Knight's tours on ringboards, deficient 4 x n chessboards and some l-boards

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 a perpendicular direction. A closed knight's tour is a sequence of knight's moves 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 RB(m,n,r) is defined to be an m x n chessboard, denoted by CB(m x n), with the middle part missing and the rim contains r rows and r columns. Next, if A is a set of two squares of CB(4 x n), then CB(4 x n) - A is the deficient board after deleting these two squares. In this dissertation, we study the existence of closed knight's tours on RB(m,n,r) and CB(4 x n) - A.