Computers and Mathematics with Applications 62 (2011) 3551–3554

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Computers and Mathematics with Applications

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

Regular connected bipancyclic spanning subgraphs of hypercubesS.A. Mane ∗, B.N. WaphareDepartment of Mathematics, University of Pune, Pune-411007, India

a r t i c l e i n f o

Article history:Received 9 April 2011Received in revised form 26 August 2011Accepted 29 August 2011

Keywords:RegularConnectedSpanningBipancyclicSubgraphsHypercubes

a b s t r a c t

An n-dimensional hypercubeQn is a Hamiltonian graph; in otherwordsQn (n ≥ 2) containsa spanning subgraph which is 2-regular and 2-connected. In this paper, we explore yetanother strong property of hypercubes. We prove that for any integer k with 3 ≤ k ≤ n,Qn (n ≥ 3) contains a spanning subgraph which is k-regular, k-connected and bipancyclic.We also obtain the result that every mesh Pm × Pn (m, n ≥ 2) is bipancyclic, which is usedto prove the property above.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

In recent decades, many interconnection networks have been developed to serve as the underlying topologies of large-scale multiprocessor systems. The hypercube is a popular interconnection network because of its various good properties.An entire chapter is devoted to the hypercube by Leighton in [1] because of its wide usage as an interconnection networkfor parallel computers. Also numerous variations of the hypercube have been investigated.

Since edge (link) failures can occur when a network is activated, it is important to consider networks with faultyedges. Hsieh et al. [2–10], Chan and Lee [11], Kuo and Hsieh [12], Tsai [13], and Tsai and Lai [14] explore manyproperties like hamiltonicity, pancyclicity, bipancyclicity and edge-bipancyclicity of various faulty networks (subgraphs).The bipancyclicity of a given network is an important factor in determining whether the network topology can be used tosimulate rings of various lengths.

One way of measuring the stability of a network (computer, communication, or transportation) is through the ease orthe cost of disrupting the network. The connectivity gives theminimum cost for disrupting the network. In amultiprocessorcomputing system, computing involves exchange of data among several of its processors. The data are transmitted from oneprocessor to another through a sequence of interlinked processors. Obviously, the transmission is faster if there are a largenumber of alternative parallel paths available. Equivalently, the demand is that in the graph of the interconnection network,there be a large number of parallel paths connecting any two vertices. Many properties of regular subgraphs or spanningsubgraphs of hypercubes have been studied by Djokovic [15], Duckworth et al. [16], Graham and Harary [17], Harary andLewinter [18,19], Kobeissi and Mollard [20], Law [21], Ramras [22] etc.

These applications and research motivated us to find proper spanning subgraphs in hypercubes which are bipancyclicand have large connectivity.

An n-dimensional hypercube, denoted by Qn, is a graph with 2n vertices, and each vertex u can be distinctly labeled withan n-bit binary string, u = u1u2 · · · un. There is an edge between two vertices if and only if their binary labels differ in exactly

∗ Corresponding author.E-mail addresses:manesmruti@yahoo.com (S.A. Mane), bnwaph@math.unipune.ac.in (B.N. Waphare).

0898-1221/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.camwa.2011.08.071

3552 S.A. Mane, B.N. Waphare / Computers and Mathematics with Applications 62 (2011) 3551–3554

one bit position.Qn hasmany attractive properties, such as being bipartite, n-regular and n-connected. It is also a bipancyclicgraph in the sense that, for every even integer k, 4 ≤ k ≤ |V (Qn)|, there exists a cycle C of length k in Qn. In particular Qn isa Hamiltonian graph; in other words Qn (n ≥ 2) contains a spanning subgraph which is 2-regular and 2-connected. In thispaper, we explore yet another strong property of hypercubes. We prove that for any integer k with 3 ≤ k ≤ n, Qn (n ≥ 3)contains a k-regular spanning subgraph that in addition is k-connected and bipancyclic.

For undefined terminology and notation, see [23].

2. Regular connected bipancyclic spanning subgraphs of hypercubes

Lemma 2.1. Let for k ≥ 2, Pk denote the path of length k − 1. Then the mesh Pm × Pn (m, n ≥ 2) contains a cycle of length l forany even integer l with 4 ≤ l ≤ mn.

Proof. Let Pm = x1 − x2 − · · · − xm and Pn = y1 − y2 − · · · − yn be paths. We prove the result by induction on n.For n = 2,G = Pm × Pn = Pm × P2 is a ladder.In this case for any kwith 2 ≤ k ≤ m, Ck = (x1, y1)− (x2, y1)− ..(xk, y1)− (xk, y2)− (xk−1, y2)−· · ·− (x1, y2)− (x1, y1)

is a cycle of length 2k. Hence Pm × P2 is bipancyclic; see Fig. 1.Now for n = 3,G = Pm × Pn = Pm × P3.As Pm × P2 is a subgraph of Pm × P3, this mesh contains cycles of even length k, for 4 ≤ k ≤ 2m. Now consider the cycle

C of length 2m in the subgraph Pm × P2 of Pm × P3. We can construct a cycle of length 2m + 2 by replacing an edge of thetype ⟨(xi, y2) − (xi+1, y2)⟩ on C by the path (xi, y2) − (xi, y3) − (xi+1, y3) − (xi+1, y2). The cycles of even length k > 2m+ 2can be obtained by replacing alternate edges of C as above; see Fig. 1.

Now let n ≥ 4 and assume that the result is true for Pm × Pn−2. Thus for any even l with 4 ≤ l ≤ m(n − 2),there is a cycle of length l. Pm × Pn is constructed from Pm × Pn−2 by adjoining a copy of Pm × P2 (with vertices labeled(xi, yj), 1 ≤ i ≤ m, n − 1 ≤ j ≤ n), together with edges ei = ⟨(xi, yn−2), (xi, yn−1)⟩, 1 ≤ i ≤ m.

Let C1 be a cycle in Pm × Pn−2 of maximum (even) length l1. Then l1 is either m(n − 2) or m(n − 2) − 1. Thus C1contains at least three of the four corner vertices of Pm × Pn−2. Hence we may assume that (x1, yn−2) ∈ C1. Thereforethe edge e = ⟨(x1, yn−2), (x2, yn−2)⟩ belongs to C1. To complete the proof we must show that for any even integer l′ withm(n − 2) + 2 ≤ l′ ≤ mn, there is a cycle C ′ of length l′ in Pm × Pn. Let k = l′ − l1. Then k is even and k = l′ − m(n − 2) orl′ − m(n − 2) + 1. Sincem(n − 2) + 2 ≤ l′ ≤ mn and l′ is even, it follows that 2 ≤ k ≤ 2m. Regarding the bottom two rowsof Pm × Pn as Pm × P2, we have seen that these two rows contain a cycle C2 of length k(4 ≤ k ≤ 2m), and we may assumethat e′

= ⟨(x1, yn−1), (x2, yn−1)⟩ belongs to C2 (for k = 2, we consider C2 = e′+ e′).

Finally, let C = (C1 − e) + (C2 − e′) + e1 + e2. Hence the length of C = (l− 1) + (k− 1) + 2 = l+ k = l′. To see that C isa cycle, start at (x1, yn) and traverse C2 counter-clockwise, stopping at (x2, yn−1) (but omitting e′), then go to (x2, yn−2) viae2, then traverse C1 counter-clockwise, stopping at (x1, yn−2) (omitting e), and follow e1 and ⟨(x1, yn−1), (x1, yn)⟩ to (x1, yn).See Fig. 2. �

We also need the following result given in [24] by Simon Spacapan.

Lemma 2.2 ([24]). For any connected graphs G and H, κ(G × H) ≥ κ(G) + κ(H) (where κ(G∗) denotes the vertex connectivityof the graph G∗).

Theorem 2.3. Let n ≥ 3 be an integer. Then for any integer k with 3 ≤ k ≤ n,Qn contains a spanning subgraph which isk-regular, k-connected and bipancyclic.

Proof. For n = 3, the proof is obvious. Now we will prove our result for n ≥ 4.Let k be an integer such that 3 ≤ k ≤ n. We claim that there exists a k-regular, k-connected and bipancyclic spanning

subgraph of Qn.We decompose Qn as Qn = Qm × Qn−m where m = k − 2. Now 1 ≤ m ≤ n − 2. Obviously 2 ≤ n − m ≤ n − 1. Let

C = t1t2t3 · · · tr1 t1 (r1 = 2n−m) be a Hamiltonian cycle in Qn−m.

Claim. The spanning subgraph G = Qm × C of Qn is k-regular, k-connected and bipancyclic.

For s ∈ V (Qm) and t ∈ V (Qn−m) we denote by (s, t) the vertex in Qn whose first m components form the tuple s andwhose last (n − m) components form the tuple t . Thus we have

V (Qn) = V (Qm) × V (Qn−m) = {(s, t) : s ∈ V (Qm), t ∈ V (Qn−m)}.

Firstly we prove that G is k-regular.As the degree of s is d(s) = m in Qm and d(t) = 2 in C , we have d(s, t) = m + 2 = k in G(Qm × C), for any vertex in G.We will prove that G is k-connected.We know that κ(G) ≤ δ(G) = m + 2. Also by Lemma 2.2, κ(G) = κ(Qm × C) ≥ κ(Qm) + κ(C) = m + 2 gives us

κ(G) = m + 2 = k.Now we will prove that G is bipancyclic for k ≥ 3.As G = Qm × C contains the mesh P2m × P2n−m as a spanning subgraph, bipancyclicity of G follows by Lemma 2.1.This completes the proof. �

S.A. Mane, B.N. Waphare / Computers and Mathematics with Applications 62 (2011) 3551–3554 3553

Fig. 1. Bipancyclicity of Pm × P2 and Pm × P3 .

Fig. 2. Bipancyclicity of Pm × Pn .

Concluding remarks.In this paper, we explore yet another strong property of the hypercubes. We prove that for any integer k with 3 ≤ k ≤

n,Qn (n ≥ 3) contains a spanning subgraph, say Gk, which is k-regular, k-connected and bipancyclic.In the case of only one edge fault, that is in G = Qn − e, by using the edge-hamiltonicity of Qn, it is easy to prove that

there exists a spanning subgraph Gk which is k-regular, k-connected and bipancyclic.But the question still remains open as to what the optimal edge-fault tolerance of Qn is, satisfying the property above.

Alsowe can ask for spanning subgraphswhich are k-regular, k-connected and bipancyclic in the case of numerous variationsof the hypercube.

Acknowledgments

The authors would like to express their gratitude to the anonymous referees for their kind suggestions and correctionsthat helped to improve the original manuscript.

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