0,1,2,3i,j,h,k

(

n

) |

h

,

k

∈

_Z₂

}

. By the inductive hypothesis, there exists a weakly 2-scheduled hamiltonian path P00of BF_0,1i,j

(

n

−

2

)

joining

h

1

,

ij0n−4

i

to

h

2

,

ijx

i

. Hence, there is at least one non-terminal level-2 vertex of P00, say

v = h

2

,

ijy

i

with y

6=

x, such that

v

is incident to a level-1 edge and a level-2 edge on P00. ByLemma 2, we have BF_0,1,2,3i,j,0,0

(

n

) = γ

₃0

◦

γ

₂0

◦

γ

₁j

(

BF₀i

(

n

−

3

)) = γ

₃0

◦

γ

₂0

(

BF_0,1i,j

(

n

−

2

))

. Thus,

γ

₃0

◦

γ

₂0

(

P00

)

is a path on BF_0,1,2,3i,j,0,0

(

n

)

joining s to

(7)

Table 2

Hamiltonian paths of BF0i,,j1(5)betweenh1,ij000iandh2,ijpqxifor any p,q,x∈Z2

hh1,ij000i, h0,ij000i, h4,ij001i, h3,ij011i, h2,ij111i, h1,ij111i, h0,ij111i, h4,ij111i, h3,ij111i, h2,ij011i, h1,ij011i, h0,ij011i, h4,ij011i, h3,ij001i, h2,ij101i, h1,ij101i, h0,ij101i, h4,ij101i, h3,ij101i, h2,ij001i, h1,ij001i, h0,ij001i, h4,ij000i, h3,ij010i, h2,ij110i, h1,ij110i, h0,ij110i, h4,ij110i, h3,ij110i, h2,ij010i, h1,ij010i, h0,ij010i, h4,ij010i, h3,ij000i, h2,ij100i, h1,ij100i, h0,ij100i, h4,ij100i, h3,ij100i, h2,ij000ii hh1,ij000i, h0,ij000i, h4,ij001i, h3,ij011i, h2,ij111i, h1,ij111i, h0,ij111i, h4,ij111i, h3,ij111i, h2,ij011i, h1,ij011i, h0,ij011i, h4,ij011i, h3,ij001i, h2,ij101i, h1,ij101i, h0,ij101i, h4,ij101i, h3,ij101i, h2,ij001i, h1,ij001i, h0,ij001i, h4,ij000i, h3,ij010i, h2,ij110i, h1,ij110i, h0,ij110i, h4,ij110i, h3,ij110i, h2,ij010i, h1,ij010i, h0,ij010i, h4,ij010i, h3,ij000i, h2,ij000i, h3,ij100i, h4,ij100i, h0,ij100i, h1,ij100i, h2,ij100ii hh1,ij000i, h0,ij000i, h4,ij001i, h3,ij011i, h2,ij111i, h1,ij111i, h0,ij111i, h4,ij111i, h3,ij111i, h2,ij011i, h1,ij011i, h0,ij011i, h4,ij011i, h3,ij001i, h2,ij101i, h1,ij101i, h0,ij101i, h4,ij101i, h3,ij101i, h2,ij001i, h1,ij001i, h0,ij001i, h4,ij000i, h3,ij000i, h2,ij000i, h3,ij100i, h2,ij100i, h1,ij100i, h0,ij100i, h4,ij100i, h3,ij110i, h4,ij110i, h0,ij110i, h1,ij110i, h2,ij110i, h3,ij010i, h4,ij010i, h0,ij010i, h1,ij010i, h2,ij010ii hh1,ij000i, h0,ij000i, h4,ij001i, h3,ij011i, h2,ij111i, h1,ij111i, h0,ij111i, h4,ij111i, h3,ij111i, h2,ij011i, h1,ij011i, h0,ij011i, h4,ij011i, h3,ij001i, h2,ij101i, h1,ij101i, h0,ij101i, h4,ij101i, h3,ij101i, h2,ij001i, h1,ij001i, h0,ij001i, h4,ij000i, h3,ij010i, h2,ij010i, h1,ij010i, h0,ij010i, h4,ij010i, h3,ij000i, h2,ij000i, h3,ij100i, h2,ij100i, h1,ij100i, h0,ij100i, h4,ij100i, h3,ij110i, h4,ij110i, h0,ij110i, h1,ij110i, h2,ij110ii hh1,ij000i, h0,ij000i, h4,ij000i, h3,ij000i, h2,ij000i, h3,ij100i, h2,ij100i, h1,ij100i, h0,ij100i, h4,ij100i, h3,ij110i, h2,ij010i, h1,ij010i, h0,ij010i, h4,ij010i, h3,ij010i, h2,ij110i, h1,ij110i, h0,ij110i, h4,ij110i, h0,ij111i, h1,ij111i, h2,ij111i, h3,ij011i, h4,ij011i, h0,ij011i, h1,ij011i, h2,ij011i, h3,ij111i, h4,ij111i, h3,ij101i, h4,ij101i, h0,ij101i, h1,ij101i, h2,ij101i, h3,ij001i, h4,ij001i, h0,ij001i, h1,ij001i, h2,ij001ii hh1,ij000i, h0,ij000i, h4,ij000i, h3,ij000i, h2,ij000i, h3,ij100i, h2,ij100i, h1,ij100i, h0,ij100i, h4,ij100i, h3,ij110i, h2,ij010i, h1,ij010i, h0,ij010i, h4,ij010i, h3,ij010i, h2,ij110i, h1,ij110i, h0,ij110i, h4,ij110i, h0,ij111i, h1,ij111i, h2,ij111i, h3,ij011i, h4,ij001i, h0,ij001i, h1,ij001i, h2,ij001i, h3,ij001i, h4,ij011i, h0,ij011i, h1,ij011i, h2,ij011i, h3,ij111i, h4,ij111i, h3,ij101i, h4,ij101i, h0,ij101i, h1,ij101i, h2,ij101ii hh1,ij000i, h0,ij000i, h4,ij000i, h3,ij000i, h2,ij000i, h3,ij100i, h2,ij100i, h1,ij100i, h0,ij100i, h4,ij100i, h3,ij110i, h2,ij010i, h1,ij010i, h0,ij010i, h4,ij010i, h3,ij010i, h2,ij110i, h1,ij110i, h0,ij110i, h4,ij110i, h0,ij111i, h1,ij111i, h2,ij111i, h3,ij111i, h4,ij111i, h3,ij101i, h4,ij101i, h0,ij101i, h1,ij101i, h2,ij101i, h3,ij001i, h2,ij001i, h1,ij001i, h0,ij001i, h4,ij001i, h3,ij011i, h4,ij011i, h0,ij011i, h1,ij011i, h2,ij011ii hh1,ij000i, h0,ij000i, h4,ij000i, h3,ij010i, h2,ij110i, h1,ij110i, h0,ij110i, h4,ij110i, h3,ij110i, h2,ij010i, h1,ij010i, h0,ij010i, h4,ij010i, h3,ij000i, h2,ij000i, h3,ij100i, h2,ij100i, h1,ij100i, h0,ij100i, h4,ij100i, h0,ij101i, h1,ij101i, h2,ij101i, h3,ij001i, h4,ij011i, h0,ij011i, h1,ij011i, h2,ij011i, h3,ij011i, h4,ij001i, h0,ij001i, h1,ij001i, h2,ij001i, h3,ij101i, h4,ij101i, h3,ij111i, h4,ij111i, h0,ij111i, h1,ij111i, h2,ij111ii

h

2

,

ij00x

i

or joining s to

h

4

,

ij00x

i

. ByCorollary 2, there is a totally scheduled hamiltonian cycle Chk_{of BF}i,j,h,k

0,1,2,3

(

n

)

including all straight edges of level 2 and level 3 for any h

,

k

∈

_Z₂.

Let F_k

= {h

2

,

ij

wi ∈

V

(

P00

_{) | h}

₂

_,

_ij

_wi

_{is not incident to any level-}

₍

_k

₊

₁

₎

_{edge on P}00

_}

_{with k}

_{∈ {}

₀

_,

₁

_}

_{. Obviously, P}00_{is a} connected spanning subgraph of BF₀i,_,j₁

(

n

−

2

)

. ByLemma 3, we have V

(γ

₃0

◦

γ

0

2

(

P00

)) =

V

(

BF

i,j,0,0

0,1,2,3

(

n

)) − (

F0

∪

F1

)

, where F0

= {h

2

,

ij00

wi | h

2

,

ij

wi ∈

F0

} ∪ {h

3

,

ij00

wi | h

2

,

ij

wi ∈

F0

}

and F1

= {h

3

,

ij00

wi | h

2

,

ij

wi ∈

F1

} ∪ {h

4

,

ij00

wi | h

2

,

ij

wi ∈

F1

}

. In addition, we have F0

∩

F1

= ∅

. If

γ

30

◦

γ

0 2

(

P

00

₎

_{joins s to}

_h

₂

_,

_ij00x

_i

_{, let P}00

₌

_γ

0 3

◦

γ

0 2

(

P 00

₎

_and

e

F0

=

F0. Otherwise, let P00

_{= h}

_s

, γ

0

3

◦

γ

20

(

P00

), h

4

,

ij00x

i

, h

3

,

ij00x

i

, h

2

,

ij00x

ii

andF

e

0

=

F0

− {h

2

,

ij00x

i

, h

3

,

ij00x

i}

. For any h

,

k

∈

Z2, let X₀hk

= {

(h

2

,

ijhk

wi, h

3

,

ijhk

wi) | h

2

,

ij00

wi

and

h

3

,

ij00

wi

are in

e

F0

}

,

Y₀hk

= {

(h

2

,

ijhk

wi, h

3

,

ijhk

¯

wi) | h

2

,

ij00

wi

and

h

3

,

ij00

wi

are in

e

F0

}

,

X₁hk

= {

(h

3

,

ijhk

wi, h

4

,

ijhk

wi) | h

3

,

ij00

wi

and

h

4

,

ij00

wi

are in F1

}

,

and Y₁hk

= {

(h

3

,

ijhk

wi, h

4

,

ijhk

¯

wi) | h

3

,

ij00

wi

and

h

4

,

ij00

wi

are in F1

}

.

Then we consider the following four cases.

Case 1: If pq

=

00, then d

= h

2

,

ij00x

i

. It is noticed that

v 6∈

F0

∪

F1. Let A

= {

(h

2

,

ij10y

i

, h

3

,

ij00y

i

), (h

2

,

ij00y

i

, h

3

,

ij10y

i

), (h

2

,

ij11y

i

, h

3

,

ij01y

i

),

(h

2

,

ij01y

i

, h

3

,

ij11y

i

), (h

3

,

ij11y

i

, h

4

,

ij10y

i

), (h

3

,

ij10y

i

, h

4

,

ij11y

i

)}

and B

= {

(h

2

,

ij00y

i

, h

3

,

ij00y

i

), (h

2

,

ij10y

i

, h

3

,

ij10y

i

), (h

2

,

ij01y

i

, h

3

,

ij01y

i

),

(h

2

,

ij11y

i

, h

3

,

ij11y

i

), (h

3

,

ij10y

i

, h

4

,

ij10y

i

), (h

3

,

ij11y

i

, h

4

,

ij11y

i

)}.

It follows from Lemma 3, that

(h

2

,

ij00y

i

, h

3

,

ij00y

i

) ∈

E

(

P00

)

_{. By} _{Corollary 2}_{, we have}

(h

₂

,

_ij10y

_i

, h

₃

,

_ij10y

_i

) ∈

E

(

C10

)

,

(h

2

,

ij01y

i

, h

3

,

ij01y

i

) ∈

E

(

C01

)

,

(h

2

,

ij11y

i

, h

3

,

ij11y

i

) ∈

E

(

C11

)

,

(h

3

,

ij10y

i

, h

4

,

ij10y

i

) ∈

E

(

C10

)

, and

(h

3

,

ij11y

i

, h

4

,

ij11y

i

) ∈

E

(

C11

₎

_{. Then the subgraph P of BF}i,j

0,1

(

n

)

, generated by

(

E

(

P00

) ∪

E

(

C10

) ∪

E

(

C01

) ∪

E

(

C11

) ∪

A

) −

B, forms a weakly 2-scheduled path of BF_0,1i,j

(

n

)

between s and d. Clearly, we have V

(

P

) =

V

(

BF_0,1i,j

(

n

)) − (

F0

∪

e

F₁

)

. Since

(8)

Fig. 5. (a) P00₌γ0

3◦γ20(P00), C10, C01, and C11; (b) the path P generated by(E(P00) ∪E(C10) ∪E(C01) ∪E(C11) ∪A) −B; (c) the path P 0

generated by

(E(P) ∪ (X00

0 ∪Y000∪Y010) ∪ (X100∪Y100∪Y101)) − (X010∪X101)to cover all vertices ofFe0∪F1. Chk_{includes all straight edges of level 2 and level 3 in BF}i,j,h,k

0,1,2,3

(

n

)

, we have X010

⊂

E

(

C10

)

and X101

⊂

E

(

C01

)

. Moreover, we have

(

X10

0

∪

X101

) ∩

B

= ∅

. Therefore, it follows that

(

X010

∪

X101

) ⊂

E

(

P

)

. Let P

0_{be the subgraph generated by}

(

E

(

P

) ∪ (

X₀00

∪

Y₀00

∪

Y₀10

) ∪ (

X₁00

∪

Y₁00

∪

Y₁01

)) − (

X₀10

∪

X₁01

)

. Then P0is a weakly 2-scheduled hamiltonian path of BF_0,1i,j

(

n

)

joining s to d. SeeFig. 5for illustration, in which

γ

₃0

◦

γ

₂0

(

P00

)

is supposed to join s and

h

2

,

ij00x

i

.

Case 2: If pq

=

10, then d

= h

2

,

ij10x

i

. Let

A

= {

(h

2

,

ij00x

i

, h

3

,

ij10x

i

), (h

2

,

ij11y

i

, h

3

,

ij01y

i

), (h

2

,

ij01y

i

, h

3

,

ij11y

i

),

(h

3

,

ij11y

i

, h

4

,

ij10y

i

), (h

3

,

ij10y

i

, h

4

,

ij11y

i

)}

and

B

= {

(h

2

,

ij10x

i

, h

3

,

ij10x

i

), (h

2

,

ij01y

i

, h

3

,

ij01y

i

), (h

2

,

ij11y

i

, h

3

,

ij11y

i

),

(h

3

,

ij10y

i

, h

4

,

ij10y

i

), (h

3

,

ij11y

i

, h

4

,

ij11y

i

)}.

Obviously, the subgraph P, generated by

(

E

(

P00

) ∪

_E

(

_C10

_{) ∪}

_E

₍

_C01

_{) ∪}

_E

₍

_C11

_{) ∪}

_A

_{) −}

_{B, forms a weakly 2-scheduled path of} BF_0,1i,j

(

n

)

between s and d. Moreover, the subgraph P0_{, generated by}

₍

_E

₍

_P

_)∪(

_X00

0

∪

Y000

∪

Y010

)∪(

X100

∪

Y100

∪

Y101

))−(

X010

∪

X101

)

, is a weakly 2-scheduled hamiltonian path of BF_0,1i,j

(

n

)

joining s to d.

(9)

Case 3: If pq

=

01, then d

= h

2

,

ij01x

i

. Let

A

= {

(h

2

,

ij00x

i

, h

3

,

ij10x

i

), (h

2

,

ij11x

i

, h

3

,

ij01x

i

), (h

3

,

ij11x

i

, h

4

,

ij10x

i

)}

and B

= {

(h

2

,

ij01x

i

, h

3

,

ij01x

i

), (h

2

,

ij11x

i

, h

3

,

ij11x

i

), (h

3

,

ij10x

i

, h

4

,

ij10x

i

)}.

Obviously, the subgraph P, generated by

(

E

(

P00

) ∪

_E

(

_C10

_{) ∪}

_E

₍

_C01

_{) ∪}

_E

₍

_C11

_{) ∪}

_A

_{) −}

_{B, forms a weakly 2-scheduled path of} BF_0,1i,j

(

n

)

, between s and d. Moreover, the subgraph P0_{, generated by}

₍

_E

₍

_P

_)∪(

_X00

0

∪

Y000

∪

Y010

)∪(

X100

∪

Y100

∪

Y101

))−(

X010

∪

X101

)

, is a weakly 2-scheduled hamiltonian path of BF_0,1i,j

(

n

)

joining s to d.

Case 4: If pq

=

11, then d

= h

2

,

ij11x

i

. Let

A

= {

(h

2

,

ij00x

i

, h

3

,

ij10x

i

), (h

3

,

ij11x

i

, h

4

,

ij10x

i

), (h

3

,

ij01y

i

, h

4

,

ij00y

i

), (h

3

,

ij00y

i

, h

4

,

ij01y

i

)}

and B

= {

(h

3

,

ij10x

i

, h

4

,

ij10x

i

), (h

3

,

ij00y

i

, h

4

,

ij00y

i

), (h

3

,

ij01y

i

, h

4

,

ij01y

i

), (h

2

,

ij11x

i

, h

3

,

ij11x

i

)}.

The subgraph P, generated by

(

E

(

P00

) ∪

_E

(

_C10

_{) ∪}

_E

₍

_C01

_{) ∪}

_E

₍

_C11

_{) ∪}

_A

_{) −}

_{B, forms a weakly 2-scheduled path of BF}i,j

0,1

(

n

)

between s and d. Moreover, the subgraph P0_{, generated by}

₍

_E

₍

_P

_{) ∪ (}

_X00

0

∪

Y000

∪

Y010

) ∪ (

X100

∪

Y100

∪

Y101

)) − (

X010

∪

X101

)

, is a weakly 2-scheduled hamiltonian path of BF_0,1i,j

(

n

)

joining s to d.

By symmetry, the next corollary can be proved in the way similar toLemma 7.

≥

4 and i

,

j

∈

_Z₂. Let s be any level-1 vertex of BF_0,1i,j

(

n

)

and d be any level-0 vertex of BF_0,1i,j

(

n

)

. Then there exists a weakly 0-scheduled hamiltonian path of BF_0,1i,j

(

n

)

joining s to d.

Lemma 8. Assume that n

≥

4. Let s

= h

1

,

0n

_i

_{, d}

1

= h

2

,

0210n−3

i

, and d2

= h

0

,

0n

i

. Then there exist two hamiltonian paths H1 and H2 of BF_0,10,0

(

n

)

, such that the following conditions are all satisfied:

(

i

)

H1 joins s to d1,

(

ii

)

H2 joins s to d2, and

(

iii

)

H1

(

1

) =

H2

(

1

) =

s and H1

(

t

) 6=

H2

(

t

)

for each 2

≤

t

≤

V

BF_0,10,0

(

n

)

=

n2 n−2_.

Proof. Let u1

=

g

(

s

) = h

2

,

0n

i

, u2

=

f

(

u1

) =

g

(

d1

) = h

3

,

0210n−3

i

, u3

=

g−1

(

d1

) = h

1

,

0210n−3

i

, u4

=

f

(

u2

)

, and u5

=

g

(

u1

) =

f

(

d1

) = h

3

,

0n

i

. Note that u4

= h

0

,

0011

i

if n

=

4 and u4

= h

4

,

02120n−4

i

if n

≥

5. We partition BF_0,10,0

(

n

)

into

{

BF_0,1,20,0,0

(

n

),

BF_0,1,20,0,1

(

n

)}

. ByCorollary 1, there is a hamiltonian cycle C0of BF_0,1,20,0,0

(

n

)

including all straight edges of level 2. Thus, we have

(

u1

,

u5

) ∈

E

(

C0

)

. ByLemma 6andCorollary 3, there is a hamiltonian cycle C1of BF_0,1,20,0,1

(

n

)

, such that

(

u2

,

u4

) ∈

E

(

C1

)

. It is noticed that s and d1are vertices of degree two in BF_0,1,20,0,0

(

n

)

and BF_0,1,20,0,1

(

n

)

, respectively. Therefore, we can write C0

= h

s

,

u1

,

u5

,

P0

,

d2

,

s

i

and C1

= h

d1

,

u2

,

u4

,

P1

,

u3

,

d1

i

. As an illustrative example,Fig. 6(a) depicts C0and C1on BF_0,10,0

(

4

)

.Fig. 6(b) illustrates the abstraction of C0and C1for general n. Since

{

(

u1

,

u2

), (

d1

,

u5

)} ⊂

E

(

BF_0,10,0

(

n

))

, we set

H1

= h

s

,

d2

,

P0−1

,

u5

,

u1

,

u2

,

u4

,

P1

,

u3

,

d1

i

and H2

= h

s

,

u1

,

u2

,

u4

,

P1

,

u3

,

d1

,

u5

,

P0

,

d2

i

.

Then it can be verified, as shown onFig. 6(c), that H1and H2satisfy the conditions.

Lemma 9. Given any k

∈ {

0

,

1

}

and n

≥

4, let

(

b1

, w

1

)

be a level-1 straight edge of BF_0,1,1,1,nk−1

(

n

)

and

(

b2

, w

2

)

be a level-0 straight edge of BF₀1,1,_,₁_,_nk₋₁

(

n

)

such that

w

1and

w

2are two distinct level-1 vertices. Then there exist two hamiltonian paths H1and H2of BF_0,11,1

(

n

)

, such that the following conditions are all satisfied:

(i) H1

(

1

) =

b1and H1

(

n2n−2

) = w

1, (ii) H2

(

1

) =

b2and H2

(

n2n−2

) = w

2, and (iii) H1

(

t

) 6=

H2

(

t

)

for each 1

≤

t

≤

n2n−2.

Proof. Without loss of generality, we assume that k

=

0. Let u₁

=

gn−3

₍

_b

1

)

, u2

=

f

(

u1

)

, u3

=

g

(

u2

)

, u4

=

g

(

u3

)

, u5

=

gn−3

(

u4

) =

g−1

(

u2

)

, u6

=

f

(

u5

) =

g−1

(w

1

)

,

v

1

=

f−1

(

b2

)

,

v

2

=

g−n+3

(v

1

)

,

v

3

=

g−1

(v

2

)

,

v

4

=

g−1

(v

3

) =

g

(v

1

)

,

v

5

=

f−1

(v

4

) =

g−1

(

b2

)

, and

v

6

=

g−n+3

(v

5

) =

g

(w

2

)

. ByCorollary 1, BF_0,1,21,1,0

(

n

)

has a totally scheduled hamiltonian cycle. ByLemma 1, BF_0,1,1,1,0_n₋₁

(

n

)

is isomorphic with BF_0,1,21,1,0

(

n

)

. Hence, there also exists a totally scheduled hamiltonian cycle C0of BF_0,1,1,1,0_n₋₁

(

n

)

. It is noticed that

w

1is adjacent to u6. Moreover,

w

1, u6, b2, and

w

2are all vertices of degree two in BF_0,1,1,1,0n−1

(

n

)

. Accordingly, C0can be written as C0

= h

w

1

,

b1

,

P0

,

u1

,

u6

, w

1

i

, where P0

= h

b1

,

P01

, v

5

,

b2

, w

2

, v

6

,

P02

,

u1

i

.

By Lemma 6, BF_0,1,21,1,1

(

4

)

has a totally scheduled hamiltonian cycle C such that e

∈

E

(

C

)

if e

∈

E

(

BF_0,1,21,1,1

(

4

)) −

{

(h

3

,

1110

i

, h

0

,

1110

i

), (h

3

,

1111

i

, h

0

,

1111

i

)}

. ByLemma 1, BF_0,1,31,1,1

(

4

)

(

4

)

. Hence, BF_0,1,31,1,1

(

4

)

has a totally scheduled hamiltonian cycle C such that e

∈

E

(

C

)

if e

∈

E

(

BF₀1,1,1_,₁_,₃

(

4

)) − {(h

2

,

1101

i

, h

3

,

1101

i

), (h

2

,

1111

i

,

h

3

,

1111

i

)}

. Obviously,

(

u5

,

u2

)

is a level-

(

n

−

3

)

for any n

≥

4. Therefore, we have

(

u5

,

u2

) ∈

E

(

BF_0,1,31,1,1

(

4

)) −

(10)

Fig. 6. Illustration forLemma 8.

Fig. 7. Illustration forLemma 9. In (a),(b1, w1) = (h2,1100i, h1,1100i)and(b2, w2) = (h0,1110i, h1,1110i)are assumed. In (c), we let R1 =

hv1,P −1 11,u4,u3,u2, ,u5,P −1 12, v2iand R0= hv5,P −1 01,b1, w1,u6,u1,P −1 02, v6i.

{

(h

2

,

1101

i

, h

3

,

1101

i

), (h

2

,

1111

i

, h

3

,

1111

i

)}

. It follows that BF₀1,1,1_,₁_,₃

(

4

)

has a totally scheduled hamiltonian cycle C1such that

(

u5

,

u2

) ∈

E

(

C1

)

. ByCorollary 3, BF_0,1,21,1,1

(

n

)

, n

≥

5, has a totally scheduled hamiltonian cycle including any required edge. Since BF_0,1,1,1,1_n₋₁

(

n

)

(

n

)

, it has a totally scheduled hamiltonian cycle C1such that

(

u5

,

u2

) ∈

E

(

C1

)

if n

≥

5. In short, byLemma 6andCorollary 3, there is a totally scheduled hamiltonian cycle C₁of BF₀1,1,1_,₁_,_n₋₁

(

n

)

, such that

(

u5

,

u2

) ∈

E

(

C1

)

. Since u2, u3,

v

3, and

v

4are vertices of degree two in BF_0,1,1,1,1n−1

(

n

)

, we write C1

= h

u3

,

u4

,

P1

,

u5

,

u2

,

u3

i

, where P1

= h

u4

,

P11

, v

1

, v

4

, v

3

, v

2

,

P12

,

u5

i

.Fig. 7(a) depicts C0and C1on BF_0,11,1

(

4

)

.Fig. 7(b) illustrates the abstraction of C0 and C1for general n. Then we set

H1

= h

b1

,

P01

, v

5

,

b2

, w

2

, v

6

,

P02

,

u1

,

u2

,

u3

,

u4

,

P11

, v

1

, v

4

, v

3

, v

2

,

P12

,

u5

,

u6

, w

1

i

and H2

= h

b2

, v

1

,

P11−1

,

u4

,

u3

,

u2

,

u5

,

P12−1

, v

2

, v

3

, v

4

, v

5

,

P01−1

,

b1

, w

1

,

u6

,

u1

,

P02−1

, v

6

, w

2

i

.

Since

w

1

6=

w

2, u2

6=

v

2, u3

6=

v

3, u4

6=

v

4, and u6

6=

v

6, it can be checked that H1and H2satisfy the conditions. SeeFig. 7(c) for illustration.