Which Chessboards have a Closed Knight's Tour within the Cube?

Abstract

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. When the chessboard is translated into graph theoretic terms the question is transformed into the existence of a Hamiltonian cycle. There are two common tours to consider on the cube. One is to tour the six exterior $n\times n$ boards that form the cube. The other is to tour within the $n$ stacked copies of the $n\times n$ board that form the cube. This paper is concerned with the latter. In this paper necessary and sufficient conditions for the existence of a closed knight's tour for the cube are proven.