Panconnectivity of locally twisted cubes

Abstract

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.

Panconnectivity and edge-pancyclicity of 3-ary N-cubes

“…That is, any path of possible length can be embedded into a panconnected graph with dilation one. 1 It is recently received much attention on the problem of finding paths of various lengths in interconnection networks [3,7,8] because this is an important measurement for determining if the topology of a network is suitable for an application in which mapping paths of various lengths into the topology is required.…”

Survey on path and cycle embedding in some networks

“…Xu and Ma 1) improved this result by proving that LT Q n is vertex-pancyclic. Further, Ma and Xu [117], Hu et al [86] improved these results. Theorem 6.1 (Hu et al [86], Ma and Xu [117]) LT Q n is edge-pancyclic for n 2.…”

Embedding hamiltonian paths in hypercubes with a required vertex in a fixed position

“…The cycle embedding problem, which deals with all possible length of the cycles in a given graph, is investigated in a lot of interconnection networks [2,7,9,11,14]. The path embedding problem, which deals with all possible length of the paths between given two vertices in a given graph, is investigated in a lot of interconnection networks [3][4][5][6]11,13,14].…”