Generalised Knight's Tours

Abstract

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