A graph G is two-disjoint-cycle-cover r-pancyclic if for any integer l satisfying r≤l≤|V(G)|-r, there exist two vertex-disjoint cycles C1 and C2 in G such that the lengths of C1 and C2 are |V(G)|-l and l, respectively, where |V(G)| denotes the total number of vertices in G. In particular, the graph G is two-disjoint-cycle-cover vertex r-pancyclic if for any two distinct vertices u and v of G, there exist two vertex-disjoint cycles C1 and C2 in G such that (i) C1 contains u, (ii) C2 contains v, and (iii) the lengths of C1 and C2 are |V(G)|-l and l, respectively, for any integer l satisfying r≤l≤|V(G)|-r. Moreover, G is two-disjoint-cycle-cover edge r-pancyclic if for any two vertex-disjoint edges (u,v) and (x,y) of G, there exist two vertex-disjoint cycles C1 and C2 in G such that (i) C1 contains (u,v), (ii) C2 contains (x,y), and (iii) the lengths of C1 and C2 are |V(G)|-l and l, respectively, for any integer l satisfying r≤l≤|V(G)|-r. In this paper, we first give Dirac-type sufficient conditions for general graphs to be two-disjoint-cycle-cover vertex/edge 3-pancyclic, and we also prove that the n-dimensional crossed cube CQn is two-disjoint-cycle-cover 4-pancyclic for n≥3, vertex 4-pancyclic for n≥5, and edge 6-pancyclic for n≥5.

In this paper, we investigate the fault-tolerant Hamiltonian problems of crossed cubes with a faulty path. More precisely, let P denote any path in an n-dimensional crossed cube CQn for n ≥ 5, and let V(P) be the vertex set of P. We show that CQn-V(P) is Hamiltonian if |V(P)|≤n and is Hamiltonian connected if |V(P)| ≤ n-1. Compared with the previous results showing that the crossed cube is (n-2)-fault-tolerant Hamiltonian and (n-3)-fault-tolerant Hamiltonian connected for arbitrary faults, the contribution of this paper indicates that the crossed cube can tolerate more faulty vertices if these vertices happen to form some specific types of structures.

In a multiprocessor system, processors are connected based on various types of network topologies. A network topology is usually represented by a graph. Let G be a graph and u, v be any two distinct vertices of G. We say that G is pancyclic if G has a cycle C of every length l(C) satisfying 3≤l(C)≤|V(G)|, where |V(G)| denotes the total number of vertices in G. Moreover, G is panpositionably pancyclic from r if for any integer m satisfying $r leq m leq rac{|V(G)|}{2}$, G has a cycle C containing u and v such that dC(u,v)=m and 2m≤l(C)≤|V(G)|, where dC(u,v) denotes the distance of u and v in C. In this paper, we investigate the panpositionable pancyclicity problem with respect to the n-dimensional locally twisted cube LTQn, which is a popular topology derived from the hypercube. Let D(LTQn) denote the diameter of LTQn. We show that for n≥4 and for any integer m satisfying $D(LTQ_n) + 2 leq m leq rac{|V(LTQ_n)|}{2}$, there exists a cycle C of LTQn such that dC(u,v)=m, where (i) 2m+1≤l(C)≤|V(LTQn)| if m=D(LTQn)+2 and n is odd, and (ii) 2m≤l(C)≤|V(LTQn)| otherwise. This improves on the recent result that u and v can be positioned with a given distance on C only under the condition that l(C)=|V(LTQn)|. In parallel and distributed computing, if cycles of different lengths can be embedded, we can adjust the number of simulated processors and increase the flexibility of demand. This paper demonstrates that in LTQn, the cycle embedding containing any two distinct vertices with a feasible distance is extremely flexible.

Dynamic Voltage/Frequency Scaling (DVFS) allows designers to improve energy efficiency through adjusting supply voltage at runtime in order to meet the workload demand. Previous works solving real-time DVFS problems often refer to the canonical schedules with the exponential length. Other solutions for online scheduling depend on empirical or stochastic heuristics, which potentially result in frequent fluctuations of voltage/speed scaling. This paper aims at increasing the schedule predictability using period transformation in the pinwheel task model and improves the control on power-awareness by decreasing the speeds of as many tasks as possible to the same level. Experimental results show the maximum energy savings of 6% over the recent Dynamic Power Management (DPM) method and 12% over other slack reclamation algorithms.

A tree 3-spanner T of a graph G is a spanning tree of G such that the distance between any two vertices in T is at most 3 times of their distance in G. Madanlal et al. have presented an O(n + m) time algorithm for finding a tree 3-spanner of a permutation graph. However, the complexity of their algorithm is not optimal, and their algorithm can not be easily parallelized. In this paper, we will propose an improved algorithm to solve the same problem in O(n) time. Moreover, our algorithm can be easily parallelized so that a tree 3-spanner of a permutation graph can be found in O(log n) time with processors on the EREW PRAM computational model.