Abstract
In [Math. Mag. 64 (1991) 325–332], Schwenk has completely determined the set of all integers m and n for which the m × n chessboard admits a closed knight's tour. In this paper, (i) we consider the corresponding problem with the knight's move generalized to ( a , b ) -knight's move (defined in the paper, Section 1). (ii) We then generalize a beautiful coloring argument of Pósa and Golomb to show that various m × n chessboards do not admit closed generalized knight's tour (Section 3). (iii) By focusing on the ( 2 , 3 ) -knight's move, we show that the m × n chessboard does not have a closed generalized knight's tour if m = 1 , 2 , 3 , 4 , 6 , 7 , 8 and 12 and determine almost completely which 5 k × m chessboards have a closed generalized knight's tour (Section 4). In addition, (iv) we present a solution to the (standard) open knight's tour problem (Section 2).
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