Closed Knight’s Tours on (m,n,r)-Ringboards

Abstract

A (legal) knight’s move is the result of moving the knight two squares horizontally or vertically on the board and then turning and moving one square in the perpendicular direction. A closed knight’s tour is a knight’s move that visits every square on a given chessboard exactly once and returns to its start square. A closed knight’s tour and its variations are studied widely over the rectangular chessboard or a three-dimensional rectangular box. For m,n>2r, an (m,n,r)-ringboard or (m,n,r)-annulus-board is defined to be an m×n chessboard with the middle part missing and the rim contains r rows and r columns. In this paper, we obtain that a (m,n,r)-ringboard with m,n≥3 and m,n>2r has a closed knight’s tour if and only if (a) m=n=3 and r=1 or (b) m,n≥7 and r≥3. If a closed knight’s tour on an (m,n,r)-ringboard exists, then it has symmetries along two diagonals.