Generalized knight's tours on rectangular chessboards

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 5k×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).