Panconnectivity of Cartesian product graphs

A graph G of order n (≥2) is said to be panconnected if for each pair (x,y) of vertices of G there exists an xy-path of length ℓ for each ℓ such that dG(x,y)≤ℓ≤n−1, where dG(x,y) denotes the length of a shortest xy-path in G. In this paper, we consider the panconnectivity of Cartesian product graphs. As a consequence of our results, we prove that the n-dimensional generalized hypercube Qn(k1,k2,…,kn) is panconnected if and only if ki≥3 (i=1,…,n), which generalizes a result of Hsieh et al. that the 3-ary n-cube \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q^{3}_{n}$\end{document} is panconnected.