Meijie Ma

Cycles in folded hypercubes

This work investigates important properties related to cycles of embedding into the folded hypercube FQ n for n ≥ 2. The authors observe that FQn is bipartite if and only if n is odd, and show that the minimum length of odd cycles is n + 1i fn is even. The authors further show that every edge of FQn lies on a cycle of every even length from 4 to 2 n ;i fn is even, every edge of FQn also lies on a cycle of every odd length from n + 1t o 2 n − 1.

Panconnectivity of locally twisted cubes

The locally twisted cube LTQn which is a newly introduced interconnection network for parallel computing is a variant of the hypercube Qn. Yang et al. [X. Yang, G.M. Megson, D.J. Evans, Locally twisted cubes are 4-pancyclic, Applied Mathematics Letters 17 (2004) 919–925] proved that LTQn is Hamiltonian connected and contains a cycle of length from 4 to 2 n for n ≥ 3. In this work, we improve this result by showing that for any two different vertices u and v in LTQn (n ≥ 3), there exists a uv-path of length l with d(u ,v )+ 2 ≤ l ≤ 2 n − 1 except for a shortest uv-path. c

Cycles embedding in exchanged hypercubes

The exchanged hypercube EH(s,t), proposed by Loh et al., is obtained by systematically removing links from a binary hypercube. This paper investigates important properties related to embedding cycles into the exchanged hypercube EH(s,t). The authors show that EH(1,t) and EH(2,2) are not bipancyclic, but EH(s,t) (2=

The domination number of exchanged hypercubes

Exchanged hypercubes (Loh et al., 2005 13]) are spanning subgraphs of hypercubes with about one half of their edges but still with many desirable properties of hypercubes. Lower and upper bounds on the domination number of exchanged hypercubes are proved which in particular imply that γ ( EH ( 2 , t ) ) = 2 t + 1 holds for any t ? 2 . Using Hamming codes we also prove that γ ( EH ( s , 2 k - 1 ) ) ≤ ( 2 s - 2 k ) γ ( Q 2 k - 1 ) + 2 2 k - 1 ( γ ( Q s - ) + 1 ) holds for s ? k ? 3 . We study a distribution of limited supply of resources into exchanged hypercube EH ( s , t ) .We give upper and lower bounds for the domination number of EH ( s , t ) .We determine exact value for the domination number of EH ( 2 , t ) .Using Hamming codes we improve an upper bound for the domination number of EH ( s , 2 k - 1 ) .

Average distance, surface area, and other structural properties of exchanged hypercubes

Exchanged hypercubes (Loh et al. in IEEE Trans Parallel Distrib Syst 16:866–874, 2005) are spanning subgraphs of hypercubes with about one half of their edges but still with many desirable properties of hypercubes. In this paper, it is shown that distance properties of exchanged hypercubes are also comparable to the corresponding properties of hypercubes. The average distance and the surface area of exchanged hypercubes are computed and it is shown that exchanged hypercubes have asymptotically the same average distance as hypercubes. Several additional metric and other properties are also deduced and it is proved that exchanged hypercubes are prime with respect to the Cartesian product of graphs.

The adversary degree-associated reconstruction number of double-brooms

A vertex-deleted subgraph of a graph G is a card. A dacard specifies the degree of the deleted vertex along with the card. The adversary degree-associated reconstruction number adrn ( G ) is the least k such that every set of k dacards determines G. We determine adrn ( D m , n , p ) , where the double-broom D m , n , p with p ? 2 is the tree with m + n + p vertices obtained from a path with p vertices by appending m leaves at one end and n leaves at the other end. We determine adrn ( D m , n , p ) for all m , n , p . For 2 ? m ? n , usually adrn ( D m , n , p ) = m + 2 , except adrn ( D m , m + 1 , p ) = m + 1 and adrn ( D m , m + 2 , p ) = m + 3 . There are exceptions when ( m , n ) = ( 2 , 3 ) or p = 4 . For m = 1 the usual value is 4, with exceptions when p ? { 2 , 3 } or n = 2 .

A note on “The super connectivity of augmented cubes”

The aim of this note is to mend a flaw in the proof of Theorem 2 in our paper [M. Ma, G. Liu, J.-M. Xu, The super connectivity of augmented cubes, Information Processing Letters 106 (2008) 59-63].

The vulnerability of the diameter of the enhanced hypercubes

For an interconnection network G, the ω-wide diameter dω(G) is the least l such that any two vertices are joined by ω internally-disjoint paths of length at most l, and the (ω − 1)-fault diameter Dω(G) is the maximum diameter of a subgraph obtained by deleting fewer than ω vertices of G. The enhanced hypercube Qn,k is a variant of the well-known hypercube. Yang, Chang, Pai, and Chan gave an upper bound for dn+1(Qn,k) andDn+1(Qn,k) and posed the problem of finding the wide diameter and fault diameter of Qn,k. By constructing internally disjoint paths between any two vertices in the enhanced hypercube, for n ≥ 3 and 2 ≤ k ≤ n we prove that Dω(Qn,k) = dω(Qn,k) = d(Qn,k) for 1 ≤ ω < n − ⌊ k 2⌋; Dω(Qn,k) = dω(Qn,k) = d(Qn,k) + 1 for n − ⌊k2⌋ ≤ ω ≤ n + 1, where d(Qn,k) is the diameter of Qn,k. These results mean that interconnection networks modelled by enhanced hypercubes are extremely robust.