Edge-pancyclicity and path-embeddability of bijective connection graphs

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Information Sciences 178 (2008) 340–351

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Edge-pancyclicity and path-embeddabilityof bijective connection graphs

Jianxi Fan a,*, Xiaohua Jia b

a College of Information Engineering, Qingdao University, Qingdao 266071, Chinab Department of Computer Science, City University of Hong Kong, Hong Kong

Received 1 February 2007; received in revised form 14 August 2007; accepted 14 August 2007

Abstract

An n-dimensional Bijective Connection graph (in brief BC graph) is a regular graph with 2n nodes and n2n�1 edges. Then-dimensional hypercube, crossed cube, Mobius cube, etc. are some examples of the n-dimensional BC graphs. In thispaper, we propose a general method to study the edge-pancyclicity and path-embeddability of the BC graphs. First, weprove that a path of length l with dist(Xn,x,y) + 2 6 l 6 2n � 1 can be embedded between x and y with dilation 1 in Xn

for x,y 2 V(Xn) with x 5 y in Xn, where Xn (n P 4) is a n-dimensional BC graph satisfying the three specific conditionsand V(Xn) is the node set of Xn. Furthermore, by this result, we can claim that Xn is edge-pancyclic. Lastly, we show thatthese results can be applied to not only crossed cubes and Mobius cubes, but also other BC graphs except crossed cubesand Mobius cubes. So far, the research on edge-pancyclicity and path-embeddability has been limited in some specificinterconnection architectures such as crossed cubes, Mobius cubes.� 2007 Elsevier Inc. All rights reserved.

Keywords: Bijective connection graph; Crossed cube; Mobius cube; Graph embedding; Dilation; Edge-pancyclicity; Path; Parallel com-puting system

1. Introduction

Graph embedding is a technique that maps a logical graph into a host graph (usually an interconnectionarchitecture). Many applications, such as architecture simulation, processor allocation, VLSI chip designetc. can be modelled as graph embedding [2,6,15–17,23,25,30].

Given a graph G, let V(G) and E(G) denote the node set and the edge set in G, respectively. For two nodes x

and y, we use dist(G,x,y) to denote the distance between x and y of G. Given two graphs G1 and G2, an embedding

from G1 to G2 is an injective mapping w: V(G1)! V(G2). The performance of an embedding can be measured bydilation. The dilation of embedding w is defined as dil(G1,G2,w) = max{dist(G2,w(u),w(v))j{u,v} 2 E(G1)}. It isdesirable to get an embedding, from G1 to G2, with the smallest dilation. Obviously, dil(G1,G2,w) P 1. And when

0020-0255/$ - see front matter � 2007 Elsevier Inc. All rights reserved.

doi:10.1016/j.ins.2007.08.012

* Corresponding author.E-mail addresses: [email protected] (J. Fan), [email protected] (X. Jia).

mailto:[email protected]

mailto:[email protected]

J. Fan, X. Jia / Information Sciences 178 (2008) 340–351 341

dil(G1,G2,w) = 1, w is surely optimal in the sense that w(v) has the smallest dilation and G1 is a subgraph of G2 inthis case.

Many graph embeddings take cycles, paths, trees, meshes, etc. as guest graphs [1,12,13,15,16,18,23,25,30],because these interconnection architectures are widely used in parallel computing systems. In particular, pathsare the common structures used to model linear arrays in parallel processing [7,8,20–22,24]. It is well knownthat cycles are popular interconnection architectures.

It is well known that hypercubes are popular interconnection architectures for parallel computing systems,owing to their smaller diameters, regularity, symmetry, etc. As variants of hypercubes, crossed cubes andMobius cubes were proposed in [9,5], respectively. All of them have the same number of nodes and edges,but have only about half of diameter as the hypercube with the same dimension. Many other features of them,such as architecture embedding [3,12,13,17,18,26–29], diagnosability [4,11,19], etc., have been studied, respec-tively. Recently, Lai et al. defined a family of interconnection architectures—matching composition networksby using the definition of graph match in [19]. Matching composition networks are like hypercubes, i.e., eachmatching composition network is a regular graph with the same node number and edge number as the hyper-cube with the same dimension (node degree). It was proved that the n-dimensional matching composition net-works include the n-dimensional hypercube, crossed cube, and Mobius cube and are n-diagnosable under thecomparison diagnostic model. In [10], another family of interconnection architectures, called bijective connec-tion graphs (in brief, BC graphs), including hypercubes, crossed cubes, and Mobius cubes, were defined byusing the definition of bijection connection. Some properties shared by all the BC graphs were studied in[10]. Furthermore, in [11], it was proved that all the n-dimensional BC graphs are all t(n,k)-diagnosable(0 6 k 6 n) under the t/k-diagnosable strategy based on the PMC diagnostic model, where tðn; kÞ ¼ðk þ 1Þn� 1

2ðk þ 1Þðk þ 2Þ þ 1. This means that the n-dimensional hypercube, crossed cube, and Mobius cube

are all ðk þ 1Þn� 12ðk þ 1Þðk þ 2Þ þ 1-diagnosable under the t/k-diagnosable strategy based on the PMC diag-

nostic model. In fact, the definitions of matching composition networks and BC graphs are equivalent to eachother. That is, each n-dimensional matching composition network is an n-dimensional BC graph and viceversa. In fact, [10,11,19] provided the important methods to study the shared properties of variants of hyper-cubes such as crossed cubes and Mobius cubes. That is, when studying some properties of variants of hyper-cubes, we do not have to only focus on a kind of specific networks such as crossed cubes, but a family ofcertain networks such as matching composition networks (or BC graphs). Once we know the properties ofthe family of networks, we can know the properties of every network in this family. Based on this idea, wepropose a general method to study the edge-pancyclicity and path-embeddability of BC graphs (or matchingcomposition networks) in this paper.

In what follows, we will adopt the item ‘‘BC graph’’ instead of ‘‘matching composition network’’. Themajor contributions of this paper are as follows:

First, we prove that a path of length l with dist(Xn,x,y) + 2 6 l 6 2n � 1 can be embedded between x and y

with dilation 1 in Xn for x,y 2 V(Xn) with x 5 y in Xn, where Xn (n P 4) is a n-dimensional BC graph satis-fying the three specific conditions (See Theorem 1). Furthermore, by this result, we can claim that Xn is edge-pancyclic. Lastly, we show that these results can be applied to not only crossed cubes and Mobius cubes, butalso other BC graphs except crossed cubes and Mobius cubes. So far, the research on edge-pancyclicity andpath-embeddability has been limited in some specific interconnection architectures such as crossed cubes,Mobius cubes, etc. [13,26,27].

The rest of this paper is organized as follows: Section 2 provides the preliminaries in this paper. Section 3discusses edge-pancyclicity and path-embeddability of the BC graphs satisfying the three specific conditions.In Section 4, we give some applications of the results obtained in Section 3.

2. Preliminaries

Given a graph G, a path P from node u to node v in G is defined as a node sequence P:u = u(0),u(1), . . . ,u(k) = v in V(G), where any two nodes (except u and v) are different. u and v are called theend nodes of P. If u = v, then P is a cycle. The length of path P is denoted by len(P), which is the numberof edges in P. The length of a shortest path between u and v is called the distance between u and v, denoted

342 J. Fan, X. Jia / Information Sciences 178 (2008) 340–351

by dist(G,u,v). The diameter of G is defined as diam(G) = max{dist(G,x,y)jx,y 2 V(G), x 5 y}. Path P canalso be denoted by: u = u(0),u(1), . . . ,u(i�1),P1,u(j+1), u(j+2), . . . ,u(k) = v, where the path P1 is called a sub-path

of P between u(i) and u(j), i.e., u(i),u(i+1), . . . ,u(j) (i 6 j). The path P1, starting from u(i) and ending with u(j),can be denoted by path(P,u(i),u(j)).

G is called a pancyclic graph, if G contains a cycle of length l for every integer l with 3 6 l 6 jVj, i.e., everycycle of length l with 3 6 l 6 jVjcan be embedded into G with dilation 1. However, there is no cycle of length 3in BC graphs. For convenience of discussion in this paper, we call G a pancyclic graph, if G contains everycycle of length l with 4 6 l 6 jVj. Similarly, we define edge-pancyclic graphs as follows:

G is called an edge-pancyclic graph if, for every edge {u,v} and every integer l with 4 6 l 6 jVj, G contains acycle C of length l such that {u,v} is in C.

If {x,y} is an edge in a cycle C, then the path between x and y in C is denoted by C � {x,y}, which is a pathbetween x and y after deleting the edge {x,y} in C.

For u,v 2 V(G), we call v to be a neighbor of u if {u,v} 2 E(G). Given U � V(G), we use G[U] to denote thesubgraph induced by U of G.

Given two graphs G 0 and G00, if there exists a bijection u from V(G 0) to V(G00) such that {u 0,v 0} 2 E(G 0) ifand only if {u(u 0),u(v 0)} 2 E(G00) for u 0,v 0 2 V(G 0), then we say that G 0 is isomorphic to G00.

A binary string x of length n is denoted by xn�1xn�2 . . . x1x0. The complement of xi is denoted byxi ¼ 1� xi.

Crossed cubes and Mobius cubes were proposed in [9,5], respectively. The n-dimensional crossed cube,0-type n-dimensional Mobius cube, and 1-type n-dimensional Mobius cube, denoted by CQn, 0 �MQn,and 1 �MQn, respectively, are all n-regular graphs of 2n nodes. Every node of CQn, 0 �MQn, and 1 �MQn

is identified by a unique binary string of length n. The followings are the formal definitions of them.

Definition 1 [9]. Two binary strings x = x1x0 and y = y1y0 of length two are said to be pair related (denoted byx � y) if and only if (x,y) 2 {(00, 00), (10, 10), (01, 11), (11,01)}.

Definition 2 [9]. CQ1 is the complete graph on two nodes whose binary strings are 0 and 1. CQn consists oftwo subcubes CQ0

n�1 and CQ1n�1. For the nodes u = un�1un�2 . . . u1u0 and v = vn�1vn�2 . . . v1v0, where un�1 = 0

and vn�1 = 1, {u,v} 2 E(CQn) if and only if

(1) un�2 = vn�2 if n is even, and(2) u2i+1u2i � v2i+1v2i, for 0 6 i < bn�1

2c.

Definition 3 [5]. 0 �MQ1 and 1 �MQ1 are both the complete graph on two nodes whose address are 0 and 1.For n P 2, both 0 �MQn and 1 �MQn contain one 0-type (n � 1)-dimensional subMobius cube M0

n�1 andone 1-type (n � 1)-dimensional subMobius cube MQ1

n�1. The leftmost bit of every node of MQ0n�1 is 0; the left-

most bit of every node of MQ1n�1 is 1. For u = u1u2 . . . un and v = v1v2 . . . vn, where u1 = 0 and v1 = 1,

(1) {u,v} 2 E(0 �MQn) if and only if ui = vi, i = 2,3, . . . ,n;(2) {u,v} 2 E(1 �MQn) if and only if xi ¼ y1; i = 2,3, . . . ,n.

Fig. 1 demonstrates the 3-dimensional crossed cube CQ3, 0-type 3-dimensional Mobius cube 0 �MQ3 and1-type 3-dimensional Mobius cube 1 �MQ3.

Before introducing the definition of BC graphs, we first give the definition of bijective connection in thefollowing:

Definition 4 [10]. Let G be a graph. If V(G) = V1 [ V2, V1 5 ;, V2 5 ;, and V1 \ V2 = ;. We say that thereexists a bijective connection between the subsets V1 and V2 in G, denoted by V 1$

GV 2, if G satisfies the two

following conditions:

(1) For every u 2 V1, there exists a unique v 2 V2 such that {u,v} 2 E(G); and(2) For every u 2 V2, there exists a unique v 2 V1 such that {u,v} 2 E(G).

Fig. 1. (a) CQ3; (b) 0 �MQ3; (c) 1 �MQ3.


An n-dimensional bijective connection graph (in brief, BC graph), denoted by Xn, is an n-regular graph with2n nodes. The set of all the n-dimensional BC graphs is called the family of the n-dimensional BC graphs,denoted by Ln. Xn and Ln may be recursively defined as below [10].

Definition 5 [10]. The 1-dimensional BC graph X1 is a complete graph on two nodes. The family of the1-dimensional BC graph is defined as L1 ¼ fX 1g. Let G be a graph. G is an n-dimensional BC graph, denotedby Xn, if there exist V0,V1 � V(G) such that the following two conditions hold:

(1) V(G) = V0 [ V1, V0 5 ;, V1 5 ;, and V0 \ V1 = ;; and(2) V 0$

GV 1, G½V 0� 2Ln�1, and G½V 1� 2Ln�1.

The family of the n-dimensional BC graphs is defined as Ln ¼ fGjG is an n-dimensional BC graphg.

Fig. 2 demonstrates two 3-dimensional BC graphs, in which (a) is isomorphic to the 3-dimensional hyper-cube Q3 and (b) is isomorphic to CQ3, 0 �MQ3, and 1 �MQ3, respectively. Fig. 3 demonstrates two 4-dimen-sional BC graphs, in which (a) is isomorphic to the 4-dimensional hypercube Q4 and (b) is isomorphic to CQ4.

According to the definitions of crossed cubes, Mobius cubes and BC graphs, if X n 2Ln, then there exist Y0,

Y 1 2Ln�1 such that V ðY 0Þ$X n V ðY 1Þ; And fCQn; 0�MQn; 1�MQng �Ln with V ðCQ0

n�1Þ $CQn V ðCQ1

n�1Þ;V ðMQ0

n�1Þ $0�MQn V ðMQ1

n�1Þ; V ðMQ0n�1Þ $

1�MQn V ðMQ1n�1Þ.

3. Edge-pancyclicity and path-embeddability of BC graphs

In this section, we study edge-pancyclicity and path-embeddability of BC graphs. We first introduce thefollowing lemmas associated with BC graphs.

Lemma 1 [10]. For every integer n P 1 and every X n 2Ln, diam(Xn) 6 n.

Lemma 2 ([3,9]). If n P 1, then diamðCQnÞ ¼ dnþ12e.

Lemma 3 [12]. If n P 3, for x,y 2 V(CQn), x 5 y, and every integer l, dnþ12e þ 1 6 l 6 2n � 1, there exists a

path of length l between x and y in CQn.

Fig. 2. Two 3-dimensional BC graphs.

Fig. 3. Two 4-dimensional BC graphs.


Lemma 4 [14]. For x, y 2 V ðCQ in�1Þ and i 2 {0,1}, dist(CQn,x,y) = distðCQ i

n�1; x; yÞ.

Lemma 5. For every integer n P 4 and every X n 2Ln with V 0$X n V 1, if dist(Xn[Vi],x,y) = dist(Xn,x,y) for

i 2 {0,1} and x,y 2 Vi with x 5 y, then there always exists a shortest path P: v(0) = u, v(1), . . . , v(l) = v between

u and v in Xn such that {v(0), v(1), . . . , v(k)} � V0 and {v(k+1), v(k+2), . . . , v(l)} � V1 for every u 2 V0 and v 2 V1

and a certain integer k with 0 6 k 6 l � 1, where l = dist(Xn,u, v).

Proof. For every u 2 V0 and v 2 V1, let P: v(0) = u,v(1), . . . ,v(l) = v be a shortest path between u and v in Xn.Since v(0) = u 2 V0 and v(l) = v 2 V1, there must exist an integer m with 0 6 m 6 l � 1 such thatv(m) 2 V(P) \ V0 and v(m+1) 2 V(P) \ V1. Let k be the smallest integer in {0,1, . . . , l � 1} such thatv(k) 2 V(P) \ V0 and v(k+1) 2 V(P) \ V1. That means {v(0),v(1), . . . ,v(k)} � V0. Since P is a shortest pathbetween u and v in Xn, path(P,v(k+1),v) is a shortest path between v(k+1) and v in Xn. As a result, dis-t(Xn[V1],v(k+1),v) = dist(Xn,v(k+1),v) = len(path(P,v(k+1),v)) = l � (k + 1). Select a shortest path P1 between

v(k+1) and v in Xn[V1]. Then len(P1) = l � (k + 1) and

vð0Þ ¼ u; vð1Þ; . . . ; vðkÞ; P 1

is a shortest path between u and v in Xn, where {v(0),v(1), . . . ,v(k)} � V0 and V(P1) � V1. Thus, we are done. h

Furthermore, we have the following result associated with a general connected graph G:

Lemma 6. For every connected graph G, if there exists a path of length l between u and v in G for u, v 2 G with

u 5 v and every integer l with dist(G, u, v) + 2 6 l 6 jV(G) � 1j, then G is edge-pancyclic.

Proof. For every {u,v} 2 E(G) and every integer l with 4 = dist(G,u,v) + 3 6 l 6 jV(G)j, by the condition inthe lemma, there is a path P of length l � 1 between u and v in G. Then,

C : P ; u

is a cycle of length l such that {u,v} is in C in G. Hence the lemma holds. h

We have the result listed in the following Theorem 1.

Theorem 1. For every integer n P 4 and any X n 2Ln with V 0$X n V 1, suppose that the following three conditions

hold:

Fig. 4. A path of length l between u and v in Xn (Corresponding to the proof of Theorem 1), where a straight line represents an edge and acurve line represents a path between two nodes.


(1) For every {x, y} 2 E(Xn) with x 2 V0 and y 2 V1, there is a cycle C of length l such that {x,y} is in C in Xn

for every integer l 2 {4,5};

(2) For every integer i 2 {0,1} and x,y 2 Vi with x 5 y, dist(Xn[Vi],x,y) = dist(Xn,x,y);(3) For x,y 2 Vi with x 5 y and two integers l 0 and i with dist(Xn[Vi],x,y) + 2 6 l 0 6 2n�1 � 1 and i 2 {0,1},

there exists a path of length l 0 between x and y in Xn[Vi].

Then, there exists a path of length l between u and v in Xn for u, v 2 Xn with u 5 v and every integer l with

dist(Xn,u, v) + 2 6 l 6 2n � 1;

Proof. For every integer n P 4 and every X n 2Ln with V 0$X n V 1, suppose that all the three conditions (1)–(3)

hold.For u,v 2 V(Xn) with u 5 v and every integer l with dist(Xn,u,v) + 2 6 l 6 2n � 1, without loss generality,

we identify the two cases that both u and v are in V0, and u and v are in V0 and V1, respectively.

Case 1. u, v 2 V0. For dist(Xn,u,v) + 2 6 l 6 2n � 1, we have the following sub-cases:Case 1.1. dist(Xn,u,v) + 2 6 l 6 2n�1 � 1. By the condition (2), dist(Xn[V0],u,v) + 2 = dist(Xn,u,v) +

2 6 l 6 2n�1 � 1. By the condition (3), there exists a path of length l between u and v inXn[V0], thus in Xn.

Case 1.2. 2n�16 l 6 2n � 1. By the condition (3), we can select a path P1 of length 2n�1 � 1 between u and v in

Xn[V0]. Therefore, for every integer l1 with 1 6 l1 6 2n�1 � 2, we can select a node s in P1 such thatlen(path(P1,u, s)) = l1. Let w and t be the neighbors, in Xn[V1], of v and s, respectively. Then, w 5 t

(See Fig. 4a). By the condition (3), for every integer l2 with dist(Xn[V1], t,w) + 2 6 l2 6 2n �1 � 1,there exists a path P2 of length l2 between t and w in Xn[V1]. By Lemma 1, dist(Xn[V1], t,w) + 5 6n + 4 6 2n�1

6 l 6 2n � 1. As a result, we can select appropriate values1 for l1 and l2 from the sets{1,2, . . . , 2n�1 � 2} and {dist(Xn[V1], t,w) + 2, dist(Xn[V1], t,w) + 3, . . . , 2n�1 � 1}, respectively,such that l1 + l2 + 2 = l, and thus

1 In this section, we use the following fact: Given four integers a, b, c, and d with a < b, c < d, for every integer e with a + c 6 e 6 b + d,there always exist two integers i and j with a 6 i 6 b and c 6 j 6 d such that i + j = e.

2 In this secta + c 6 f 6 b +


pathðP 1; u; sÞ; P 2; v

is a path of length l1 + l2 + 2 = l between u and v in Xn.
Case 2. u 2 V0 and v 2 V1. We deal with the sub-cases as follows:
Case 2.1. dist(Xn,u,v) = 1. Then {u,v} 2 E(Xn). For l 2 {3,4} = {dist(Xn,u,v) + 2, dist(Xn,u,v) + 3}, by thecondition (1), there is a cycle C of length l + 1 such that {u,v} is in C in Xn. Hence,

C � fu; vgis a path of length l between u and v in Xn.For 5 6 l 6 2n � 1, by the condition (1), we can letu,v, s,w,u be a cycle of length 4 in Xn. By Definition 5, we can deduce that s 2 V1 and w 2 V0

(See Fig. 4b). Clearly dist (Xn[V0], u,w) = dist(Xn[V1], s,v) = 1. For two integers l1 and l2 withl1 2 {1, 4, . . . , 2n�1 � 1} and l2 2 {3,4, . . . , 2n�1 � 1}, by the condition (3), there exists a path P1

of length l1 between u and w in Xn[V0] and a path P2 of length l2 between s and v in Xn[V1]. Obvi-ously, we can select appropriate values2 for l1 and l2 from the sets {1,3,4, . . . , 2n�1 � 1} and{3, 4, . . . , 2n�1 � 1}, respectively, such that l1 + l2 + 1 = l. Thus,

P 1; P 2

Case 2.2. dist(Xn,u,v) 5 1. For dist(Xn,u,v) + 2 6 l 6 2n � 1, we identify the following sub-cases:
Case 2.2.1. dist(Xn,u,v) + 2 6 l 6 2n�1. By Lemma 5, let P: u = u(0),u(1), . . . ,u(k) = v be a shortest pathbetween u and v in Xn such that {u(0),u(1), . . . ,u(j)} � V0 and {u(j+1),u(j+2), . . . ,u(k)} � V1 for a cer-tain integer j with 0 6 j 6 k � 1, where len(P) = k (See Fig. 4 (c)). Without loss of generality, wecan further assume that len(path(P,u(j+1),v)) P len(path(P,u,u(j))). Then, k � j P 2. Obviously,path(P,u(j+1),v) is a shortest path between u(j+1) and v in Xn. By the condition (2),dist(Xn[V1], u(j+1),v) = dist(Xn,u(j+1),v) = dist(Xn,u,v) � dist(Xn,u,u(j+1)) = k � j � 1 P 1 and dis-t(Xn,u,u(j+1)) = j + 1 P 1. Let l 0 = l � dist(Xn,u,u(j+1)). Hence, dist(Xn[V1], u(j+1),v) + 2 = [dis-t(Xn,u,v) � dist(Xn,u,u(j+1))] +2 = [dist(Xn,u,v) + 2] � dist(Xn,u,u(j+1)) 6 l � dist(Xn,u,u(j+1)) =l 0 6 2n � dist(Xn,u,u(j+1)) 6 2n�1 � 1. By the condition (3), there exists a path P 0 of length l 0

between u(j+1) and v in Xn[V1]. Thus,

pathðP ; u; uðjÞÞ; P 0

is a path of length l 0 + (j + 1) = l between u and v in Xn.
Case 2.2.2. 2n�1 + 1 6 l 6 2n � 1. By Definition 5, we can select w 2 V0 and s 2 V1 � {v} such that
{u,w} 2 E(Xn[V0]), {w, s} 2 E(Xn) and {w,v} 62 E(Xn) (See Fig. 4d). Obviously, dis-t(Xn[V0],u,w) = 1. By the condition (3), there is a path P1 of length l1 between u and w inXn[V0] and a path P2 of length l2 between s and v in Xn[V1] for two integers l1 and l2 with 3 = dis-t(Xn[V0],u,w) + 2 6 l1 6 2n�1 � 1 and dist(Xn[V1], s,v) + 2 6 l2 6 2n�1 � 1. By Lemma 1, dis-t(Xn[V1], s,v) + 6 6 n + 5 6 2n�1 + 1 6 l 6 2n � 1. Therefore, we can select appropriate valuesfor l1 and l2 from the sets {3,4, . . . , 2n�1 � 1} and {dist(Xn[V1], s,v) + 2, dis-t(Xn[V1], s,v) + 3, . . . , 2n�1 � 1}, respectively, such that l1 + l2 + 1 = l. Thus,

P 1; P 2


So far, we have completed the proof of the theorem. h

Theorem 2. For every integer n P 4 and every X n 2Ln with V 0$X n V 1, if the three conditions in Theorem 1 hold,

then Xn is edge-pancyclic.

Proof. By Lemma 6 and Theorem 1, Xn is edge-pancyclic. h

ion, we use the following fact: Given five integers a, b, c, d, and e with a < e < b and c < d, for every integer f withd, there always exist two integers i and j with a 6 i 6 b, i 5 e, and c 6 j 6 d such that i + j = f.


4. Applications

In this section, we will show that the results in last section can be used to induce the path-embeddability andedge-pancyclicity of various BC graphs such as crossed cubes and Mobius cubes, etc.

We have the following lemmas associated with crossed cubes and Mobius cubes:

Lemma 7. For every integer n P 3, every Xn 2 {CQn,0 �MQn,1 �MQn}, and every {x, y} 2 E(Xn) with x 2 V0

and y 2 V1, there is a cycle C of length l such that {x, y} is in C in Xn for every integer l 2 {4,5}, where

V i ¼ V ðCQin�1Þ if Xn = CQn and V i ¼ V ðMQi

n�1Þ if Xn 2 {0 �MQn,1 �MQn} for every integer i 2 {0,1}.

Proof. By the definitions of crossed cubes, Mobius cubes, and BC graphs, we have V 0$X n V 1. We identify the

following cases:

Case 1. Xn = CQn. For every x 2 V ðCQ0n�1Þ, y 2 V ðCQ1

n�1Þ with {x,y} 2 E(CQn). Let x = 0 xn�2 . . . x1x0 andy = 1yn�2 . . . y1y0. By the Definition 2, (x1x0,y1y0) 2 {(00, 00), (10, 10), (01, 11), (11, 01)}.Letz ¼ 1yn�2 . . . y2y1y0 and u ¼ 0xn�2 . . . x2x1x0. Then, by the Definition 2, C: x,y,z,u,x is a cycle oflength 4 such that {x,y} is in C in CQn.If (x1x0,y1y0) 2 {(00, 00),(10, 10)}, let z ¼ 1yn�2 . . . y1y0

and u ¼ 0xn�2 . . . x3x2y1y0, and v ¼ 0xn�2 . . . x3x2y1y0. Then, by the Definition 2, C : x,y,z, u,v,x isa cycle of length 5 such that {x,y} is in C in CQn.Otherwise, (x1x0,y1y0) 2 {(01,11), (11,01)}, letz ¼ 1yn�2 . . . y1y0; u ¼ 0xn�2 . . . x3x2y1y0, and v = 0xn�2 . . . x3x2y1y0. Then, by the Definition 2,C : x,y,z,u,v,x is a cycle of length 5 such that {x,y} is in C in CQn.

Case 2. Xn 2 {0 �MQn, 1 �MQn}. For every x 2 V ðMQ0n�1Þ, y 2 V ðMQ1

n�1Þ with {x,y} 2 E(MQn). Letx = x1x2 . . . xn and y = y1y2 . . . yn.Let z ¼ y1y2 . . . yn�2yn and u ¼ x1x2 . . . xnxn. By the Definition 3,C : x,y,z,u,x is a cycle of length 4 such that {x,y} is in C in Xn.Furthermore, we prove that thereis a cycle of length 5 such that {x,y} is in C in Xn. We deal with the following sub-cases.

Case 2.1. Xn = 0 �MQn. Let z ¼ y1y2y3 . . . yn and u ¼ y1y2 . . . yn. Furthermore, if y2 = 0, letv ¼ y1y2y3y4 . . . yn; Otherwise, let v ¼ y1y2y3y4 . . . yn. By the Definition 3, C : x,y,z,u,v,x is a cycleof length 5 such that {x,y} is in C in 0 �MQn.

Case 2.2. Xn = 1 �MQn. Let z ¼ y1y2y3 . . . yn and u ¼ y1y2y3 . . . yn. Furthermore, if y2 = 0, letv ¼ y1y2y3y4 . . . yn; Otherwise, let v ¼ y1y2y3y4 . . . yn. By the Definition 3, C : x,y,z,u,v,x is a cycleof length 5 such that {x,y} is in C in 1 �MQn. h

Lemma 8. CQ3 ffi 0 �MQ3 and CQ3 ffi 1 �MQ3.

Proof. By observing Fig. 1a–c, we can easily verify this lemma. h

In [5], a routing algorithm was provided to find a shortest path between two different nodes in MQn. By thisrouting algorithm, we can easily verify the following lemma.

Lemma 9. For two integers n P 2 and i 2 {0,1} and u; v 2 V ðMQin�1Þ with u 5 v, we have distðMQi

n�1; u; vÞ ¼distð0�MQn; u; vÞ ¼ distð1�MQn; u; vÞ.

With these lemmas, we give the following two theorems.

Theorem 3. For every integer n P 3 and every Xn 2 {CQn,0 �MQn,1 �MQn}, the following result holds:

For x,y 2 V(Xn) with x 5 y and every integer l with dist(Xn,x,y) + 2 6 l 6 2n � 1, there is a path of length l

between x and y in Xn.

Proof. For every integer n P 3 and every Xn 2 {CQn, 0 �MQn, 1 �MQn}, we prove the theorem by inductionon n.

We first prove that the theorem holds for n = 3. For x,y 2 V(CQ3) with x 5 y, by Lemma 3, there exists apath of length l 0 between x and y in CQ3 for every integer l 0 with 3 ¼ dnþ1

2 e þ 1 6 l0 6 7. By Lemma 2,dist(CQ3,x,y) 2 {1,2}. Thus, there exists a path of length l between x and y in CQ3 for every integer l withdist(CQ3,x,y) + 2 6 l 6 7. By Lemma 8, the theorem holds when n = 3.


Suppose that the theorem holds when n = s � 1 (s P 4). We will prove that the theorem holds when n = s.For Xs 2 {CQs, 0 �MQs, 1 �MQs}, we have V 0$

X n V 1, where

(1) V 0 ¼ V ðCQ0s�1Þ and V 1 ¼ V ðCQ1

s�1Þ for Xs = CQs; and

(2) V 0 ¼ V ðMQ0s�1Þ and V 1 ¼ V ðMQ1

s�1Þ for Xs 2 {0 �MQs, 1 �MQs}.

By Lemma 7, Xs satisfies the condition (1) in Theorem 1. Furthermore, by Lemmas 4 and 9, Xs satisfies thecondition (2) in Theorem 1. By the induction hypothesis, Xs satisfies the condition (3) in Theorem 1.Consequently, by Theorem 1, the theorem holds when s.

Thus, we have proven the theorem. h

Theorem 4. For every integer n P 2 and every Xn 2 {CQn,0 �MQn,1 �MQn}, Xn is edge-pancyclic.

Proof. The theorem obviously holds for n = 2. By Lemma 6 and Theorem 3, the theorem holds forn P 3. h

In fact, in [13,14,26,27], it was respectively proved that for Xn 2 {CQn, 0 �MQn, 1 �MQn}, Xn is edge-pan-cyclic (n P 2) and contains a path of length l between x and y for x,y 2 V(Xn) with x 5 y and every integer l

with dist(Xn,x,y) + 2 6 l 6 2n � 1 (n P 3) by their own specific properties. However, by the above discussion,we can find that the method given in Section 1 is more general, which is applicable for both crossed cubes andMobius cubes; And, by using the results in Section 1, the proofs of the edge-pancyclicilies and path-embedd-abilities of crossed cubes and Mobius cubes become much simpler. In what follows, we will further show thatthe results in Section 1 are also applicable for some other BC graphs in Ln.

Lemma 10. Let A4 be the graph shown in Fig. 5. Then, A4 2L4, A4 m CQ4, A4 m 0 �MQ4, and A4 m 1 �MQ4.

Proof. Clearly, V ðCQ03Þ$

A4 V ðCQ13Þ. Hence, A4 2L4.

First, we decide whether A4 is isomorphic to CQ4, 0 �MQ4, and 1 �MQ4. In fact, in order to solve thesedecision problems, we only need to compute the numbers of cycles of lengths 4 in CQ4,0 �MQ4,1 �MQ4,and A4. Next, we can easily decide whether the number of cycles of lengths 4 in A4 is the same as that of cyclesof lengths 4 in Xn 2 {CQ4,0 �MQ4,1 �MQ4}. If the answer is NO, then the answer of the correspondingdecision problem is also NO. In the following, we will compute the number of cycles of lengths 4 in A4, CQ4,0 �MQ4, and 1 �MQ4.

Considering that V ðCQ03Þ $

CQ4 V ðCQ13Þ; V ðMQ0

3Þ $0�MQ4 V ðMQ1

3Þ; V ðMQ03Þ $1�MQ4 V ðMQ1

3Þ, and V ðCQ03Þ$

A4

V ðCQ13Þ, by Definition 5, the number of cycles of lengths 4 in X4 is composed of the following parts for

every X4 2 {CQ4,0 �MQ4,1 �MQ4,A4}:

(1) The number of cycles of lengths 4 in CQ03;

(2) The number of cycles of lengths 4 in CQ13;

(3) The number of cycles of lengths 4 in X4, where each of these cycles has an edge {x,y} in CQ03 and {z,u} in

CQ13 with {y,z} and {u,x}.

Fig. 5. A 4-dimensional BC graph A4.


The numbers of cycles, in the parts (1) and (2), of lengths 4 are the same in CQ4, 0 �MQ4, 1 �MQ4, andA4. As a result, we only need to compute the number of cycles, in the part (3), of lengths 4 in CQ4, 0 �MQ4,1 �MQ4, and A4, respectively. In order to complete this task, we only need to use the following simplealgorithm to CQ4, 0 �MQ4, 1 �MQ4, and A4, respectively:

Step 1. Set the counter I = 0 and the edge set E0 ¼ EðCQ03Þ.

Step 2. Arbitrarily selecting an edge {x,y} in E 0.Step 3. Let u and z be the neighbors, in CQ1

3, of x and y, respectively. If u is adjacent to z, then let I increasesby 1; Otherwise, I is not changed. Let E 0 = E 0 � {{x,y}}.

Step 4. Arbitrarily selecting an edge {x,y} in E 0. Repeat Steps 3 and 4 until E 0 = ;.

After the algorithm ends, the value of I is the number of cycles, in the part (3), of lengths 4 in the graphselected from CQ4, 0 �MQ4, 1 �MQ4, and A4.

By using the above algorithm, we can find that the numbers of cycles, in the part (3), of lengths 4 in CQ4,0 �MQ4, 1 �MQ4, and A4 are 8, 8, 4, and 5, respectively. Since the number of cycles, in the part (3), oflengths 4 in A4 is not the same as those in CQ4, 0 �MQ4, and 1 �MQ4, we have A4 m CQ4, A4 m 0 �MQ4,and A4 m 1 �MQ4. h

Lemma 11. For every {x, y} 2 E(Xn) with x 2 V ðCQ03Þ and y 2 V ðCQ1

3Þ, there is a cycle C of length l such that

{x, y} is in C in A4, where l 2 {4,5}.

Proof. In fact, for every {x,y} 2 A4 with x 2 V ðCQ03Þ and y 2 V ðCQ1

3Þ, all the five cycles of length 4 gotten byusing the algorithm given in the proof of Lemma 10 are as follows:

0000; 1010; 1011; 0100; 0000;

0001; 1001; 1111; 0111; 0001;

0010; 1101; 1100; 0110; 0010;

0011; 1000; 1001; 0001; 0011;

0101; 1110; 1111; 0111; 0101:

We can easily find that {x,y} lies in at least one of the above five cycles. Therefore, A4 contains a cycle C oflength 4 such that {x,y} is in C for every {x,y} 2 E(Xn) with x 2 V ðCQ0

3Þ and y 2 V ðCQ13Þ.

Furthermore, we can easily verify the following claim:

Claim

For every {x, y} 2 A4 with x 2 V ðCQ03Þ and y 2 V ðCQ1

3Þ, there always exist z 2 V ðCQ03Þ and u 2 V ðCQ1

3Þ with

fx; zg 2 EðCQ03Þ, {z,u} 2 E(A4), and fu; yg 62 EðCQ1

3Þ.

By this claim, for every {x,y} 2 A4 with x 2 V ðCQ03Þ and y 2 V ðCQ1

3Þ, select z 2 V ðCQ03Þ and u 2 V ðCQ1

3Þ withfx; zg 2 EðCQ0

3Þ, {z,u} 2 E(A4), and fu; yg 62 EðCQ13Þ. Then, distðCQ1

3; u; yÞP 2: By Lemma 2, diamðCQ13Þ ¼

diamðCQ3Þ ¼ 2 and thus distðCQ13; u; yÞ 6 2. Hence, distðCQ1

3; u; yÞ ¼ 2 and we can select a node v 2V ðCQ1

3Þ such that fu; vg; fv; yg 2 EðCQ13Þ. Then, C : x,y,v,u,z,x is a cycle of length 5 such that {x,y} is in

C in A4. h

Lemma 12. For every integer i 2 {0,1} and x; y 2 V ðCQi3Þ with x 5 y, distðA4½V ðCQi

3Þ�; x; yÞ ¼ distðA4; x; yÞ.

Proof. For x; y 2 V ðCQi3Þ with x 5 y, we have distðA4½V ðCQi

3Þ�; x; yÞ 2 f1; 2g. It is trivial fordistðA4½V ðCQi

3Þ�; x; yÞ ¼ 1. For distðA4½V ðCQi3Þ�; x; yÞ ¼ 2, obviously, distðA4; x; yÞ 6 distðA4½V ðCQi

3Þ�; x; yÞ ¼2. If dist(A4,x,y) < 2, then dist(A4,x,y) = 1, contradicting that distðA4½V ðCQi

3Þ�; x; yÞ ¼ 2. Hence, dis-t(A4,x,y) 5 1 and thus distðA4; x; yÞ ¼ 2 ¼ distðA4½V ðCQi

3Þ�; x; yÞ. h


Theorem 5. For two nodes x,y 2 V(A4) with x 5 y and every integer l with dist(A4,x,y) + 2 6 l 6 15, there is a

path of length l between x and y in A4.

Proof. Let V i ¼ V ðCQi3Þ for i 2 {0,1}. Then V 0$

A4 V 1. By Lemmas 11, 12, and Theorem 3, A4 satisfies the threeconditions in Theorem 1. By Theorem 1, the theorem holds. h

Theorem 6. A4 is edge-pancyclic.

Proof. By Lemma 6 and Theorem 5, the theorem holds. h

Remark. Since Qn is a bipartite graph, for every integer n P 3, Qn is not edge-pancyclic. Furthermore, fromLemmas 6, 10, and Theorem 5, we can deduce:

Let S ¼ fX njn P 3;X n 2Ln; and X n is edge-pancyclicg and S0 ¼ fX njn P 3;X n 2Ln; for u; v 2 V ðX nÞwith u 6¼ v; and every integer l withdistðX n; u; vÞ þ 2 6 l 6 2n � 1; X n contains a path of length l betweenu and vg. Then fCQnjn P 3g[ f0�MQn j n P 3g [ f1�MQn j n P 3g � S � S0 � [1n¼3Ln.

Acknowledgement

This work was supported by a grant from Research Grants Council of Hong Kong [Project No.CityU*1149/04E*].

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Edge-pancyclicity and path-embeddability of bijective connection graphs