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Geodesic pancyclicity and balanced pancyclicity of Augmented cubes
Information Processing Letters 101 (2007) 227–232 www.elsevier.com/locate/ipl
Geodesic pancyclicity and balanced pancyclicity of Augmented cubes ✩ Hong-Chun Hsu a,∗ , Pao-Lien Lai b , Chang-Hsiung Tsai c a Department of Medical Informatics, Tzu Chi University, Hualien, Taiwan 970, ROC b Department of Computer Science and Information Engineering, National Dong Hwa University, Shoufeng, Hualien, Taiwan 97401, ROC c Department of Computer and Information Science, National Hualien University of Education, Hualien, Taiwan 970, ROC
Received 8 September 2006; received in revised form 26 October 2006; accepted 27 October 2006 Available online 11 December 2006 Communicated by L. Boasson
Abstract For two distinct vertices u, v ∈ V (G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC (u, v) = max{dC (x, y) | x, y ∈ V (C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271–278] if it contains a cycle of every length from 3 to |V (G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181–187] (respectively, balanced pancyclic) if for each pair of vertices u, v ∈ V (G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG (u, v), 3} l |V (G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251] proved that the n-dimensional Augmented cube, AQn , is pancyclic in the sense that a cycle of length l exists, 3 l |V (AQn )|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n 2. © 2006 Elsevier B.V. All rights reserved. Keywords: Augmented cubes; Panconnected; Pancyclic; Geodesic pancyclic; Balanced pancyclic; Interconnection networks
1. Introduction Interconnection networks play a major role in the performance of modern parallel computers. There are a lot of mutually conflicting demands in designing the topology of computer networks. It is almost impossible ✩ This work was supported in part by the National Science Council of the Republic of China under Contract NSC 94-2213-E-126-012. * Corresponding author. E-mail address: [email protected] (H.-C. Hsu).
0020-0190/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2006.10.013
to design a network which is optimum in all perspectives. Low latency and load balancing are two factors of all important issues. In order to get the low latency and load balancing for data transfer, one method is to transfer data in multiple disjoint paths. Obviously, if the paths have the same length, the transfer latency and load balancing are optimal. In general, the existence of multiple disjoint paths or short path(s) is beneficial to data transfer. In this paper, a network is represented as a loopless undirected graph. For the graph definitions and nota-
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tions, we follow [1]. Let G be a graph if V (G) is a finite set and E(G) is a subset of {(u, v) | (u, v) is an unordered pair of V }. We say that V (G) is the vertex set and E(G) is the edge set. Two vertices u and v are adjacent if (u, v) ∈ E(G). A path is a sequence of adjacent vertices, written as v0 , v1 , v2 , . . . , vm , in which all the vertices v0 , v1 , . . . , vm are distinct except possibly v0 = vm . We also write the path v0 , P , vm , where P = v0 , v1 , . . . , vm . The length of a path P , l(P ), is the number of edges in P . We also use (v0 , vm )i to represent the path joining v0 and vm with length i. Let u and v be two vertices of G. Let the distance between u and v, denoted by dG (u, v), be the length of the shortest path of G joining u and v. The diameter of G is the maximum distance between any pair of vertices of G. The path embedding problems have attracted much attention of research [8,11]. One of the most popular path embedding problem is Hamiltonian path problem. A path is a Hamiltonian path if its vertices are distinct and span V (G). A graph G is Hamiltonian connected if there exists a Hamiltonian path joining any two vertices of G. A graph G is k-fault Hamiltonian connected if G − F remains Hamiltonian connected for every F ⊂ V (G) ∪ E(G) with |F | k. Another path embedding problem deals with all the possible lengths of the paths. A graph G is panconnected if each pair of distinct vertices u, v are joined by a path of length l, dG (u, v) l |V (G)| − 1. A cycle is a path with at least three vertices such that the first vertex is the same as the last one. 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. One of the most popular problem is Hamiltonian problem. A Hamiltonian cycle of G is a cycle that traverses every vertex of G exactly once. A graph is Hamiltonian if it has a Hamiltonian cycle. A Hamiltonian graph G is k-fault Hamiltonian if G − F remains Hamiltonian for every F ⊂ V (G) ∪ E(G) with |F | k. Another ring embedding problem, which deals with all the possible lengths of the cycles, is investigated in a lot of interconnection networks [2,6,7,9–11]. In general, a graph is pancyclic if it contains a cycle of every length from 3 to |V (G)| inclusive. There are a lot of interconnection network topologies proposed in literature. Among these topologies, the n-dimensional hypercube, denoted by Qn , is a popular one. Several variations of the hypercubes have been investigated to improve the efficiency of the hypercubes [1,2]. The Augmented cube AQn , recently proposed by Choudum and Sunitha [3], is one of such variations. For any positive integer n, AQn is a vertex transitive,
(2n − 1)-regular, and (2n − 1)-connected graph with 2n vertices. Recently, Chan et al. [2] proposed the geodesic pancyclic embedding problem of a graph G. In this paper, we consider the geodesic and balanced cycle embedding problems in AQn . Herein, we will prove that AQn is geodesic pancyclic and balanced pancyclic for n 2. The rest of this paper is organized as follows. In next section, we study necessary definitions and discuss some useful properties of the augmented cubes. Section 3 shows that AQn is geodesic pancyclic. Section 4 proves that AQn is also balanced pancyclic. Finally, we conclude this paper. 2. Preliminaries Let n 1 be an integer. The graph of the n-dimensional Augmented cube [3], denoted by AQn , has 2n vertices, each labeled by an n-bit binary string V (AQn ) = {un−1 un−2 . . . u1 u0 | ui ∈ {0, 1} for all 0 i n − 1}. AQ1 is the complete graph K2 with vertex set {0, 1}. For n 2, AQn can be recursively constructed by two copies of AQn−1 , denoted by AQ0n and AQ1n , and by adding 2n edges between AQ0n and AQ1n as follows: Let V (AQ0n ) = {0un−2 un−3 . . . u0 | ui ∈ {0, 1} for 0 i n − 2} and V (AQ1n ) = {1vn−2 vn−3 . . . v0 | vi ∈ {0, 1} for 0 i n − 2}. A vertex u = 0un−2 un−3 . . . u0 of AQ0n is joined to a vertex v = 1vn−2 vn−3 . . . v0 of AQ1n if and only if either (i) ui = vi , for 0 i n − 2; in this case, (u, v) is called a hypercube edge and we set v = uh , or (ii) ui = 1 − vi , for 0 i n − 2; in this case, (u, v) is called a complement edge and we set v = uc . The Augmented cubes AQ1 , AQ2 , and AQ3 are illustrated in Fig. 1. It is proved in [3] that AQn is a vertex transitive, (2n − 1)-regular, and (2n − 1)-connected graph with 2n vertices for any positive integer n. Lemma 1. [3] The diameter of AQn is n/2 . Lemma 2. [3] Suppose that two distinct vertices u and v are both in AQbn with b ∈ {0, 1}. Then dAQn (u, v) = dAQb (u, v). n
Lemma 3. [3] Suppose that the vertex u in AQbn and the vertex v in AQn1−b with b ∈ {0, 1}. Then there exist two shortest paths P1 and P2 such that all vertices of P1 in AQbn except v and all vertices of P2 in AQn1−b except u when d(u, v) 2 and n 3.
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Fig. 1. AQn for n = 1, 2, 3.
Lemma 4. Suppose that (u, v) is an edge of AQn such that u is in AQ0n and v is in AQ1n . Then the paths u, u , v and u, v , v exist where u in AQ1n and v in AQ0n . Proof. For convenience, we use x¯ to represent 1 − x for x ∈ {0, 1}. Suppose that u = un−1 un−2 . . . u1 u0 . Then v ∈ {u¯ n−1 un−2 . . . u1 u0 , u¯ n−1 u¯ n−2 . . . u¯ 1 u¯ 0 }. Without loss of generality, assume that v = u¯ n−1 un−2 . . . u1 u0 . Obviously, u = u¯ n−1 u¯ n−2 . . . u¯ 1 u¯ 0 and v = un−1 u¯ n−2 . . . u¯ 1 u¯ 0 are common neighbors of u and v. 2 Lemma 5. [3] AQn is pancyclic for n 2. Lemma 6. [5] AQn is panconnected for n 1. Theorem 1. [4] AQn is (2n − 3)-fault Hamiltonian and (2n − 4)-fault Hamiltonian connected for n 2. 3. Geodesic pancyclicity of Augmented cubes This section is devoted to illustrating the geodesic pancyclic property of Augmented cubes. Next, geodesic cycle and geodesic pancyclic are formally defined and discussed. Then AQn , n 2, is demonstrated to be geodesic pancyclic. Definition 1. Let G be a graph. For two vertices u, v ∈ V (G), a cycle is called a geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle. A geodesic l-cycle with u and v in G, denoted by gC l (u, v; G), is a geodesic cycle of length l. Let C = u, Ps , v, Pc , u be a geodesic cycle with vertices u and v, where Ps and Pc are two disjoint paths joining u and v on C with l(Ps ) = dAQn (u, v) and l(Ps ) l(Pc ). We call Ps and Pc as s-path and c-path of C, respectively. Definition 2. Let G be a graph. For two vertices u, v ∈ V (G), they are called geodesic pancyclic on u and v if
for every integer l satisfying max{2dG (u, v), 3} l |V (G)|, the geodesic cycle gC l (u, v; G) exists. Definition 3. Let G be a graph. G is called geodesic pancyclic if any distinct two vertices on G are geodesic pancyclic on them. Lemma 7. AQ2 and AQ3 are geodesic pancyclic. Proof. Since AQ2 is the complete graph K4 , AQ2 is geodesic pancyclic. Since AQ3 is vertex-transitive. We assume the starting vertex is u = 000 and consider the destination vertex v as the three cases: (1) v ∈ {001, 010, 011, 100}, (2) v = 111, and (3) v ∈ {101, 110}. By the symmetry of AQ3 , there is only one vertex discussed for each case and related geodesic cycles of the three cases are listed in Table 1. 2 Theorem 2. AQn is geodesic pancyclic for n 2. Proof. We prove this theorem by induction on n. By Lemma 7, this theorem holds for n = 2, 3. Assume that this theorem is true for AQm with every 3 m < n. Then we consider AQn . Let u, v be any two distinct vertices of AQn . By the relative position of u and v, we have the following cases. Case 1. u, v ∈ V (AQbn ) with b ∈ {0, 1}. Without loss of generality, assume that u, v ∈ V (AQ0n ). By induction hypothesis, AQ0n is geodesic pancyclic. Hence, all the cycles gC l (u, v; AQn ) with max{2d(u, v), 3} l 2n−1 exist. Then we divide the remainder into l = 2n−1 + 1 and 2n−1 + 2 l 2n . n−1 Subcase 1.1. l = 2n−1 + 1. Clearly, gC 2 −1 (u, v; 0 AQn ) = u, Ps , v, Pc , u exists. Let Ps be the s-path n−1 and Pc be the c-path of gC 2 −1 (u, v; AQ0n ). Then, l(Ps ) = dAQ0 (u, v) and l(Pc ) = 2n−1 − dAQ0 (u, v) − 1. n
n
Since dAQ0 (u, v) (n − 1)/2 , l(Pc ) 2n−1 − (n − n 1)/2 − 1 2 when n 3. Let (v, w) be an edge on Pc . Therefore, we write Pc as v, w, Pc , u where l(Pc ) 1. By the definition of AQn , (v h , w h ) is an edge of AQ1n . Let C = u, Ps , v, v h , w h , w, Pc , u. Since
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Table 1 Summary of the geodesic cycles with u = 000 and v in AQ3 (geodesic cycle) (even length)
v 001 001 001 001 101 101 101 111 111 111 111
(geodesic cycle) (odd length) (000, 001, 010, 000) (000, 001, 101, 110, 010, 000) (000, 001, 110, 101, 111, 011, 010, 000)
(000, 001, 011, 010, 000) (000, 001, 101, 111, 011, 010, 000) (000, 001, 110, 111, 100, 101, 010, 011, 000) (000, 001, 101, 010, 000) (000, 001, 101, 111, 011, 010, 000) (000, 001, 101, 100, 110, 111, 011, 010, 000)
(000, 001, 101, 110, 010, 000) (000, 001, 101, 110, 111, 011, 010, 000) (000, 111, 011, 000) (000, 111, 110, 101, 001, 000) (000, 111, 110, 100, 101, 001, 011, 000)
(000, 111, 110, 010, 000) (000, 111, 110, 101, 100, 011, 000) (000, 111, 110, 100, 101, 001, 011, 010, 000)
dAQn (u, v) = dAQ0 (u, v), l(C) = dAQn (u, v)+3+2n−1 − n
dAQn (u, v) − 2 = 2n−1 + 1. Thus, C is a geodesic cycle n−1 gC 2 +1 (u, v; AQn ). Subcase 1.2. 2n−1 + 2 l 2n . Base on the previn−1 ous subcase, we have the geodesic cycle gC 2 +1 (u, v; AQn ) = u, Ps , v, w, w h , v h , v, Pc , u. Since AQ1n is panconnected, there exist the (v h , w h )i paths with 1 i 2n−1 −1 in AQ1n . Let C = u, Ps , v, w, w h , (w h , v h )i , v h , v, Pc , u. Since dAQn (u, v) = dAQ0 (u, v), l(C) = n
dAQn (u, v) + 2 + i + 1 + 2n−1 − dAQn (u, v) − 2 = 2n−1 + 1 + i. Hence, we can find C with 2n−1 + 2 l(C) 2n . Therefore, C is a format of the geodesic cycles gC l (u, v; AQn ) for 2n−1 + 2 l 2n . Case 2. u ∈ V (AQbn ) and v ∈ V (AQn1−b ) with b ∈ {0, 1}. Without loss of generality, assume that u ∈ V (AQ0n ) and v ∈ V (AQ1n ). Subcase 2.1. dAQn (u, v) = 1. By Lemma 4, there exists a path u, w, v such that w is a neighbor of u in AQ0n . Since AQ0n is panconnected, there exist the paths (u, w)i with 1 i 2n−1 − 1 in AQ0n . Thus, we can construct (a) gC l (u, v; AQn ) for 3 l 2n−1 + 1 with the form u, (u, w)i , w, v, u with 1 i 2n−1 − 1. By the structure of AQn , there exists a neighbor x of w such that (x, v) ∈ E(AQ1n ). Since AQ1n is panconnected, there exist (x, v)i paths with 1 i 2n−1 − 1 in AQ1n . Thus, we can find the geodesic cycle (b) gC l (u, v; AQn ) for 2n−1 − 2 l 2n with the form u, (u, w)2n−1 −1 , w, x, (x, v)i , v, u for 1 i 2n−1 − 1. Clearly, with (a) and (b) we can find all geodesic cycles in this subcase. Subcase 2.2. dAQn (u, v) 2. By Lemma 3, there exist two disjoint shortest paths joining u and v with the form u, P1 , v , v and u, u , P2 , v such that all vertices of u, P1 , v are in AQ0n and all vertices of u , P2 , v are in AQ1n . Clearly, gC l (u, v; AQn ) with l = 2dAQn (u, v) exists. Then we consider the case of 2dAQn (u, v) + 1 l 2n .
By induction hypothesis, there exist gC l (u, v ; AQ0n ) with the form u, Ps , v , Pc , u for all max{2dAQ0 (u, v ), n
3} l 2n−1 where Ps is the s-path and Pc is the cpath. Herein, 2 max{2dAQ0 (u, v ), 3} − dAQ0 (u, v ) n
n
l(Pc ) 2n−1 − dAQ0 (u, v ). Thus, we can write Pc as n v , w, Pc , u. Since l(Pc ) = l(Pc ) − 1, max 2dAQ0 (u, v ), 3 − dAQ0 (u, v ) − 1 l Pc
n
n
2n−1 − dAQ0 (u, v ) − 1. n
By the structure of Augmented cube, one of {w h , w c } is an neighbor of v in AQ1n . Let w ∈ {w h , w c }, and (w 1 , v) is also an edge of AQ1n . By Lemma 6, there exist (w , v)i paths in AQ1n for all 1 i 2n−1 − 1. Let C = u, Ps , v , v, (v, w )i , w , w, Pc , u. Thus, l(C) = dAQ0n (u, v ) + 1 + i + 1 + l(Pc ). Since dAQn (u, v) = dAQ0 (u, v ) + 1 and max{2dAQ0 (u, v ), 3} − dAQn (u, v) n
n
l(Pc ) 2n−1 − dAQn (u, v), dAQn (u, v) + 1 + 1 + max 2dAQ0 (u, v ), 3 n
− dAQn (u, v) l(C) dAQn (u, v) + 2n−1 − 1 + 1 + 2n−1 − dAQn (u, v). Hence, max{2dAQ0 (u, v ), 3} + 2 2dAQn (u, v) + 1 n l(C) 2n . Hence, C is a form of the geodesic cycles gC l (u, v, AQn ) for all 2dAQn (u, v) + 1 l 2n . This completes the proof. 2 4. Balanced pancyclicity of Augmented cubes This section is devoted to illustrating the balanced pancyclic property of augmented cubes. First, we introduce the definitions of balanced cycle and balanced pancyclic. Then, AQn for n 2 is showed to be balanced pancyclic. Definition 4. Let G be a graph. For two vertices u, v ∈ V (G), a cycle C is called a balanced cycle with u and v
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Table 2 Some results of the balanced cycles with u = 000 and v in AQ3 v
balanced cycle
001 001 001 001 001 001 101 101 101 101 101 111 111 111 111 111 111
(000, 001, 010, 000) (000, 010, 001, 011, 000) (000, 010, 001, 101, 100, 000) (000, 010, 011, 001, 101, 100, 000) (000, 010, 011, 001, 110, 101, 100, 000) (000, 010, 110, 101, 001, 011, 111, 100, 000) (000, 001, 101, 010, 000) (000, 001, 101, 110, 010, 000) (000, 001, 010, 101, 111, 011, 000) (000, 001, 010, 101, 110, 111, 011, 000) (000, 001, 011, 010, 101, 111, 110, 100, 000) (000, 111, 011, 000) (000, 011, 111, 100, 000) (000, 011, 111, 110, 100, 000) (000, 010, 011, 111, 110, 100, 000) (000, 010, 011, 111, 110, 101, 100, 000) (000, 010, 001, 011, 111, 110, 101, 100, 000)
if dC (u, v) = max{dC (x, y) | x, y ∈ V (C)}. A balanced l-cycle with u and v in G, denoted by bC l (u, v; G), is a cycle of length l with dbC l (u,v;G) (u, v) = l/2 . Let C = u, bP0 , v, bP1 , u be a balanced cycle of length l with vertices u and v, where |bP0 | |bP1 |. If l is even, |bP0 | = |bP1 |; otherwise, |bP0 | = |bP1 | − 1. Herein, bP0 and bP1 are named as b0-path and b1-path of C, respectively. Definition 5. Let G be a graph. For two vertices u, v ∈ V (G), they are called balanced pancyclic on u and v if for every integer l satisfying max{2dG (u, v), 3} l |V (G)|, the balanced cycle bC l (u, v; G) exists. Definition 6. Let G be a graph. G is called balanced pancyclic if any distinct two vertices on G are balanced pancyclic on them. Lemma 8. AQ3 and AQ4 are both balanced pancyclic. Proof. We verify this small case directly using computer. However, we only list some results in this paper to reduce the space. Please check Tables 2 and 3. (Note that we use decimal number to represent the binary string in Table 3.) 2 Theorem 3. The n-dimensional Augmented cube is balanced pancyclic for n 2. Proof. We prove this theorem by induction on n. Since AQ2 is complete graph K4 , AQ2 is balanced pancyclic. By Lemma 8, the theorem holds for n = 3, 4. The in-
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Table 3 Some results of the balanced cycles with u = 0 and v = 1 in AQ4 (0, 2, 1, 3, 0) (0, 2, 1, 5, 4, 0) (0, 2, 3, 1, 5, 4, 0) (0, 2, 3, 1, 6, 5, 4, 0) (0, 2, 6, 5, 1, 9, 11, 3, 0) (0, 2, 6, 5, 1, 14, 15, 7, 3, 0) (0, 2, 3, 7, 5, 1, 9, 14, 6, 4, 0) (0, 2, 3, 7, 5, 1, 9, 13, 14, 6, 4, 0) (0, 2, 3, 7, 5, 6, 1, 9, 13, 14, 12, 4, 0) (0, 2, 3, 7, 5, 6, 1, 9, 13, 15, 14, 12, 4, 0) (0, 2, 3, 7, 5, 4, 6, 1, 14, 13, 10, 11, 9, 8, 0) (0, 2, 3, 7, 5, 4, 6, 1, 14, 15, 13, 10, 11, 9, 8, 0) (0, 2, 3, 7, 5, 4, 6, 14, 1, 9, 13, 15, 12, 11, 10, 8, 0)
duction hypothesis is that AQn−1 is balanced pancyclic. Then we consider the case of AQn . Let u and v be any two distinct vertices of AQn . Then we divide the cases by the relative position between u and v. Case 1: u, v ∈ AQbn for b ∈ {0, 1}. Without loss of generality, we can assume that u, v ∈ AQ0n . By induction hypothesis, AQn−1 is balanced pancyclic. Hence, there exist bC l (u, v; AQn ) with max{2dAQn (u, v), 3} l 22−1 . Then we consider the remainder into two subcases. Subcase 1.1. 2n−1 + 1 l 2n−1 . By Lemma 1, dAQ0 (u, v) (n−1)/2 . Hence, dAQ0 (u, v) < 2n−2 for n
n
n 3. Since AQ0n is panconnected, there exist (u, v)i paths for 2n−2 i 2n−1 − 1 in AQ0n . Also there exist (uh , v h )j paths for 2n−2 − 1 j 2n−1 − 2 in AQ1n . Suppose we choose j = i − 2 or j = i − 1. Let C = u, (u, v)i , v, v h , (uh , v h )j , uh , u. Hence, 2n−2 + 1 + 2n−2 − 1 + 1 = 2n−1 + 1 l(C) 2n−1 − 1 + 1 + 2n−1 − 2 + 1 = 2n − 1. The cycle C is a form of bC l (u, v; AQn ) for all 2n−1 + 1 l 2n − 1. See Fig. 2(a) for illustration. Subcase 1.2. l = 2n . Then we want to choose a neighbor x of v in AQ0n such that x must satisfy that there exists y ∈ {x h , x c } such that there exists w ∈ {y h , y c } with w ∈ / {u, x, v}. Let X be the set of the neighbors of of v in v in AQ0n . Since there are 2n − 3 7 neighbors AQ0n for n 5, |X| 7. Let Y be the set xi ∈X {xih , xic }. Clearly |Y | 7 also. Let W be the set yi ∈Y {yih , yic }. Therefore, |W | 7. Hence, we can choose the neighbor x of v in AQ0n such that there exist y ∈ {x h , x c } and w ∈ / {y h , y c } with w ∈ {u, x, v}. Since AQn−1 is k-fault Hamiltonian connected by Theorem 1 with k = 2n − 4 2 when n 3, there exists a Hamiltonian path (u, w)i of AQ0n − {x, v} with i = 2n−1 − 3, and there exists a Hamiltonian path (uh , v h )j of AQ1n − {y} with j = 2n−1 − 2. Then, we can construct a cycle D = u, (u, w)2n−1 −3 , w, y, x, v, v h , (uh , v h )2n−1 −2 , uh ,
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Fig. 2. The illustration for Case 1.
u. Since the path u, (u, w)2n−1 −3 , w, y, x, v has length 2n−1 − 3 + 1 + 1 + 1 = 2n−1 and the path v, v h , (uh , v h )2n−1 −2 , uh , u has length 1 + 2n−1 − 2 + 1 = 2n−1 , D is a balanced cycle bC 2n (u, v; AQn ). See Fig. 2(b) for illustration. Case 2. u ∈ AQbn and v ∈ AQn1−b with b ∈ {0, 1}. Without loss of generality, we can assume that u ∈ AQ0n and v ∈ AQ1n . Suppose that (u, v) is an edge of AQn . By Lemma 4, the paths u, v , v and v, u , u exist where v ∈ {v h , v c } and u ∈ {uh , uc }. Thus, u, v, v , u is a balanced cycle bC 3 (u, v; AQn ) and u, u , v, v , u is a balanced cycle bC 4 (u, v; AQn }. By Lemma 6, there exist the paths (v , u)i and (u , v)j for all 1 i 2n−1 − 1 and 1 j 2n−1 − 1. Suppose we choose i = j or i = j − 1. Clearly, u, u , (u , v)i , v, v , (v , u)j , u is a form of the balanced cycles bC l (u, v; AQn ) for all 3 l 2n . Suppose that (u, v) is not an edge of AQn . We know dAQn (u, v) 2 in this case. By Lemma 3, there exist two disjoint shortest paths of AQn joining u and v with the form u, (u, v )s , v , v and u, u , (u , v)s , v where all vertices of (u, v )s are in AQ0n and all vertices of (v, u )s are in AQ1n with s = dAQn (u, v) − 1 1. Since AQ0n is panconnected, there exist the paths (u, v )i with dAQ0 (u, v ) i 2n−1 − 1 in AQ0n . Similarly, there n
exist (v, u )j for dAQ1 (v, u ) j 2n−1 − 1 in AQ1n . n Suppose we choose i = j or i = j − 1. Let the cycle C = u, (u, v )i , v , v, (v, u )j , u , u. Hence, C is a form of the balanced cycles bC l (u, v; AQn ) for all 2dAQn (u, v) l 2n . This completes the proof. 2 5. Conclusions
In this paper, we study the existence of cycles with some requirements in AQn . For given two vertices u and v, the cycle is called geodesic cycle if it contains the shortest path between u and v. Clearly, a geodesic cycle can minimize the transmission delay from u to v. Herein, we prove that AQn is geodesic pancyclic.
We also discuss the balanced pancyclic property for load balancing in AQn . It is of interest to find a cycle passing through u and v such that lengths of two disjoint paths between u and v in this cycle are as equal as possible. AQn is shown to be balanced pancyclic. Acknowledgements The authors would like to thank the referees for their comments and suggestions. These comments and suggestions were very helpful for improving the quality of this paper. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North-Holland, New York, 1980. [2] H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesicpancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181– 187. [3] S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71–84. [4] H.C. Hsu, L.C. Chiang, J.J.M. Tan, L.H. Hsu, Fault hamiltonicity of augmented cubes, Parallel Computing 31 (2005) 131–145. [5] P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251. [6] T.K. Li, C.H. Tsai, J.J.M. Tan, L.H. Hsu, Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes, Information Processing Letters 87 (1) (2003) 107–110. [7] J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and Applications (1982) 271–278. [8] O. Ore, Hamiltonian connected graph, Journal de Mathématiques Pures et Appliquées 42 (1963) 121–127. [9] C.H. Tsai, H.C. Hsu, P.L. Lai, Geodesic and balanced bipancyclicity of hyper-cubes, Submitted for publication. [10] X.F. Yang, G.M. Megson, On the double-vertex-cycle-connectivity of crossed cubes, Parallel Algorithms and Applications 19 (2004) 11–17. [11] X.F. Yang, G.M. Megson, On the path-connectivity, vertexpancyclicity, and edge-pancyclicity of crossed cubes, Neural, Parallel & Scientific Computations 13 (2005) 107–118.
Geodesic pancyclicity and balanced pancyclicity of Augmented cubes ✩ Hong-Chun Hsu a,∗ , Pao-Lien Lai b , Chang-Hsiung Tsai c a Department of Medical Informatics, Tzu Chi University, Hualien, Taiwan 970, ROC b Department of Computer Science and Information Engineering, National Dong Hwa University, Shoufeng, Hualien, Taiwan 97401, ROC c Department of Computer and Information Science, National Hualien University of Education, Hualien, Taiwan 970, ROC
Received 8 September 2006; received in revised form 26 October 2006; accepted 27 October 2006 Available online 11 December 2006 Communicated by L. Boasson
Abstract For two distinct vertices u, v ∈ V (G), a cycle is called geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle; and a cycle C is called balanced cycle with u and v if dC (u, v) = max{dC (x, y) | x, y ∈ V (C)}. A graph G is pancyclic [J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and applications (1982) 271–278] if it contains a cycle of every length from 3 to |V (G)| inclusive. A graph G is called geodesic pancyclic [H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesic-pancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181–187] (respectively, balanced pancyclic) if for each pair of vertices u, v ∈ V (G), it contains a geodesic cycle (respectively, balanced cycle) of every integer length of l satisfying max{2dG (u, v), 3} l |V (G)|. Lai et al. [P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251] proved that the n-dimensional Augmented cube, AQn , is pancyclic in the sense that a cycle of length l exists, 3 l |V (AQn )|. In this paper, we study two new pancyclic properties and show that AQn is geodesic pancyclic and balanced pancyclic for n 2. © 2006 Elsevier B.V. All rights reserved. Keywords: Augmented cubes; Panconnected; Pancyclic; Geodesic pancyclic; Balanced pancyclic; Interconnection networks
1. Introduction Interconnection networks play a major role in the performance of modern parallel computers. There are a lot of mutually conflicting demands in designing the topology of computer networks. It is almost impossible ✩ This work was supported in part by the National Science Council of the Republic of China under Contract NSC 94-2213-E-126-012. * Corresponding author. E-mail address: [email protected] (H.-C. Hsu).
0020-0190/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ipl.2006.10.013
to design a network which is optimum in all perspectives. Low latency and load balancing are two factors of all important issues. In order to get the low latency and load balancing for data transfer, one method is to transfer data in multiple disjoint paths. Obviously, if the paths have the same length, the transfer latency and load balancing are optimal. In general, the existence of multiple disjoint paths or short path(s) is beneficial to data transfer. In this paper, a network is represented as a loopless undirected graph. For the graph definitions and nota-
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tions, we follow [1]. Let G be a graph if V (G) is a finite set and E(G) is a subset of {(u, v) | (u, v) is an unordered pair of V }. We say that V (G) is the vertex set and E(G) is the edge set. Two vertices u and v are adjacent if (u, v) ∈ E(G). A path is a sequence of adjacent vertices, written as v0 , v1 , v2 , . . . , vm , in which all the vertices v0 , v1 , . . . , vm are distinct except possibly v0 = vm . We also write the path v0 , P , vm , where P = v0 , v1 , . . . , vm . The length of a path P , l(P ), is the number of edges in P . We also use (v0 , vm )i to represent the path joining v0 and vm with length i. Let u and v be two vertices of G. Let the distance between u and v, denoted by dG (u, v), be the length of the shortest path of G joining u and v. The diameter of G is the maximum distance between any pair of vertices of G. The path embedding problems have attracted much attention of research [8,11]. One of the most popular path embedding problem is Hamiltonian path problem. A path is a Hamiltonian path if its vertices are distinct and span V (G). A graph G is Hamiltonian connected if there exists a Hamiltonian path joining any two vertices of G. A graph G is k-fault Hamiltonian connected if G − F remains Hamiltonian connected for every F ⊂ V (G) ∪ E(G) with |F | k. Another path embedding problem deals with all the possible lengths of the paths. A graph G is panconnected if each pair of distinct vertices u, v are joined by a path of length l, dG (u, v) l |V (G)| − 1. A cycle is a path with at least three vertices such that the first vertex is the same as the last one. 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. One of the most popular problem is Hamiltonian problem. A Hamiltonian cycle of G is a cycle that traverses every vertex of G exactly once. A graph is Hamiltonian if it has a Hamiltonian cycle. A Hamiltonian graph G is k-fault Hamiltonian if G − F remains Hamiltonian for every F ⊂ V (G) ∪ E(G) with |F | k. Another ring embedding problem, which deals with all the possible lengths of the cycles, is investigated in a lot of interconnection networks [2,6,7,9–11]. In general, a graph is pancyclic if it contains a cycle of every length from 3 to |V (G)| inclusive. There are a lot of interconnection network topologies proposed in literature. Among these topologies, the n-dimensional hypercube, denoted by Qn , is a popular one. Several variations of the hypercubes have been investigated to improve the efficiency of the hypercubes [1,2]. The Augmented cube AQn , recently proposed by Choudum and Sunitha [3], is one of such variations. For any positive integer n, AQn is a vertex transitive,
(2n − 1)-regular, and (2n − 1)-connected graph with 2n vertices. Recently, Chan et al. [2] proposed the geodesic pancyclic embedding problem of a graph G. In this paper, we consider the geodesic and balanced cycle embedding problems in AQn . Herein, we will prove that AQn is geodesic pancyclic and balanced pancyclic for n 2. The rest of this paper is organized as follows. In next section, we study necessary definitions and discuss some useful properties of the augmented cubes. Section 3 shows that AQn is geodesic pancyclic. Section 4 proves that AQn is also balanced pancyclic. Finally, we conclude this paper. 2. Preliminaries Let n 1 be an integer. The graph of the n-dimensional Augmented cube [3], denoted by AQn , has 2n vertices, each labeled by an n-bit binary string V (AQn ) = {un−1 un−2 . . . u1 u0 | ui ∈ {0, 1} for all 0 i n − 1}. AQ1 is the complete graph K2 with vertex set {0, 1}. For n 2, AQn can be recursively constructed by two copies of AQn−1 , denoted by AQ0n and AQ1n , and by adding 2n edges between AQ0n and AQ1n as follows: Let V (AQ0n ) = {0un−2 un−3 . . . u0 | ui ∈ {0, 1} for 0 i n − 2} and V (AQ1n ) = {1vn−2 vn−3 . . . v0 | vi ∈ {0, 1} for 0 i n − 2}. A vertex u = 0un−2 un−3 . . . u0 of AQ0n is joined to a vertex v = 1vn−2 vn−3 . . . v0 of AQ1n if and only if either (i) ui = vi , for 0 i n − 2; in this case, (u, v) is called a hypercube edge and we set v = uh , or (ii) ui = 1 − vi , for 0 i n − 2; in this case, (u, v) is called a complement edge and we set v = uc . The Augmented cubes AQ1 , AQ2 , and AQ3 are illustrated in Fig. 1. It is proved in [3] that AQn is a vertex transitive, (2n − 1)-regular, and (2n − 1)-connected graph with 2n vertices for any positive integer n. Lemma 1. [3] The diameter of AQn is n/2 . Lemma 2. [3] Suppose that two distinct vertices u and v are both in AQbn with b ∈ {0, 1}. Then dAQn (u, v) = dAQb (u, v). n
Lemma 3. [3] Suppose that the vertex u in AQbn and the vertex v in AQn1−b with b ∈ {0, 1}. Then there exist two shortest paths P1 and P2 such that all vertices of P1 in AQbn except v and all vertices of P2 in AQn1−b except u when d(u, v) 2 and n 3.
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Fig. 1. AQn for n = 1, 2, 3.
Lemma 4. Suppose that (u, v) is an edge of AQn such that u is in AQ0n and v is in AQ1n . Then the paths u, u , v and u, v , v exist where u in AQ1n and v in AQ0n . Proof. For convenience, we use x¯ to represent 1 − x for x ∈ {0, 1}. Suppose that u = un−1 un−2 . . . u1 u0 . Then v ∈ {u¯ n−1 un−2 . . . u1 u0 , u¯ n−1 u¯ n−2 . . . u¯ 1 u¯ 0 }. Without loss of generality, assume that v = u¯ n−1 un−2 . . . u1 u0 . Obviously, u = u¯ n−1 u¯ n−2 . . . u¯ 1 u¯ 0 and v = un−1 u¯ n−2 . . . u¯ 1 u¯ 0 are common neighbors of u and v. 2 Lemma 5. [3] AQn is pancyclic for n 2. Lemma 6. [5] AQn is panconnected for n 1. Theorem 1. [4] AQn is (2n − 3)-fault Hamiltonian and (2n − 4)-fault Hamiltonian connected for n 2. 3. Geodesic pancyclicity of Augmented cubes This section is devoted to illustrating the geodesic pancyclic property of Augmented cubes. Next, geodesic cycle and geodesic pancyclic are formally defined and discussed. Then AQn , n 2, is demonstrated to be geodesic pancyclic. Definition 1. Let G be a graph. For two vertices u, v ∈ V (G), a cycle is called a geodesic cycle with u and v if a shortest path of G joining u and v lies on the cycle. A geodesic l-cycle with u and v in G, denoted by gC l (u, v; G), is a geodesic cycle of length l. Let C = u, Ps , v, Pc , u be a geodesic cycle with vertices u and v, where Ps and Pc are two disjoint paths joining u and v on C with l(Ps ) = dAQn (u, v) and l(Ps ) l(Pc ). We call Ps and Pc as s-path and c-path of C, respectively. Definition 2. Let G be a graph. For two vertices u, v ∈ V (G), they are called geodesic pancyclic on u and v if
for every integer l satisfying max{2dG (u, v), 3} l |V (G)|, the geodesic cycle gC l (u, v; G) exists. Definition 3. Let G be a graph. G is called geodesic pancyclic if any distinct two vertices on G are geodesic pancyclic on them. Lemma 7. AQ2 and AQ3 are geodesic pancyclic. Proof. Since AQ2 is the complete graph K4 , AQ2 is geodesic pancyclic. Since AQ3 is vertex-transitive. We assume the starting vertex is u = 000 and consider the destination vertex v as the three cases: (1) v ∈ {001, 010, 011, 100}, (2) v = 111, and (3) v ∈ {101, 110}. By the symmetry of AQ3 , there is only one vertex discussed for each case and related geodesic cycles of the three cases are listed in Table 1. 2 Theorem 2. AQn is geodesic pancyclic for n 2. Proof. We prove this theorem by induction on n. By Lemma 7, this theorem holds for n = 2, 3. Assume that this theorem is true for AQm with every 3 m < n. Then we consider AQn . Let u, v be any two distinct vertices of AQn . By the relative position of u and v, we have the following cases. Case 1. u, v ∈ V (AQbn ) with b ∈ {0, 1}. Without loss of generality, assume that u, v ∈ V (AQ0n ). By induction hypothesis, AQ0n is geodesic pancyclic. Hence, all the cycles gC l (u, v; AQn ) with max{2d(u, v), 3} l 2n−1 exist. Then we divide the remainder into l = 2n−1 + 1 and 2n−1 + 2 l 2n . n−1 Subcase 1.1. l = 2n−1 + 1. Clearly, gC 2 −1 (u, v; 0 AQn ) = u, Ps , v, Pc , u exists. Let Ps be the s-path n−1 and Pc be the c-path of gC 2 −1 (u, v; AQ0n ). Then, l(Ps ) = dAQ0 (u, v) and l(Pc ) = 2n−1 − dAQ0 (u, v) − 1. n
n
Since dAQ0 (u, v) (n − 1)/2 , l(Pc ) 2n−1 − (n − n 1)/2 − 1 2 when n 3. Let (v, w) be an edge on Pc . Therefore, we write Pc as v, w, Pc , u where l(Pc ) 1. By the definition of AQn , (v h , w h ) is an edge of AQ1n . Let C = u, Ps , v, v h , w h , w, Pc , u. Since
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Table 1 Summary of the geodesic cycles with u = 000 and v in AQ3 (geodesic cycle) (even length)
v 001 001 001 001 101 101 101 111 111 111 111
(geodesic cycle) (odd length) (000, 001, 010, 000) (000, 001, 101, 110, 010, 000) (000, 001, 110, 101, 111, 011, 010, 000)
(000, 001, 011, 010, 000) (000, 001, 101, 111, 011, 010, 000) (000, 001, 110, 111, 100, 101, 010, 011, 000) (000, 001, 101, 010, 000) (000, 001, 101, 111, 011, 010, 000) (000, 001, 101, 100, 110, 111, 011, 010, 000)
(000, 001, 101, 110, 010, 000) (000, 001, 101, 110, 111, 011, 010, 000) (000, 111, 011, 000) (000, 111, 110, 101, 001, 000) (000, 111, 110, 100, 101, 001, 011, 000)
(000, 111, 110, 010, 000) (000, 111, 110, 101, 100, 011, 000) (000, 111, 110, 100, 101, 001, 011, 010, 000)
dAQn (u, v) = dAQ0 (u, v), l(C) = dAQn (u, v)+3+2n−1 − n
dAQn (u, v) − 2 = 2n−1 + 1. Thus, C is a geodesic cycle n−1 gC 2 +1 (u, v; AQn ). Subcase 1.2. 2n−1 + 2 l 2n . Base on the previn−1 ous subcase, we have the geodesic cycle gC 2 +1 (u, v; AQn ) = u, Ps , v, w, w h , v h , v, Pc , u. Since AQ1n is panconnected, there exist the (v h , w h )i paths with 1 i 2n−1 −1 in AQ1n . Let C = u, Ps , v, w, w h , (w h , v h )i , v h , v, Pc , u. Since dAQn (u, v) = dAQ0 (u, v), l(C) = n
dAQn (u, v) + 2 + i + 1 + 2n−1 − dAQn (u, v) − 2 = 2n−1 + 1 + i. Hence, we can find C with 2n−1 + 2 l(C) 2n . Therefore, C is a format of the geodesic cycles gC l (u, v; AQn ) for 2n−1 + 2 l 2n . Case 2. u ∈ V (AQbn ) and v ∈ V (AQn1−b ) with b ∈ {0, 1}. Without loss of generality, assume that u ∈ V (AQ0n ) and v ∈ V (AQ1n ). Subcase 2.1. dAQn (u, v) = 1. By Lemma 4, there exists a path u, w, v such that w is a neighbor of u in AQ0n . Since AQ0n is panconnected, there exist the paths (u, w)i with 1 i 2n−1 − 1 in AQ0n . Thus, we can construct (a) gC l (u, v; AQn ) for 3 l 2n−1 + 1 with the form u, (u, w)i , w, v, u with 1 i 2n−1 − 1. By the structure of AQn , there exists a neighbor x of w such that (x, v) ∈ E(AQ1n ). Since AQ1n is panconnected, there exist (x, v)i paths with 1 i 2n−1 − 1 in AQ1n . Thus, we can find the geodesic cycle (b) gC l (u, v; AQn ) for 2n−1 − 2 l 2n with the form u, (u, w)2n−1 −1 , w, x, (x, v)i , v, u for 1 i 2n−1 − 1. Clearly, with (a) and (b) we can find all geodesic cycles in this subcase. Subcase 2.2. dAQn (u, v) 2. By Lemma 3, there exist two disjoint shortest paths joining u and v with the form u, P1 , v , v and u, u , P2 , v such that all vertices of u, P1 , v are in AQ0n and all vertices of u , P2 , v are in AQ1n . Clearly, gC l (u, v; AQn ) with l = 2dAQn (u, v) exists. Then we consider the case of 2dAQn (u, v) + 1 l 2n .
By induction hypothesis, there exist gC l (u, v ; AQ0n ) with the form u, Ps , v , Pc , u for all max{2dAQ0 (u, v ), n
3} l 2n−1 where Ps is the s-path and Pc is the cpath. Herein, 2 max{2dAQ0 (u, v ), 3} − dAQ0 (u, v ) n
n
l(Pc ) 2n−1 − dAQ0 (u, v ). Thus, we can write Pc as n v , w, Pc , u. Since l(Pc ) = l(Pc ) − 1, max 2dAQ0 (u, v ), 3 − dAQ0 (u, v ) − 1 l Pc
n
n
2n−1 − dAQ0 (u, v ) − 1. n
By the structure of Augmented cube, one of {w h , w c } is an neighbor of v in AQ1n . Let w ∈ {w h , w c }, and (w 1 , v) is also an edge of AQ1n . By Lemma 6, there exist (w , v)i paths in AQ1n for all 1 i 2n−1 − 1. Let C = u, Ps , v , v, (v, w )i , w , w, Pc , u. Thus, l(C) = dAQ0n (u, v ) + 1 + i + 1 + l(Pc ). Since dAQn (u, v) = dAQ0 (u, v ) + 1 and max{2dAQ0 (u, v ), 3} − dAQn (u, v) n
n
l(Pc ) 2n−1 − dAQn (u, v), dAQn (u, v) + 1 + 1 + max 2dAQ0 (u, v ), 3 n
− dAQn (u, v) l(C) dAQn (u, v) + 2n−1 − 1 + 1 + 2n−1 − dAQn (u, v). Hence, max{2dAQ0 (u, v ), 3} + 2 2dAQn (u, v) + 1 n l(C) 2n . Hence, C is a form of the geodesic cycles gC l (u, v, AQn ) for all 2dAQn (u, v) + 1 l 2n . This completes the proof. 2 4. Balanced pancyclicity of Augmented cubes This section is devoted to illustrating the balanced pancyclic property of augmented cubes. First, we introduce the definitions of balanced cycle and balanced pancyclic. Then, AQn for n 2 is showed to be balanced pancyclic. Definition 4. Let G be a graph. For two vertices u, v ∈ V (G), a cycle C is called a balanced cycle with u and v
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Table 2 Some results of the balanced cycles with u = 000 and v in AQ3 v
balanced cycle
001 001 001 001 001 001 101 101 101 101 101 111 111 111 111 111 111
(000, 001, 010, 000) (000, 010, 001, 011, 000) (000, 010, 001, 101, 100, 000) (000, 010, 011, 001, 101, 100, 000) (000, 010, 011, 001, 110, 101, 100, 000) (000, 010, 110, 101, 001, 011, 111, 100, 000) (000, 001, 101, 010, 000) (000, 001, 101, 110, 010, 000) (000, 001, 010, 101, 111, 011, 000) (000, 001, 010, 101, 110, 111, 011, 000) (000, 001, 011, 010, 101, 111, 110, 100, 000) (000, 111, 011, 000) (000, 011, 111, 100, 000) (000, 011, 111, 110, 100, 000) (000, 010, 011, 111, 110, 100, 000) (000, 010, 011, 111, 110, 101, 100, 000) (000, 010, 001, 011, 111, 110, 101, 100, 000)
if dC (u, v) = max{dC (x, y) | x, y ∈ V (C)}. A balanced l-cycle with u and v in G, denoted by bC l (u, v; G), is a cycle of length l with dbC l (u,v;G) (u, v) = l/2 . Let C = u, bP0 , v, bP1 , u be a balanced cycle of length l with vertices u and v, where |bP0 | |bP1 |. If l is even, |bP0 | = |bP1 |; otherwise, |bP0 | = |bP1 | − 1. Herein, bP0 and bP1 are named as b0-path and b1-path of C, respectively. Definition 5. Let G be a graph. For two vertices u, v ∈ V (G), they are called balanced pancyclic on u and v if for every integer l satisfying max{2dG (u, v), 3} l |V (G)|, the balanced cycle bC l (u, v; G) exists. Definition 6. Let G be a graph. G is called balanced pancyclic if any distinct two vertices on G are balanced pancyclic on them. Lemma 8. AQ3 and AQ4 are both balanced pancyclic. Proof. We verify this small case directly using computer. However, we only list some results in this paper to reduce the space. Please check Tables 2 and 3. (Note that we use decimal number to represent the binary string in Table 3.) 2 Theorem 3. The n-dimensional Augmented cube is balanced pancyclic for n 2. Proof. We prove this theorem by induction on n. Since AQ2 is complete graph K4 , AQ2 is balanced pancyclic. By Lemma 8, the theorem holds for n = 3, 4. The in-
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Table 3 Some results of the balanced cycles with u = 0 and v = 1 in AQ4 (0, 2, 1, 3, 0) (0, 2, 1, 5, 4, 0) (0, 2, 3, 1, 5, 4, 0) (0, 2, 3, 1, 6, 5, 4, 0) (0, 2, 6, 5, 1, 9, 11, 3, 0) (0, 2, 6, 5, 1, 14, 15, 7, 3, 0) (0, 2, 3, 7, 5, 1, 9, 14, 6, 4, 0) (0, 2, 3, 7, 5, 1, 9, 13, 14, 6, 4, 0) (0, 2, 3, 7, 5, 6, 1, 9, 13, 14, 12, 4, 0) (0, 2, 3, 7, 5, 6, 1, 9, 13, 15, 14, 12, 4, 0) (0, 2, 3, 7, 5, 4, 6, 1, 14, 13, 10, 11, 9, 8, 0) (0, 2, 3, 7, 5, 4, 6, 1, 14, 15, 13, 10, 11, 9, 8, 0) (0, 2, 3, 7, 5, 4, 6, 14, 1, 9, 13, 15, 12, 11, 10, 8, 0)
duction hypothesis is that AQn−1 is balanced pancyclic. Then we consider the case of AQn . Let u and v be any two distinct vertices of AQn . Then we divide the cases by the relative position between u and v. Case 1: u, v ∈ AQbn for b ∈ {0, 1}. Without loss of generality, we can assume that u, v ∈ AQ0n . By induction hypothesis, AQn−1 is balanced pancyclic. Hence, there exist bC l (u, v; AQn ) with max{2dAQn (u, v), 3} l 22−1 . Then we consider the remainder into two subcases. Subcase 1.1. 2n−1 + 1 l 2n−1 . By Lemma 1, dAQ0 (u, v) (n−1)/2 . Hence, dAQ0 (u, v) < 2n−2 for n
n
n 3. Since AQ0n is panconnected, there exist (u, v)i paths for 2n−2 i 2n−1 − 1 in AQ0n . Also there exist (uh , v h )j paths for 2n−2 − 1 j 2n−1 − 2 in AQ1n . Suppose we choose j = i − 2 or j = i − 1. Let C = u, (u, v)i , v, v h , (uh , v h )j , uh , u. Hence, 2n−2 + 1 + 2n−2 − 1 + 1 = 2n−1 + 1 l(C) 2n−1 − 1 + 1 + 2n−1 − 2 + 1 = 2n − 1. The cycle C is a form of bC l (u, v; AQn ) for all 2n−1 + 1 l 2n − 1. See Fig. 2(a) for illustration. Subcase 1.2. l = 2n . Then we want to choose a neighbor x of v in AQ0n such that x must satisfy that there exists y ∈ {x h , x c } such that there exists w ∈ {y h , y c } with w ∈ / {u, x, v}. Let X be the set of the neighbors of of v in v in AQ0n . Since there are 2n − 3 7 neighbors AQ0n for n 5, |X| 7. Let Y be the set xi ∈X {xih , xic }. Clearly |Y | 7 also. Let W be the set yi ∈Y {yih , yic }. Therefore, |W | 7. Hence, we can choose the neighbor x of v in AQ0n such that there exist y ∈ {x h , x c } and w ∈ / {y h , y c } with w ∈ {u, x, v}. Since AQn−1 is k-fault Hamiltonian connected by Theorem 1 with k = 2n − 4 2 when n 3, there exists a Hamiltonian path (u, w)i of AQ0n − {x, v} with i = 2n−1 − 3, and there exists a Hamiltonian path (uh , v h )j of AQ1n − {y} with j = 2n−1 − 2. Then, we can construct a cycle D = u, (u, w)2n−1 −3 , w, y, x, v, v h , (uh , v h )2n−1 −2 , uh ,
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Fig. 2. The illustration for Case 1.
u. Since the path u, (u, w)2n−1 −3 , w, y, x, v has length 2n−1 − 3 + 1 + 1 + 1 = 2n−1 and the path v, v h , (uh , v h )2n−1 −2 , uh , u has length 1 + 2n−1 − 2 + 1 = 2n−1 , D is a balanced cycle bC 2n (u, v; AQn ). See Fig. 2(b) for illustration. Case 2. u ∈ AQbn and v ∈ AQn1−b with b ∈ {0, 1}. Without loss of generality, we can assume that u ∈ AQ0n and v ∈ AQ1n . Suppose that (u, v) is an edge of AQn . By Lemma 4, the paths u, v , v and v, u , u exist where v ∈ {v h , v c } and u ∈ {uh , uc }. Thus, u, v, v , u is a balanced cycle bC 3 (u, v; AQn ) and u, u , v, v , u is a balanced cycle bC 4 (u, v; AQn }. By Lemma 6, there exist the paths (v , u)i and (u , v)j for all 1 i 2n−1 − 1 and 1 j 2n−1 − 1. Suppose we choose i = j or i = j − 1. Clearly, u, u , (u , v)i , v, v , (v , u)j , u is a form of the balanced cycles bC l (u, v; AQn ) for all 3 l 2n . Suppose that (u, v) is not an edge of AQn . We know dAQn (u, v) 2 in this case. By Lemma 3, there exist two disjoint shortest paths of AQn joining u and v with the form u, (u, v )s , v , v and u, u , (u , v)s , v where all vertices of (u, v )s are in AQ0n and all vertices of (v, u )s are in AQ1n with s = dAQn (u, v) − 1 1. Since AQ0n is panconnected, there exist the paths (u, v )i with dAQ0 (u, v ) i 2n−1 − 1 in AQ0n . Similarly, there n
exist (v, u )j for dAQ1 (v, u ) j 2n−1 − 1 in AQ1n . n Suppose we choose i = j or i = j − 1. Let the cycle C = u, (u, v )i , v , v, (v, u )j , u , u. Hence, C is a form of the balanced cycles bC l (u, v; AQn ) for all 2dAQn (u, v) l 2n . This completes the proof. 2 5. Conclusions
In this paper, we study the existence of cycles with some requirements in AQn . For given two vertices u and v, the cycle is called geodesic cycle if it contains the shortest path between u and v. Clearly, a geodesic cycle can minimize the transmission delay from u to v. Herein, we prove that AQn is geodesic pancyclic.
We also discuss the balanced pancyclic property for load balancing in AQn . It is of interest to find a cycle passing through u and v such that lengths of two disjoint paths between u and v in this cycle are as equal as possible. AQn is shown to be balanced pancyclic. Acknowledgements The authors would like to thank the referees for their comments and suggestions. These comments and suggestions were very helpful for improving the quality of this paper. References [1] J.A. Bondy, U.S.R. Murty, Graph Theory with Applications, North-Holland, New York, 1980. [2] H.C. Chan, J.M. Chang, Y.L. Wang, S.J. Horng, Geodesicpancyclic graphs, in: Proceedings of the 23rd Workshop on Combinatorial Mathematics and Computation Theory, 2006, pp. 181– 187. [3] S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2) (2002) 71–84. [4] H.C. Hsu, L.C. Chiang, J.J.M. Tan, L.H. Hsu, Fault hamiltonicity of augmented cubes, Parallel Computing 31 (2005) 131–145. [5] P.L. Lai, J.W. Hsue, J.J.M. Tan, L.H. Hsu, On the panconnected properties of the Augmented cubes, in: Proceedings of the 2004 International Computer Symposium, 2004, pp. 1249–1251. [6] T.K. Li, C.H. Tsai, J.J.M. Tan, L.H. Hsu, Bipanconnectivity and edge-fault-tolerant bipancyclicity of hypercubes, Information Processing Letters 87 (1) (2003) 107–110. [7] J. Mitchem, E. Schmeichel, Pancyclic and bipancyclic graphs a survey, Graphs and Applications (1982) 271–278. [8] O. Ore, Hamiltonian connected graph, Journal de Mathématiques Pures et Appliquées 42 (1963) 121–127. [9] C.H. Tsai, H.C. Hsu, P.L. Lai, Geodesic and balanced bipancyclicity of hyper-cubes, Submitted for publication. [10] X.F. Yang, G.M. Megson, On the double-vertex-cycle-connectivity of crossed cubes, Parallel Algorithms and Applications 19 (2004) 11–17. [11] X.F. Yang, G.M. Megson, On the path-connectivity, vertexpancyclicity, and edge-pancyclicity of crossed cubes, Neural, Parallel & Scientific Computations 13 (2005) 107–118.