The Genus of a Minimal Connected Knight’s Graph

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.