Generalized knight's tour on 3D chessboards

In [G.L. Chia, Siew-Hui Ong, Generalized knight's tours on rectangular chessboards, Discrete Applied Mathematics 150 (2005) 80-98], Chia and Ong proposed the notion of the generalized knight's tour problem (GKTP). In this paper, we address the 3D GKTP, that is, the GKTP on 3D chessboards of size L x M x N, where L <= M <= N. We begin by presenting several sufficient conditions for a 3D chessboard not to admit a closed or open generalized knight's tour (GKT) with given move patterns. Then, we turn our attention to the 3D GKTP with (1, 2, 2) move. First, we show that a chessboard of size L x M x N does not have a closed GKT if either (a) L <= 2 or L = 4, or (b) L = 3 and M <= 7. Then, we constructively prove that a chessboard of size 3 x 45 x 4t with s >= 2 and t >= 2 must contain a closed GKT. (C) 2010 Elsevier B.V. All rights reserved.