Leaper Tours

Let \(p\) and \(q\) be positive integers. The \((p, q)\)-leaper \(L\) is a generalised knight which leaps \(p\) units away along one coordinate axis and \(q\) units away along the other. Consider a free \(L\), meaning that \(p + q\) is odd and \(p\) and \(q\) are relatively prime. We prove that \(L\) tours the board of size \(4pq \times n\) for all sufficiently large positive integers \(n\). Combining this with the recently established conjecture of Willcocks which states that \(L\) tours the square board of side \(2(p + q)\), we conclude that furthermore \(L\) tours all boards both of whose sides are even and sufficiently large. This, in particular, completely resolves the question of the Hamiltonicity of leaper graphs on sufficiently large square boards.