Closed Almost Knight’s Tours on 2D and 3D Chessboards

Abstract

Let a (generalized) chessboard in two or three dimensions be given. A closed knight’s tour is defined as a Hamiltonian cycle over all cells of the chessboard where all moves are knight’s moves, i. e. have length $\sqrt{5}$. It is well-characterized for which chessboard sizes it is not possible to construct a closed knight’s tour. On such chessboards a closed almost knight’s tour is defined as a Hamiltonian cycle of minimal length if only moves of length at least $\sqrt{5}$ are allowed. The problem of determining closed (almost) knight’s tours on 2D and 3D chessboards is equivalent to the so called traveling salesman problem with forbidden neighborhoods on regular 2D and 3D grids if the radius of the forbidden neighborhood is two and hence the minimal feasible distance of two consecutive vertices in the tour equals the length of a knight’s move. In this paper we present explicit construction schemes for closed almost knight’s tours by extending ideas used for the construction of knight’s tours.