Decomposition of augmented cubes into regular connected pancyclic subgraphs (original) (raw)

AKCE International Journal of Graphs and Combinatorics, 2020

The n-dimensional augmented cube AQ n is a variation of the hypercube Q n : It is a ð2n À 1Þ-regular and ð2n À 1Þ-connected graph on 2 n vertices. One of the fundamental properties of AQ n is that it is pancyclic, that is, it contains a cycle of every length from 3 to 2 n : In this paper, we generalize this property to k-regular subgraphs for k ¼ 3 and k ¼ 4: We prove that the augmented cube AQ n with n ! 4 contains a 4-regular, 4-connected and pancyclic subgraph on l vertices if and only if 8 l 2 n : Also, we establish that for every even integer l from 4 to 2 n , there exists a 3-regular, 3-connected and pancyclic subgraph of AQ n on l vertices.

Regular connected bipancyclic spanning subgraphs of hypercubes

Computers & Mathematics with Applications, 2011

An n-dimensional hypercube Q n is a Hamiltonian graph; in other words Q n (n ≥ 2) contains a spanning subgraph which is 2-regular and 2-connected. In this paper, we explore yet another strong property of hypercubes. We prove that for any integer k with 3 ≤ k ≤ n, Q n (n ≥ 3) contains a spanning subgraph which is k-regular, k-connected and bipancyclic.

Partial cubes as subdivision graphs and as generalized Petersen graphs

Discrete Mathematics, 2003

Isometric subgraphs of hypercubes are known as partial cubes. The subdivision graph of a graph G is obtained from G by subdividing every edge of G. It is proved that for a connected graph G its subdivision graph is a partial cube if and only if every block of G is either a cycle or a complete graph. Regular partial cubes are also considered. In particular it is shown that among the generalized Petersen graphs P (10, 3) and P (2n, 1), n ≥ 2, are the only (regular) partial cubes.

On the k-subgraphs of the generalized n-cubes

Graphs are used in modeling interconnections networks and measuring their properties. Knowing and understanding the graph theoretical/combinatorial properties of the underlying networks are necessary in developing more efficient parallel algorithms as well as fault-tolerant communication/routing algorithms [1] The hypercube is one of the most versatile and efficient networks yet discovered for parallel computation. One generalization of the hypercube is the n-cube Q(n,m) which is a graph whose vertices are all the binary n-tuples, such that two vertices are adjacent whenever they differ in exactly m coordinates. The k-subgraph of the Generalized n-cube Q k (n,m) is the induced subgraph of the n-cube Q(n,m) where q=2, such that a vertex v ∈ V(Q k (n,m)) if and only if v ∈ V(Q(n,m)) and v is of parity k. This paper presents some degree properties of Q k (n,m) as well as some isomorphisms it has with other graphs, namely: 1)) 2 , (1 n Q n− is isomorphic to Kn 2)) 2 , (i n Q k is isomor...

Constructing Two Edge-Disjoint Hamiltonian Cycles and Two-Equal Path Cover in Augmented Cubes

2012

The n-dimensional hypercube network Qn is one of the most popular interconnection networks since it has simple structure and is easy to implement. The n-dimensional augmented cube AQn, an important variation of the hypercube, possesses several embedding properties that hypercubes and other variations do not possess. The advantages of AQn are that the diameter is only about half of the diameter of Qn and it is node-symmetric. Recently, some interesting properties of AQn have been investigated in the literature. The presence of edge-disjoint Hamiltonian cycles provides an advantage when implementing algorithms that require a ring structure by allowing message traffic to be spread evenly across the interconnection network. A network G contains two-equal path cover and is called two-equal path coverable if for any two distinct pairs of nodes �µ s ,µ tand �υ s ,υ tof G, there exist two node-disjoint paths P and Q satisfying that (1) P joins µs and µt, and Q joins υs and υt, (2) |P | = |Q...

Even-Pancyclic Subgraphs of Meshes

widulski.net

A graph G of order n is even-pancyclic if it contains cycles of all possible even lengths 4, 6, 8,. .. , 2 n 2 ¡. The 2-dimensional mesh M (m, n) is the Cartesian product of the two paths Pm and Pn. We present several results on even-pancyclic subgraphs of meshes.

Geodesic-pancyclicity and fault-tolerant panconnectivity of augmented cubes

Applied Mathematics and Computation, 2009

Choudum and Sunitha [S.A. Choudum, V. Sunitha, Augmented cubes, Networks 40 (2002) 71-84] proposed the class of augmented cubes as a variation of hypercubes and showed that augmented cubes possess several embedding properties that the hypercubes and other variations do not possess. Recently, Hsu et al. [H.-C. Hsu, P.-L. Lai, C.-H. Tsai, Geodesic-pancyclicity and balanced pancyclicity of augmented cubes, Information Processing Letters 101 (2007) 227-232] showed that the n-dimensional augmented cube AQ n , n P 2, is weakly geodesic-pancyclic, i.e., for each pair of vertices u; v 2 AQ n and for each integer ' satisfying maxf2dðu; vÞ; 3g 6 ' 6 2 n where d(u, v) denotes the distance between u and v in AQ n , there is a cycle of length ' that contains a u-v geodesic. In this paper, we obtain a stronger result by proving that AQ n , n P 2, is indeed geodesic-pancyclic, i.e., for each pair of vertices u; v 2 AQ n and for each integer ' satisfying maxf2dðu; vÞ; 3g 6 ' 6 2 n , every u-v geodesic lies on a cycle of length '. To achieve the result, we first show that AQ n À f , n P 3, remains panconnected (and thus is also edge-pancyclic) if f 2 AQ n is any faulty vertex. The result of fault-tolerant panconnectivity is the best possible in the sense that the number of faulty vertices in AQ n , n P 3, cannot be increased.

The Splitting Number of the 4-Cube

Lecture Notes in Computer Science

The splitting number of a graph G is the smallest integer k greater than or equal to 0 such that a planar graph can be obtained from G by k splitting operations. Such operation replaces v by two nonadjacent vertices v(1) and v(2), and attaches the neighbors of v either to v(1) or to v(2). The n-cube has a distinguished place in Computer Science. Dean and Richter devoted an article to proving that the minimum number of crossings in an optimum drawing of the 4-cube is 8, but no results about splitting number of other nonplanar n-cubes are known. In this note we give a proof that the splitting number of the 4-cube is 4. In addition, we give the lower bound 2(n-2) for the splitting number of the n-cube. It is known that the splitting number of the n-cube is O(2(n)), thus our result implies that the splitting number of the n-cube is Theta(2n).

Partial cubes: structures, characterizations, and constructions

Discrete Mathematics, 2008

Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djoković's and Winkler's relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.