Improving the panconnectedness property of locally twisted cubes

Abstract

The n-dimensional locally twisted cube LTQn is a promising alternative to the hypercube because of its great properties. Not only is LTQn n-connected, but also meshes, torus, and edge-disjoint Hamiltonian cycles can embed in it. Ma and Xu [Panconnectivity of locally twisted cubes, Appl. Math. Lett. 19 (2006), pp. 681–685] investigated the panconnectivity of LTQn for flexible routing. In this paper, we combine panconnectivity with Hamiltonian connectedness to define Hamiltonian r-panconnectedness: a graph G of m vertices, m≥3, is Hamiltonian r-panconnected if for any three distinct vertices x, y, and z of G there exists a Hamiltonian path P of G such that P(1)=x, P(l+1)=y, and P(m)=z for every r≤l≤m−1−r, where P(i) denotes the ith vertex of P for 1≤i≤m. Then, we show that LTQn is Hamiltonian n-panconnected for n≥5. This property admits the path embedding via an intermediate node at any prescribed position, and our result achieves an improvement over that of Ma and Xu.