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×M×NL×M×N, where L≤M≤NL≤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×M×NL×M×N does not have a closed GKT if either (a) L≤2L≤2 or L=4L=4, or (b) L=3L=3 and M≤7M≤7. Then, we constructively prove that a chessboard of size 3×4s×4t3×4s×4t with s≥2s≥2and t≥2t≥2 must contain a closed GKT.