ANALYSIS OF THE EXISTENCE OF THE KNIGHT’S TOUR ON PLUS-SHAPED BOARDS

Abstract

The Knight’s Tour is a well-known mathematical problem in combinatorics and graph theory, involving a sequence of moves of a knight on a chessboard such that every square is visited exactly once. This study investigates the existence of Hamiltonian cycles and Hamiltonian paths, referred to as closed knight tours (CKT) and open knight tours (OKT), respectively, on a special class of boards called plus-shape boards. A plus-shape board, denoted by , is defined as a configuration in which the central region consists of  squares, and each of its four arms is attached as a rectangular section of  squares, where  and  are positive integers. The results reveal that whenever a CKT exists, an OKT also exists. In particular,  admits both CKT and OKT. Furthermore, it is shown that every  admits a CKT, and that both  and  always admit an OKT for any positive integers  and . These findings contribute to the combinatorial characterization of knight’s movement patterns on non-standard chessboard geometries.