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

The locally twisted cube LTQ n which is a newly introduced interconnection network for parallel computing is a variant of the hypercube Q n. 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 LTQ n 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 LTQ n (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.

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To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

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