4 problems

Let $G$ denote the number of geometrically distinct open knight tours on the standard $8\times 8$ chessboard. The number of open tour diagrams is $T=8G$, and the number of open tou…

Let $a_1,\ldots,a_s\in\mathbb{N}$, let $r>s$, and consider the $(a_1,\ldots,a_s)$-knight's graph on an $n_1\times\cdots\times n_r$ chessboard. An $(a_1,\ldots,a_s)$-tour is a tour…

Let $K_{a,b}(n_1,n_2,\ldots,n_r)$ be the $(a,b)$-knight's graph on an $n_1\times n_2\times\cdots\times n_r$ chessboard, and let an $(a,b)$-tour be a tour using its edges. Conjectur…

Let $K_{a,b}(n,m)$ be the graph whose vertices are the squares of an $n\times m$ chessboard, with edges joining squares related by an $(a,b)$-knight move. An $(a,b)$-tour is a tour…