Generalised Knight's Tours

Abstract

The problem of existence of closed knight's tours in $[n]^d$, where $[n]=\{0, 1, 2, \dots, n-1\}$, 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 $n$.