Generalised Knight's tours

Abstract

The problem of existence of closed knight's tours in [n]d, where [n] = {0, 1, 2,..., n - 1}, was recently solved by Erde, Golénia, and Golénia. They raised the same question for a generalised, (a, b) knight, which is allowed to move along any two axes of [n]d by a and b unit lengths respectively. Given an even number a, we show that the [n]d grid admits an (a, 1) knight's tour for sufficiently large even side length n.