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. In 1991 Schwenk completely classified the rectangular chessboards that admit a closed knight's tour. For a rectangular chessboard that does not contain a closed knight's tour, this paper determines the minimum number of squares that must be removed in order to admit a closed knight's tour. Furthermore, constructions that generate a closed tour once appropriate squares are removed are provided.
DeMaio, J., & Hippchen, T. (2009). Closed knight's tours with minimal square removal for all rectangular boards. Mathematics Magazine, 82(3), 219-225. doi:10.4169/193009809X468869