ON THE GEODESIC PANCYCLICITY OF MO¨ BIUS CUBES

ON THE GEODESIC PANCYCLICITY OF M ¨ OBIUS CUBES

Chang-Hsiung Tsai, Jheng-Cheng Chen Institute of Learning Technology National Hualien University of Education

Hualien, Taiwan 970, R.O.C.

[email protected]

Hong-Chun Hsu

Department of Medical Informatics Tzu Chi University

Haulien, Taiwan R.O.C.

[email protected] Pao-Lien Lai

Department of Computer Science and Information Engineering National Dong Hwa University

Shoufeng, Hualien, Taiwan, R.O.C.

[email protected]

Abstract

For two vertices X, Y ∈ V (G), a cycle is called a geodesic cycle with X and Y if a shortest path joining X and Y lies on the cycle. A graph G is called to be geodesic k-pancyclic if any two vertices X, Y on G have such geodesic cycle of length l that 2dG(X, Y ) + k ≤ l ≤ |V (G)|.

In this paper, we show that the n-dimensional M¨obius cube M Q_n is geodesic 3-pancyclic for n ≥ 3. This result is near optimal because there is no geodesic 1-cycle with two adjacent vertices in M Q_n.

Keywords: geodesic pancyclic, M¨obius cubes, panconnected, pancyclic, shortest path.

1 Introduction

Interconnection networks are essential for paral- lel and distributed computing. A ring structure is often used as a interconnection architecture for local area network and as a control and data flow

structure in distributed networks due to its good properties. To carry out a ring-structure algo- rithm on a specific multicomputer or a distributed system, the processes of the parallel algorithm need to be mapped to the nodes of the intercon- nection network in the system such that any two adjacent processes in the ring are mapped to two adjacent node of the network. For this purpose, it is desired that the targeted interconnection net- work posses a hamiltonian cycle, i.e., a cycle that passes every node of the network exactly once if the number of processes in the ring-structure parallel algorithm equals the number of nodes of the interconnection network. When the number of processes is less than the number of nodes of the network, the pancyclic property of the net- work with n nodes is desired, that is, there exists a cycle of length l in the network for each inte- ger l with4 ≤ l ≤ n. The hypercube is one of the most popular interconnection networks since it has simple structure and is easy to implement.

1

M¨obius cubes form a class of hypercube variants that give better performance with the same num- ber of edges and vertices [2]. Cull et al. [2]

proved that the n-dimensional M¨obius cube, de- noted by M Q_n, has several better properties than the n-dimensional hypercube, denoted by Q_n, for example, the diameter of M Q_n is about one half that of Q_n and graph embedding capability of M Q_nis better than Q_n.

With regard to the pancyclicity of M¨obuis cubes, many related results have received con- siderable attention [3, 4, 5, 7, 10, 12, 13]. Fan [3] showed that an n-dimensional M¨obius cube is pancyclic. Xu et al. [10] proved that an n- dimensional M¨obius cube is edge-pancyclic, that is, every edge lies on a cycle of length l for each integer l with 4 ≤ l ≤ n. Hu et al.

[7] found that an n-dimensional M¨obius cube is node-pancyclic, that is, every node lies on a cy- cle of length l for each integer l with 4 ≤ l ≤ n. As concerns the fault-tolerant pancyclicity of M¨obius cubes, Hsieh and Chen [4] proved that an n-dimensional M¨obius cube with up to n−2 edge faults is pancyclic. After, Yang et al. [13] pro- posed that an n-dimensional M¨obius cube is pan- cyclic in the presence of up to n− 2 faulty nodes.

When concerns pancyclicity of M¨obius cubes in the presence of faulty nodes and/or edges, Yang et al. [12] proved that an n-dimensional M¨obius cube is still pancyclic even if it has up to n− 2 node and/or edge faults.

Here, we consider the geodesic cycle em-

bedding problem that have been studied in [1, 6, 8] in M¨obius cubes. In other words, for any two vertices, we want to find all the possible lengths of cycles including a shortest path joining them.

A graph G is called geodesic k-pancyclic if any two vertices X, Y on G have such geodesic cycle of length l that 2dG(X, Y ) + k ≤ l ≤ |V (G)|

where d_G(X, Y ) is the distance from X to Y in G. Hsu et al. [6] proved that an n-dimensional Augmented cube contains a geodesic pancyclic of length from max{2d(X, Y ), 3} ≤ l ≤ 2ⁿ. Lai et al. [8] proposed that an n-dimensional Crossed cube is geodesic4-pancyclic. In this pa- per, we prove that M Q_n is geodesic 3-pancyclic for n≥ 3.

This paper is organized as follows. In Sec- tion 2, we give some definitions and properties of M¨obius cubes. In Section 3, we prove that M Q_n is geodesic 3-pancyclic. The final section con- cludes this papers.

2 M¨obius cubes

Let the interconnection network be modeled by an undirected graph G = (V, E) where the set of vertices V(G) represents the processing ele- ments of the network and the set of edges E(G) represents the communication links. Through- out this paper, for the graph theoretic defini- tions and notations we follow [9]. Two ver- tices are adjacent when they are incident with a common edge. A path of length k from X to Y is a finite sequence of adjacent vertices

written as X₁, X₂, . . . , X_k+1, where X₁ = X, X_k+1= Y , and all the vertices X1, X₂, . . . , X_k+1 are distinct except possibly X₁ = Xk+1. For convenience, we use the sequence X1, . . . , X_i, P(Xi, X_j), Xj, . . . , X_k+1 to denote the path

X₁, X₂, . . . , X_k+1, where P (X_i, X_j) = X_i, X_i+1, . . ., X_j and the two vertices Xi and X_j are called the end-vertices of P(X_i, X_j). We call that P(X_i, X_j) is a sub-path of the path from X to Y . Sometimes, we also use P to denote a path P(Xi, X_j). Let l(P (Xi, X_j)) denote the length of the path P that is the number of edges in P . The distance between X and Y in G is denoted by d_G(X, Y ), which is the length of a shortest path between X and Y in G. A cycle C is a spe- cial path with at least three vertices such that the first vertex is the same as the last one. A cycle of length k is called a k-cycle. A path (respectively, cycle) which traverses each vertex of G exactly once is hamiltonian path (respectively, hamilto- nian cycle).

The n-dimensional M¨obius cube M Q_n, proposed first by Cull and Larson [2], consists of 2ⁿ vertices and each vertex has a unique n- component binary vector for an address. Each vertex has n neighbors as follows. A vertex X = x1x₂. . . x_n connects to its ith neighbor, denoted by N_i(X), for 2 ≤ i ≤ n, Ni(X) = x₁x₂. . . x_i−1x_ix_i+1. . . x_n if x_i−1 = 0 or Ni(X)

= x₁x₂. . . x_i−1x_ix_i+1. . . x_nif x_i−1= 1.

For i= 1, since there is no bit on the left of x₁, N₁(X) can be defined as the first neighbor of

0001 0011

0101 0111

0100 0110

0000 0010

1001 1011

1101 1111

1000 1010

1100 1110

(a) MQ4⁰

0001 0011

0101 0111

0100 0110

0000 0010

1001 1011

1101 1111

1000 1010

1100 1110

(b) MQ4¹

Figure 1. (a) A 0-type 4-dimensional M¨obius cube. (b) A 1-type 4-dimensional M¨obius cube.

X can be denoted as x₁x₂. . . x_n or x₁x₂. . . x_n. If we assume that the zeroth bit of every ver- tex of M Q_n is 0, we call the network a 0-type n-dimensional M¨obius cube, denoted by M Q⁰_n; and if we assume that the zeroth bit of every ver- tex of M Q_n is 1, we call the network a 1-type n-dimensional M¨obius cube, denoted by M Q¹_n. Either M Q⁰_nor M Q¹_n may be denoted by M Q_n. The example of M Q⁰₄ and M Q¹₄ are shown in Fig 1.

Therefore, M Q_n is an n-regular graph and can be recursively defined as follows: Both M Q⁰₁ and M Q¹₁ are complete graph K₂ with one ver- tex labeled 0 and the other 1. MQ⁰_n and M Q¹_n are both composed of a sub-M¨obius cube M Q⁰_n−1 and a sub-M¨obius cube M Q¹_n−1. Each vertex

X = 0x₂x₃. . . x_n−1x_n ∈ V (MQ⁰_n−1) connects to 1x2x₃. . . x_n−1x_n ∈ MQ¹_n−1 in M Q⁰_n and to 1x2x₃. . . x_n−1x_n in M Q¹_n. For convenience, we say that M Q⁰_n−1 and M Q¹_n−1 are two sub- M¨obius cubes of M Q_n, where M Q⁰_n−1 (respec- tively, M Q¹_n−1) is an(n − 1)-dimensional 0-type M¨obius cube (respectively, 1-type M¨obius cube) which includes all vetices 0x₂x₃. . . x_n−1x_n (re- spectively,1x₂x₃. . . x_n−1x_n), x_i ∈ {0, 1}.

Let e_i be the n-dimensional (0, 1) vector with only its ith component equal to 1. Let E_ibe the n-dimensional(0, 1) vector with ith through nth components equal to 1. Let{Z2}ⁿbe the n- dimensional vector space over {0, 1} with addi- tion and scalar multiplication mod 2. It is clear that both{e_i | 1 ≤ 1 ≤ n} and {E_i | 1 ≤ i ≤ n}

are bases for this space. Hence{e_i, E_i | 1 ≤ i ≤ n} forms a redundant basis for this vector space.

Any vector X can be represented as a linear sum of these basis vectors:

X=^n

i=1

(αie_i+ βiE_i), (1)

where α_i ∈ {0, 1} and βi ∈ {0, 1}. Clearly, we can represent a vector X by the set of vectors e_i, E_i that have nonzero coefficients in the above sum. For any vertex X in M Q_n, the ith neighbor of X, N_i(X), is formed by X + e_i if x_i−1= 0, or X+ E_i if x_i−1= 1. It is clearly that every vertex X of M Q_ncan be formulated as the above sum.

Definition 1 [2] A set S of e_i, E_i, where1 ≤ i ≤ n, is an expansion of X if and only if the equality

in(1) is true, where α_i = 1 if and only if e_i ∈ S and β_i = 1 if and only if Ei ∈ S. Also, any t ∈ S is called a term of this expansion of X.

Because we are using a redundant basis, there can be more than one expansion of a vec- tor. For a vector X, the weight of an expansion S of X is the cardinality of set S, denoted by|S|

and a minimal expansion of X is an expansion with least weight.

Lemma 1 Let X be a vertex of M Q⁰_nwith n≥ 3 and Y = N_i(X). Then d_MQ⁰_n(N₁(X), N₁(Y )) = 1 if 3 ≤ i ≤ n and d_MQ⁰_n(N₁(X), N₁(Y )) = 2 if i= 2.

Proof. Let X = x1x₂. . . x_i−1x_ix_i+1. . . x_n where x_j ∈ {0, 1} for 1 ≤ j ≤ n.

Since Y is an ith neighbor of X, Y = x₁x₂. . . x_i−1x_ix_i+1. . . x_n if x_i−1 = 0 or Y = x₁x₂. . . x_i−1x_ix_i+1. . . x_nif x_i−1 = 1.

Case 1: i= 2.

Suppose that x₁ = 0. Then, N₁(X) = 1x₂x₃. . . x_n and N₁(Y ) = 1x₂x₃. . . x_n. By definition, d_MQ0

n(N₁(X), N₁(Y )) > 1. If x₂ = 0, N1(Y ) + E3 = 1x2 x₃. . . x_n. Hence N₁(Y ) + E3 + E2 = 1x2x₃. . . x_n. Hence d_MQ0n(N1(X), N1(Y )) = 2. If x2 = 1, N1(X) + E3 = 1x2x₃. . . x_n. Hence N₂(X) + E3 + E2 = 1x2x₃. . . x_n. Therefore, d_MQ0n(N1(X), N1(Y )) = 2.

Suppose that x₁ = 1. Then, N₁(X) = 0x₂x₃. . . x_n and N₁(Y ) = 0x₂x₃. . . x_n. By definition, d_MQ0

n(N₁(X), N₁(Y )) > 1. If

x₂ = 0, N₁(Y ) + E₃ = 0x₂x₃. . . x_n. Hence N₁(Y ) + E3 + e2 = 0x2x₃. . . x_n. Hence d_MQ0n(N1(X), N1(Y )) = 2. If x2 = 1, N1(X) + E3 = 0x2x₃. . . x_n. Hence N₁(X) + E3 + e2 = 0x2x₃. . . x_n. Therefore, d_MQ0n(N₁(X), N₁(Y )) = 2.

Case 2: 3 ≤ i ≤ n.

Suppose that x_i−1 = 0. N₁(X) = x₁x₂. . . x_i−20x_i. . . x_n and N₁(Y ) = x₁x₂ . . . x_i−20x_ix_i+1. . . x_n. It is obvious that N₁(Y )+

e_i = N₁(X). Hence d_MQ⁰_n(N₁(X), N₁(Y )) = 1.

Suppose that x_i−1 = 1. N₁(X) = x₁x₂. . . x_i−21xi. . . x_n and N₁(Y ) = x1x₂ . . . x_i−21xi . . . x_n. It is obvious that N₁(Y ) + E_i = N₁(X). Hence d_MQ⁰_n(N₁(X), N₁(Y )) = 1.

The lemma is proved.

Lemma 2 Let X be a vertex of M Q¹_nwith n≥ 3 and Y = Ni(X). Then dMQ¹_n(N1(X), N1(Y )) = 1 if i = n and d_MQ¹_n(N₁(X), N₁(Y )) = 2 if 2 ≤ i≤ n − 1.

Proof. Let X = x1x₂. . . x_i−1x_ix_i+1. . . x_n where x_j ∈ {0, 1} for 1 ≤ j ≤ n.

Since Y is an ith neighbor of X, Y = x₁x₂. . . x_i−1x_ix_i+1. . . x_n if x_i−1 = 0 or Y = x₁x₂. . . x_i−1x_ix_i+1. . . x_nif x_i−1 = 1.

Case 1: 2 ≤ i ≤ n − 1.

Suppose that x_i−1 = 0. N₁(X) = x₁x₂ . . . x_i−21xi . . . x_n and N₁(Y ) = x1

x₂ . . . x_i−2 1x_ix_i+1 . . . x_n. By definition, d_MQ0

n(N₁(X), N₁(Y )) > 1. If x_i = 0, N₁(X) + E_i+1 = x₁x₂ . . . x_i−21x_ix_i+1 . . . x_n. Hence

N₁(X) + E_i+1 + E_i = x₁x₂ . . . x_i−21x_ix_i+1 . . . x_n. Hence d_MQ0n(N1(X), N1(Y )) = 2. If x_i = 1, N1(Y ) + Ei+1 = x1x₂x₃ . . . x_i−21xix_i+1 . . . x_n. Hence N₁(Y ) + Ei+1 + Ei = N1(X).

Therefore, d_MQ0n(N1(X), N1(Y )) = 2.

Suppose that x_i−1 = 1. N₁(X) = x₁x₂ . . . x_i−20x_i . . . x_n and N₁(Y ) = x₁x₂ . . . x_i−2 0xix_i+1. . . x_n. By definition, d_MQ0n(N1(X), N1(Y )) > 1. If x_i = 0, N₁(X) + Ei+1 = x1x₂ . . . x_i−20xix_i+1. . . x_n. Hence N₁(X) + Ei+1 + ei = N1(Y ). Hence d_MQ0n(N1(X), N1(Y )) = 2. If xi = 1, N1(Y ) + E_i+1 = x1x₂ . . . x_i−20xix_i+1. . . x_n. Hence N₁(Y ) + E_i+1 + e_i = N₁(X). Therefore, d_MQ0

n(N₁(X), N₁(Y )) = 2.

Case 2: i= n.

N₁(X) = x₁x₂. . . x_n−2x_n−1x_n and N₁(Y ) = x₁x₂ . . . x_n−2x_n−1x_n. It is ob- vious that N₁(Y ) + e_n = N₁(X). Hence d_MQ0

n(N₁(X), N₁(Y )) = 1.

Cull and Larson [2] proposed an algorithm to generate a minimal expansion of a vector of X using only components x_i through x_n as S(X, i) for1 ≤ i ≤ n.

Algorithm S(X, i)

Input: A vector X and an integer i with1 ≤ i ≤ n.

Output: A minimal expansion of X using com- ponents x_i through x_n.

begin

if X = () then return empty set.

if X = (1) then return {Ei}.

if X = (0X^) then return S(X^, i+ 1).

if X = (10X^) then return {e_i}∪S(X^, i+ 2).

if X = (11X^) then return {Ei}∪S(X^, i+ 2).

end

Lemma 3 [2] The ”greedy” algorithm given in

correctly produces a minimal expansion of X, by computing S(X, 1).

Lemma 4 [2] Let X and Y be two vertices of M Q_n, and S be a minimal expansion of X + Y produced by the ”greedy” minimal expansion al- gorithm. Then d_MQn(X, Y ) = |S| or |S| + 1.

Lemma 5 Let X and Y be two distinct vertices in M Q_n. Then d_MQn(X, Y ) = d_MQn(N₁(X), N₁(Y )) ±k where k = 0, 1.

Proof. Let X = x₁x₂. . . x_n and Y = y₁y₂. . . y_n. Also let S be a minimal expansion of X+ Y produced by the ”greedy” minimal ex- pansion algorithm. It is observed that e₁ and E₁ doesn’t be contained in S. Suppose that X and Y are in M Q⁰_n. Hence N₁(X) = x1x₂. . . x_n and N₁(Y ) = y₁y₂. . . y_n. Suppose that X and Y are in M Q¹_n. First neighbors of X and Y are N₁(X) = x₁x₂. . . x_nand N₁(Y ) = y₁y₂. . . y_n, respectively. One may observe that X + Y = N₁(X) + N₁(Y ). Consequently, S is a minimal

expansion of N₁(X) + N₁(Y ). By Lemma 4, we have that d_MQn(X, Y ) = |S| or dMQn(X, Y ) =

|S| + 1, and dMQn(N1(X), N1(Y )) = |S| or d_MQn(N1(X), N1(Y )) = |S| + 1. Therefore, d_MQn(X, Y ) = dMQn(N1(X), N1(Y )) ± k where k = 0, 1.

Lemma 6 Let X and Y be two vertices in the

same sub-M¨obius M Qⁱ_n−1of M Q_nwith i= 0, 1.

Then every shortest path P_s(X, Y ) joining X and Y in M Q_nsatisfies that all vertices on P_s(X, Y ) belong to M Qⁱ_n−1.

Proof. Without loss of generality, we assume that X and Y are two vertices in M Q⁰_n−1. Let P_s(X, Y ) be a shortest path joining X and Y in M Q_n. Suppose that there exists a sub-path of P_s(X, Y ) in MQ¹_n−1. Let P_s(X, Y ) is formed by

X, Ps(X, U), U, W, Ps(W, Z), Z, S, Ps(S, Y ), Y  where W = N1(U), Z = N1(S), and Ps(W, Z) lies on M Q¹_n−1. Hence U, S ∈ V (MQ⁰_n−1).

Since the path P_s(X, Y ) is a short- est path joining X and Y in M Q_n, the path U, W, Ps(W, Z), Z, S is a shortest path between U, S in M Q_n. Therefore, d_MQn(U, S) = dMQn(W, Z) + 2. Since W = N1(U), Z = N1(S), and by Lemma 5, d_MQn(U, S) = dMQn(W, Z) ± k where k = 0, 1.

This is contradiction. Consequently, there is no sub-path of P_s(X, Y ) in MQ¹_n−1. The lemma follows.

Lemma 7 Let X ∈ V (MQⁱ_n−1) and Y ∈ V(MQ¹⁻ⁱ_n−1) be two vertices in MQ_n. Then there

exists a shortest path P_s(X, Y ) joining X and Y forms of X, N1(X), Ps(N1(X), Y ), Y  or X, P_s(X, N1(Y )), N1(Y ), Y  where Ps(X, N1(Y )) in M Qⁱ_n−1and P_s(N1(X), Y ) in MQ¹⁻ⁱ_n−1.

Proof. Let S be a minimal expansion of X + Y produced by the ”greedy” minimal expansion algorithm. Assume that the lowest index term in S has index i. It is clearly that routing along any edge (X, X + tj), j > i doesn’t affect bit x_i and routing along any edge (X, X + tj), j < i doesn’t lead to minimal path from X to Y where t_j ∈ S. So the shortest path algorithm must eventually rout along only one of the edges (Z, Z + ei) or (Z, Z + Ei) for some vertex Z on the path between X and Y . Since X ∈ V (MQⁱ_n−1) and Y ∈ V (MQ¹⁻ⁱ_n−1), X = x₁x₂x₃. . . x_n and Y = x₁y₂y₃. . . y_n. Hence the lowest index term of S has index 1.

Therefore, an exact minimal routing algorithm given in [2] can determine a shortest path P_s(X, Y ) between X and Y such that Ps(X, Y ) forms of X, N1(X), Ps(N1(X), Y ), Y  or

X, Ps(X, N1(Y )), Y  where Ps(X, N1(Y )) in M Qⁱ_n−1 and P_s(N1(X), Y ) in MQ¹⁻ⁱ_n−1. The lemma follows.

A graph G is panconnected if each pair of distinct vertices X and Y are joined by a path of length l where d_G(X, Y ) ≤ l ≤ |V (G)| − 1. The following panconnected property of MQn

are useful in the proof of next section.

Lemma 8 [11] If n≥ 3 then for any two distinct

vertices X and Y in M Q_n, there exists a path of every length from d_MQn(X, Y ) + 2 to 2ⁿ− 1.

The diameter D(G) of G is the maximal value of distances between all pairs of vertices in G. It is clearly that D(MQ₃) = 2.

Lemma 9 [2] The diameter of the n- dimensional M ¨obius cube M Q_n is D(MQ⁰_n) =

ⁿ⁺²₂ for n ≥ 4 and D(MQ¹_n) = ⁿ⁺¹₂ for n≥ 1.

3 M Q

is geodesic 3-pancyclic

Definition 2 Let G be a graph. For two vertices X, Y ∈ V (G), a cycle is called a geodesic cycle with X and Y if a shortest path joining X and Y lies on the cycle. A geodesic l-cycle with X and Y in G, denoted by gC^l(X, Y ; G), is a geodesic cycle of length l.

Definition 3 Let G be a graph. For two ver- tices X, Y ∈ V (G), they are called geodesic k- pancyclic on X and Y if for every integer l satis- fying2dG(X, Y ) + k ≤ l ≤ |V (G)|, the geodesic cycle gC^l(X, Y ; G) exists.

Definition 4 The graph G is called geodesic k-

pancyclic if any distinct two vertices on G are geodesic k-pancyclic on them. The geodesic- pancyclicity of G, denoted by gpc(G), is defined as the minimum integer k such that G is geodesic k-pancyclic.

This section is dedicated to illustrating the geodesic pancyclic property of M¨obius

cubes. We first propose that M Q₃ is geodesic 2-pancyclic. Finally, we prove that 2 ≤ gpc(MQn) ≤ 3 for n ≥ 4.

Lemma 10 M Q₃is geodesic 2-pancyclic.

Proof. Since M Q⁰₃ and M Q¹₃ are iso- morphic, we only prove the case of M Q⁰₃. Since M Q₃ is vertex-transitive. We assume that X = 000 and consider Y as the four cases: (1) Y ∈ {100, 010}, (2) Y = 001, (3) Y = {110, 111}, and (4) Y ∈ {101, 011}. By the symmetry of M Q₃, there is only one vertex is discussed for each case and related geodesic cycles are listed as Table 1.

Theorem 1 M Q_n is geodesic 3-pancyclic for n≥ 3.

Proof. The theorem is proved by induc- tion on n. By Lemma 10, M Q₃ is geodesic 2- pancyclic. This implies that M Q₃ is geodesic 3- pancyclic. The theorem holds for n = 3. As- sume that the theorem is true for every integer 3 ≤ m < n. We now consider m = n as follows.

Let X and Y be two vertices in M Q_n. By the relative position of X and Y , the proof is divided into two parts: (1) X and Y are in the same sub- M¨obius M Qⁱ_n−1 and (2) X ∈ V (MQⁱ_n−1) and Y ∈ V (MQ¹⁻ⁱ_n−1) for i = 0, 1.

Case 1: X, Y ∈ V (MQⁱ_n−1) for i = 0, 1.

Let d_MQn(X, Y ) = d. Without loss of generality, we may assume that X, Y ∈ V(MQ⁰_n−1). By the induction hypothesis, we

X Y

N (X)₁ N (Y)₁

P_s(X,Y)

P_c(N₁(Y),N₁(X))

MQ⁰n-1 MQ¹_n-1 (a)

W P (X,Y)_s

X

P (W,X)c

N (Y)₁

P (N (Y),N (W))_{c 1 1}

MQ_n-1⁰ MQ¹_n-1 Y

N (W)₁

(b)

Figure 2. Two examples for case 1 of Theorem 1

have the geodesic cycle gC^l(X, Y ; MQ⁰_n−1) for all2d_MQ⁰_n−1(X, Y )+3 ≤ l ≤ 2ⁿ⁻¹. By Lemma 6, d_MQn(X, Y ) = d_MQ⁰_n−1(X, Y ) = d. There- fore, the geodesic cycle gC^l(X, Y ; MQ_n) for all 2d + 3 ≤ l ≤ 2ⁿ⁻¹follows.

We now construct the geodesic cycle gC^l(X, Y ; MQn) for all 2ⁿ⁻¹ + 1 ≤ l ≤ 2ⁿ⁻¹. By Lemma 5, d_MQ1

n−1(N₁(X), N₁(Y )) = d_MQn(N₁(X), N₁(Y )) = d or d + 1. By Lemma 8, there exists a path of N₁(Y ) ,P_c(N₁(Y ), N₁(X)), N₁(X) in MQ¹_n−1 where d + 3 ≤ l(Pc(N1(Y ), N1(X)) ≤ 2ⁿ⁻¹ − 1.

Let cycle C = X, Ps(X, Y ), Y , N1(Y ), P_c(N1(Y ), N1(X)), N1(X), X. Then, 2d + 5 ≤ l(C) ≤ 2ⁿ⁻¹ + d + 1. Since d ≤ D(MQn−1), 2d + 5 ≤ 2ⁿ⁻¹ + 1 for n ≥ 4. Therefore, the geodesic cycle gC^l(X, Y ; MQn) exists where 2ⁿ⁻¹+ 1 ≤ l ≤ 2ⁿ⁻¹+ d + 1. (See Figure 2 (a).) It is difficult to prove that the geodesic cycle gC⁵(X, Y : MQ3) exists for any X, Y in M Q₃. Since 2ⁿ − 3 > 2 × D(MQn) + 3 for n ≥ 4, there exists the geodesic cy- cle gC²ⁿ⁻³(X, Y ; MQ_n) on any two distinct vertices X and Y in M Q_n for n ≥ 3.

Let gC²ⁿ⁻¹⁻³(X, Y ; MQ⁰_n−1)=X, P_s(X, Y ), Y ,

Table 1. Summary of the geodesic cycles with X = 000 and Y in MQ⁰₃. Y geodesic cyclic (even length) geodesic cyclic (odd length) 100 000, 100, 101, 001, 000 000, 100, 111, 011, 010, 000

100 000, 100, 111, 110, 101, 001, 000 000, 100, 101, 110, 111, 011, 001, 000

100 000, 100, 111, 011, 001, 101, 110, 010, 000

001 000, 001, 011, 010, 000 000, 001, 011, 111, 100, 000

001 000, 001, 011, 111, 110, 010, 000 000, 001, 011, 111, 110, 101, 100, 000

001 000, 001, 011, 111, 100, 101, 110, 010, 000

110 000, 010, 110, 111, 100, 000

110 000, 010, 110, 111, 011, 001, 000 000, 010, 110, 111, 100, 101, 001, 000

110 000, 010, 110, 101, 001, 011, 111, 100, 000

011 000, 001, 011, 010, 000 000, 001, 011, 111, 100, 000

011 000, 001, 011, 111, 110, 010, 000 000, 001, 011, 111, 110, 101, 100, 000

011 000, 001, 011, 111, 100, 101, 110, 010, 000

P_c(Y, X), X where l(Ps(X, Y )) = d. Let W and Y be two adjacent vertices on P_c(Y, X).

Hence P_c(Y, X) = Y, W, Pc(W, X), X where W = Nj(Y ) for some j. Since W = N_j(Y ) for some 2 ≤ j ≤ n and by Lemma 5, d_MQn(N1(Y ), N1(W )) ≤ 2. By Lemma 8, there exists a path of N₁(Y ), P_c(N₁(Y ), N₁(W )), N₁(W ) in MQ¹_n−1 where 4 ≤ l(P_c(N₁(Y ), N₁(W ))) ≤ 2ⁿ⁻¹ − 1.

Let cycle C = X, P_s(X, Y ), Y, N₁(Y ), P_c(N1(Y ), N1(W )), N1(W ), W, Pc(W, X), X.

Then 2ⁿ⁻¹ + 3 ≤ l(C) ≤ 2ⁿ − 3. Since d ≥ 1, we have the geodesic cycle gC^l(X, Y ; MQn) for all 2ⁿ⁻¹ + d + 2 ≤ l ≤ 2ⁿ − 3 with format C. (See Figure 2 (b).) Similarly, the geodesic cycle gC²ⁿ(X, Y ; MQn) for all 2ⁿ − 2 ≤ l ≤ 2ⁿ may be obtained if the geodesic cycle gC²ⁿ⁻¹(X, Y ; MQ⁰_n−1) is used in the con- struction method. Hence, this case holds.

Case 2: X ∈ V (MQⁱ_n−1) and Y ∈ V (MQ¹⁻ⁱ_n−1) for i= 0, 1.

Without loss of generality, let X ∈ V(MQ⁰_n−1) and Y ∈ V (MQ¹_n−1). According to relationship of X and Y , the proof of this case is divided into two parts: (1) Y = N1(X), i.e., X and Y are adjacent. (2) Y = N1(X), i.e., X and Y are not adjacent.

Subcase 2.1 Y = N₁(X).

By Lemma 1-2, N_n(Y ) and N_n(X) are adjacent. By Lemma 8, any path of N_n(Y ), P_c(N_n(Y ), Y ), Y  exists in MQ¹_n−1where1, 3 ≤ l(P_c(N_n(Y ), Y ) ≤ 2ⁿ⁻¹ − 1 and there exists a path of X, Pc(X, Nn(X)), Nn(X) in MQ⁰_n−1 where1, 3 ≤ l(Pc(X, Nn(X)) ≤ 2ⁿ⁻¹ − 1. Let cycle C = X, Pc(X, Nn(X)), Nn(X), Nn(Y ), P_c(Nn(Y ), Y ), Y, X. Then 4, 6 ≤ l(C) ≤ 2ⁿ. By Lemma 1-2, d_MQn(Y, N1(N2(X)) = 2, the geodesic cycle gC⁵(X, Y ; MQn) can be found. Hence, we have the geodesic cycle gC^l(X, Y ; MQ_n) for all 4 ≤ l ≤ 2ⁿ.

Subcase 2.2 Y = N₁(X).

By Lemma 7, without loss of general-

X

Y N (X)₁

N (Y)₁

P_s(X,Y) P_c(N₁(Y),X)

MQ⁰n-1 MQ¹_n-1 (a)

W X

Y N (X)1

P_s(N₁(X),Y) P_c(Y,W) N (W)₁

P_c(N₁(W),X)

MQ⁰_n-1 (b) MQ¹_n-1

Figure 3. Two examples for subcase 2.2 of Theo- rem 1

ity, we may assume there exists a shortest path P_s(X, Y ) with format of X, N1(X), P_s(N1(X), Y ), Y  where Ps(N1(X), Y ) is a shortest path joining N₁(X) and Y in MQ¹_n−1 (See Figure ?? (a)). Let d_MQn(X, Y ) = d. Hence d_MQn(N1(X), Y ) = d − 1. By Lemma 5, we have that d_MQn(X, N1(Y )) = d − 1 or d. By Lemma 8, there exists a path P_c(N₁(Y ), X) in M Q⁰_n−1 where d + 2 ≤ l(P_c(N₁(Y ), X)) ≤ 2ⁿ⁻¹ − 1. Let cycle C = X, P_s(X, Y ), Y , N₁(Y ), Pc(N1(Y ), X), X. Then, 2d + 3 ≤ l(C) ≤ 2ⁿ⁻¹+ d. Consequently, there exists the geodesic cycle gC^l(X, Y ; MQn) for all 2d + 3 ≤ l ≤ 2ⁿ⁻¹+ d with format C.

We now construct the geodesic cycle gC^l(X, Y ; MQ_n) for all 2ⁿ⁻¹+ d + 1 ≤ l ≤ 2ⁿ (See Figure ?? (b)). By the induction hypothe- sis, the geodesic cycle gC^l(N1(X), Y ; MQ¹_n−1) for all 2(d − 1) + 3 ≤ l ≤ 2ⁿ⁻¹ exits. It is ob- served that for any two distinct vertices A, B in M Q_nwith n≥ 3, the cycle gC²ⁿ⁻²(A, B; MQn) exists. Let gC²ⁿ⁻¹⁻²(N1(X), Y ; MQ¹_n−1) =

N₁(X), P_s(N₁(X), Y ), Y , P_c(Y, N₁(X)), N₁(X). Let W be the adjacent vertex of N₁(X) on P_c(Y, N₁(X)). Hence P_c(Y, N₁(X))

= Y , P_c(Y, W ), W , N₁(X). By Lemma 1- 2, d_MQn(X, N1(W )) ≤ 2. By Lemma 8, there exists a path P_c(N1(W ), X) in MQ⁰_n−1 where 4 ≤ l(Pc) ≤ 2ⁿ⁻¹ − 1. Let cycle C = X, N1(X), Ps(N1(X), Y ), Y, Pc(Y, W ), W, N₁(W ), P_c(N₁(W ), X), X. Then the length of cycle C is l(Ps(N1(X), Y )) + l(Pc(Y, W )) + l(P_c(N₁(W ), X)) + 2. It is obvious that 2ⁿ⁻¹ + 3 ≤ l(C) ≤ 2ⁿ − 2. Since d ≥ 2, there exists the geodesic cycle gC^l(X, Y ; MQn) for all 2ⁿ⁻¹ + d + 1 ≤ l ≤ 2ⁿ − 2. Sim- ilarly, the geodesic cycle gC²ⁿ⁻¹(X, Y ; MQn) and gC²ⁿ(X, Y ; MQn) may be obtained if the geodesic cycle gC²ⁿ⁻¹(N1(X), Y ; MQ¹_n−1) is used in this construction method. Hence, this case holds.

It is well known that there is no triangle cy- cle in M Q_n. Therefore, there is no geodesic 1- cycle with two adjacent vertices in M Q_n. Hence gpc(MQn) ≥ 2. Then the following corollary holds.

Corollary 1 The geodesic-pancyclicity of M Q_n

is gpc(MQ₃) = 2 and 2 ≤ gpc(MQ_n) ≤ 3 for n≥ 4.

4 Conclusions

In this paper, we demonstrate that for any two vertices X and Y in M Q_n for n ≥ 3, there exists a geodesic l-cycle on them where 2d_MQn(X, Y ) + 3 ≤ l ≤ 2ⁿ. We show that 2 ≤ gpc(MQ_n) ≤ 3 for n ≥ 3. This result is near optimal because there is no geodesic 1-cycle

with two adjacent vertices in M Q_n. We have a conjecture that gpc(MQn) = 2 because not all pair of vertices X and Y does not exist a path of length d_MQn(X, Y ) + 1 between them.

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