The Genus of a Minimal Connected Knight’s Graph

Abstract

It has been proved that the smallest rectangular board on which a $(p,q)$-knight’s graph is connected has sides $p+q$ by $2q$ when $p<q$. It has also been proved that these minimal connected knight's graphs are of genus $0$ or $1$, and that they are of genus $0$ when $p$ and $q$ are of the form $Md+1$ and $(M+1)d+1$, with $M$ a non-negative integer and $d$ a positive odd integer. It is proved in this paper that the minimal connected knight's graph is of genus $1$ in all other cases.