Optimizing Hamiltonian panconnectedness for the crossed cube architecture

A graph G of k vertices is panconnected if for any two distinct vertices x and y, it has a path of length l joining x and y for any integer l satisfying d(G) (x, y) <= l <= k - 1, where dG (x, y) denotes the distance between x and y in G. In particular, when k >= 3, G is called Hamiltonian r-panconnected if for any three distinct vertices x, y, and z, there exists a Hamiltonian path P of G with d P (x, y) = l such that P(1) = x, P(l + 1) = y, and P(k) = z for any integer l satisfying r <= l <= k - r - 1, where P (i) denotes the i th vertex of path P for 1 <= i <= k. Then, this paper shows that the n-dimensional crossed cube, which is a popular variant of the hypercube topology, is Hamiltonian ([n+1/2] + 1)-panconnected for n >= 4. The lower bound [n+1/2] + 1 on the path length is sharp, which is the shortest that can be embedded between any two distinct vertices with dilation 1 in the n-dimensional crossed cube. (c) 2018 Elsevier Inc. All rights reserved.