Information Processing Letters 110 (2010) 203–205

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Information Processing Letters

www.elsevier.com/locate/ipl

Unpaired many-to-many vertex-disjoint path covers of a classof bipartite graphs ✩

Xie-Bin Chen

Department of Mathematics and Information Science, Zhangzhou Teachers College, Zhangzhou, Fujian 363000, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 August 2009Received in revised form 3 December 2009Accepted 8 December 2009Available online 16 December 2009Communicated by M. Yamashita

Keywords:HypercubeVertex-disjoint pathsHamiltonian pathMatchingFolded hypercubeInterconnection network

Let Wn denote any bipartite graph obtained by adding some edges to the n-dimensionalhypercube Q n , and let S and T be any two sets of k vertices in different partite sets of Wn .In this paper, we show that the graph Wn has k vertex-disjoint (S, T )-paths containing allvertices of Wn if and only if k = 2n−1 or the graph Wn − (S ∪ T ) has a perfect matching.Moreover, if the graph Wn − (S ∪ T ) has a perfect matching M , then the graph Wn hask vertex-disjoint (S, T )-paths containing all vertices of Wn and all edges in M . And somecorollaries are given.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

One of the most central issues in various interconnec-tion networks is to find node-disjoint paths concernedwith a routing among nodes. The problem of node-disjointpaths has received much attention because of its numer-ous applications in high performance communication net-works, fault-tolerant routings, and so on. Given two dis-joint sets of k labeled vertices S = {s1, s2, . . . , sk} andT = {t1, t2, . . . , tk} in a graph G , called sources and sinks,respectively. Assume there exist k vertex-disjoint pathsP1, P2, . . . , Pk in G , such that Pi connects si and tψ(i) foreach i = 1,2, . . . ,k, where ψ is some permutation on theset {1,2, . . . ,k}. We call these paths to be paired if ψ is theidentical permutation and unpaired otherwise. A collectionof (un)paired k vertex-disjoint (S, T )-paths in a graph Gis said to be an (un)paired many-to-many k-disjoint pathcover of G if these k paths contain all vertices of G .

✩ The work was supported by NNSF of China (No. 10671191).E-mail address: chenxbfz@yahoo.com.cn.

0020-0190/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.ipl.2009.12.004

The problem of vertex-disjoint path cover of a graphoriginated from applications of interconnection networksin which the full utilization of nodes is important. For anembedding of linear arrays in a network, the cover im-plies every node can be participated in a pipeline com-putation [12]. Indeed, a paired many-to-many 1-disjointpath cover of a graph is just a Hamiltonian path betweentwo vertices. A large amount of work on Hamiltonian pathembedding in various networks appeared in the litera-ture. However, for k � 2, the problem of many-to-manyk-disjoint path cover of networks is rather difficult. Theresearch attention focused on some special classes of well-known networks. For paired many-to-many disjoint pathcover, see Refs. [2,4,8,10,12,13], and for unpaired one, seeRefs. [3,11].

Let Wn denote any bipartite graph obtained by addingsome edges to the n-dimensional hypercube Q n , and let Sand T be any two sets of k vertices in different partite setsof Wn . In this paper, we obtain a necessary and sufficientcondition for the graph Wn to have an unpaired many-to-many k-disjoint (S, T )-path cover.

204 X.-B. Chen / Information Processing Letters 110 (2010) 203–205

2. Preliminaries

We follow [1] for terminology and notation on graphtheory. A graph G = (V , E) means a simple graph, whereV = V (G) is its vertex-set and E = E(G) is its edge-set.Assume V 0 is a subset of the vertex-set of a graph G , thesymbol G − V 0 denotes the graph obtained from G by re-moving all vertices in V 0 and all edges incident with V 0.A graph P = (v1, v2, . . . , vk) is called a path if k verticesv1, v2, . . . , vk are distinct and (vi, vi+1) is an edge of P fori = 1,2, . . . ,k − 1, two vertices v1 and vk are called end-vertices of the path P . A path containing all vertices of agraph is called a Hamiltonian path. If P = (v1, v2, . . . , vk)

with k � 3, and (v1, vk) is also an edge in G , then C =P ∪ (v1, vk) is called a cycle. A cycle containing all verticesof a graph is called a Hamiltonian cycle. A set of vertex-disjoint edges in a graph G is called a matching of G; amatching M of a graph G is called a perfect matching ifV (M) = V (G).

It is well known that the n-dimensional hypercube(the n-cube) Q n is one of the most popular and effi-cient interconnection networks, it possesses many excel-lent properties. There is a large amount of literature ongraph-theoretical properties of the n-cube and their ap-plications in parallel computing, e.g., see [9,14], a recentsurvey [15] and references therein. The n-cube is a graphwith 2n vertices, its any vertex is denoted by an n-bit bi-nary string. Two vertices of Q n are adjacent if and only iftheir binary strings differ in exactly one-bit position. It isclear that Q n is a bipartite graph of valence n.

The n-dimensional folded hypercube, denoted by FQn(n � 2), is an attractive variation of the n-cube Q n , it canbe constructed from an n-cube by adding an edge to everypair of vertices with distance n. FQn has many good prop-erties, such as it is a Cayley graph, its connectivity is equalto its valence n + 1, its diameter is �n/2�, about half ofthe diameter of Q n , etc., see [6], a recent survey [15] andreferences therein.

For an unpaired many-to-many disjoint path cover ofthe n-cube, the following result was obtained in Ref. [3].

Lemma 1. (See [3].) Let 1 � k � n − 1, and let S and T beany two sets of k vertices in different partite sets of the n-cubeQ n. Then Q n has a fault-free unpaired many-to-many k-disjoint(S, T )-path cover provided that it contains at most n − k − 1faulty edges, and the number of tolerant faulty edges is sharp.

The proof of Theorem 1 in Section 3 is based on thefollowing interesting result.

Lemma 2. (See [7].) Let Q n, n � 2, be the n-cube and K (Q n) bethe complete graph on its vertex-set being V (Q n). Then, for ev-ery perfect matching M of K (Q n), there exists a perfect match-ing R of Q n, such that M ∪ R is a Hamiltonian cycle of K (Q n). Inparticular, every perfect matching of Q n is contained in a Hamil-tonian cycle.

3. Main results

Let Wn denote any bipartite graph obtained by addingsome edges to the n-dimensional hypercube Q n . In the fol-

lowing theorem, we give a necessary and sufficient condi-tion for the graphs Wn to have an unpaired many-to-manydisjoint path cover.

Theorem 1. Let S and T be any two sets of k vertices in differ-ent partite sets of the bipartite graph Wn. Then the graph Wnhas an unpaired many-to-many k-disjoint (S, T )-path cover ifand only if k = 2n−1 or the graph Wn − (S ∪ T ) has a per-fect matching. Moreover, if the graph Wn − (S ∪ T ) has aperfect matching M, then Wn has an unpaired many-to-manyk-disjoint (S, T )-path cover containing all edges in M.

Proof. Let S = {s1, s2, . . . , sk} and T = {t1, t2, . . . , vk}, andlet

En = {e ∈ E(Q n)

∣∣ n-th bits of two end-vertices of e

are different}.

Clearly, En is a perfect matching of Q n . Since Q n is a span-ning subgraph of Wn , then En is also a perfect matchingof Wn .

If k = 2n−1, it is easy to see that En is an unpairedmany-to-many k-disjoint (S, T )-path cover of the graphWn . Assume k < 2n−1 below, then the graph Wn − (S ∪ T )

contains 2n − 2k vertices.Necessity. Assume that the graph Wn has an unpaired

many-to-many k-disjoint (S, T )-path cover. Let these kpaths be P1, P2, . . . , Pk . Note that the two sets S and Tare in different partite sets of the bipartite graph Wn and|S| = |T | = k < 2n−1. Clearly, for i = 1,2, . . . ,k, all verticesof the path Pi are alternately in different partite sets ofWn , and one end-vertex of Pi is in S , and the other isin T . Hence, the path Pi itself has a perfect matching fori = 1,2, . . . ,k. Since these k paths contain all vertices ofWn . It is easy to see that the graph Wn − (S ∪ T ) has aperfect matching.

Sufficiency. Let M be a perfect matching of the graphWn − (S ∪ T ), and let ei = (si, ti) be an edge in the com-plete graph K (Q n) for i = 1,2, . . . ,k. Then M ′ = M ∪ (e1 ∪e2 ∪ · · · ∪ ek) is a perfect matching of K (Q n). By Lemma 2,there exists a perfect matching R of Q n (R is also a perfectmatching of Wn), such that C = M ′ ∪ R is a Hamiltoniancycle of K (Q n). Clearly, C − (e1 ∪ e2 ∪ · · · ∪ ek) = R ∪ M is acollection of k vertex-disjoint paths in the graph Wn , andevery path of the collection has two end-vertices belong-ing to S ∪ T . Note that the graph Wn is a bipartite graphand its perfect matching R is contained in the collection, itfollows that every path of the collection contains an evennumber of vertices, then one end-vertex of every path isin S and the other is in T . Hence, the collection of these kpaths is an unpaired many-to-many k-disjoint (S, T )-pathcover of Wn and contains all edges in the matching M . �

If n � 3 is odd, then any two vertices with distance nin the n-cube Q n are in different partite sets of Q n , andso the n-dimensional folded hypercube FQn is a bipartitegraph and is a special class of Wn . And the following resultfollows.

Corollary 1. If n � 3 is odd, then Theorem 1 holds for then-dimensional folded hypercube FQn.

X.-B. Chen / Information Processing Letters 110 (2010) 203–205 205

By Lemma 1 and Theorem 1, we have the following re-sult.

Corollary 2. Let 1 � k � n − 1, and let S and T be any two setsof k vertices in different partite sets of the bipartite graph Wn.Then the graph Wn − (S ∪ T ) has a perfect matching. If M is anyperfect matching in the graph Wn − (S ∪ T ), then the graph Wn

has an unpaired many-to-many k-disjoint (S, T )-path covercontaining all edges in M. On the other hand, if 2n−1 > k � n �3, then it is possible that the graph Wn − (S ∪ T ) has no perfectmatching and the graph Wn has no unpaired many-to-manyk-disjoint (S, T )-path cover.

Proof. Note that the n-cube Q n is a spanning subgraph ofthe bipartite graph Wn . Since 1 � k � n − 1, and S andT are two sets of k vertices in different partite sets ofthe n-cube Q n , by Lemma 1, the n-cube Q n has an un-paired many-to-many k-disjoint (S, T )-path cover, and sodoes the graph Wn . By Theorem 1, the graph Wn − (S ∪ T )

has a perfect matching, and the graph Wn has an unpairedmany-to-many k-disjoint (S, T )-path cover containing alledges in M . Assume 2n−1 > k � n � 3 and the graph Wn

has a vertex v of degree at most k. Let S be a set of kvertices in a partite set of Wn such that S contains allneighbors of the vertex v , and let T be a set of k verticesin the other partite set of Wn such that v does not belongto T . Clearly, the graph Wn − (S ∪ T ) has an isolated ver-tex v , and then has no perfect matching. Hence, the graphWn has no unpaired many-to-many k-disjoint (S, T )-pathcover by Theorem 1. �

For k = 1,2, it was known (see, e.g., [5]) that the n-cube Q n,n � 2, has a paired many-to-many k-disjoint pathcover. The following two corollaries are stronger results.

Corollary 3. Let s and t be any two vertices in different partitesets of the bipartite graph Wn, where n � 2, and let M be a per-fect matching in Wn − {s, t} (by Corollary 2). Then there existsa Hamiltonian path P between s and t such that all edges of Mare contained in P .

Corollary 4. Assume n � 3. Let {s1, t1, s2, t2} be four vertices ofthe bipartite graph Wn, such that two vertices si and t j are indifferent partite sets of Wn for i, j ∈ {1,2}, and let M be a per-fect matching in the graph Wn −{s1, t1, s2, t2} (by Corollary 2).Then there exist two vertex-disjoint paths Pi whose end-verticesare si and ti for i = 1,2, such that V (P1) ∪ V (P2) = V (Wn)

and all edges of M are contained in E(P1) ∪ E(P2), that is,the graph Wn has a paired many-to-many 2-disjoint path covercontaining all edges in the matching M.

Proof. Let e1 = (s1, t2) and e2 = (s2, t1) be two edges inthe complete graph K (Q n). Similar to the proof of suffi-ciency of Theorem 1, there exists a perfect matching R inthe graph Wn , such that C = e1 ∪ e2 ∪ M ∪ R is a Hamil-tonian cycle of K (Q n). Then R ∪ M is a collection of twovertex-disjoint paths P1 and P2 in the graph Wn , where

two end-vertices of the path Pi are si and ti for i = 1,2,and the conclusion follows. �4. Conclusion remarks

Let the graph Wn and two sets S and T be as before.In this paper, we show that the graph Wn has an unpairedmany-to-many k-disjoint (S, T )-path cover if and only ifk = 2n−1 or the graph Wn − (S ∪ T ) has a perfect matching.It is well known that the complexity of Hungarian Algo-rithm (see, e.g., [1]) for finding a maximun matching in abipartite graph G is O (pq), where p and q is the numberof vertices and edges of G , respectively. By Theorem 1, wededuce that the complexity of the problem of the bipartitegraph Wn to have an unpaired many-to-many k-disjoint(S, T )-path cover is also O (pq), where p and q is thenumber of vertices and edges of the graph Wn − (S ∪ T ),respectively. The problem of an (un)paired many-to-manydisjoint path cover of a network is of interest and shouldbe further investigated.

Acknowledgements

The author would like to express his gratitude to theanonymous referees for their kind suggestions and carefulcorrections that helped improve the original manuscript.

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