Which Chessboards have a Closed Knight\u27s Tour within the Cube?

A closed knight\u27s 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 x n boards that form the cube. The other is to tour within the n stacked copies of the n x 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\u27s tour for the cube are proven

Closed Knight\u27s Tours with Minimal Square Removal for All Rectangular Boards

A closed knight\u27s tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. In 1991 Schwenk completely classified the rectangular chessboards that admit a closed knight\u27s tour. For a rectangular chessboard that does not contain a closed knight\u27s tour, this paper determines the minimum number of squares that must be removed in order to admit a closed knight\u27s tour. Furthermore, constructions that generate a closed tour once appropriate squares are removed are provided