Leaper Tours

TL;DR

This paper proves that certain generalized chess pieces called leapers can tour large rectangular boards, and together with existing results, it shows they can tour all sufficiently large even-sided boards, resolving a key question in leaper graph Hamiltonicity.

Contribution

It establishes that all sufficiently large boards of certain dimensions are tourable by leapers, confirming Hamiltonicity for large even-sided boards and extending previous conjectures.

Findings

Leapers tour large rectangular boards of size 4pq by n.

Leapers tour all sufficiently large even-sided boards.

Complete resolution of Hamiltonicity for large square leaper graphs.

Abstract

Let $p$ and $q$ be positive integers. The $(p, q)$-leaper $L$ is a generalised knight which leaps $p$ units away along one coordinate axis and $q$ units away along the other. Consider a free $L$, meaning that $p + q$ is odd and $p$ and $q$ are relatively prime. We prove that $L$ tours the board of size $4pq \times n$ for all sufficiently large positive integers $n$. Combining this with the recently established conjecture of Willcocks which states that $L$ tours the square board of side $2(p + q)$, we conclude that furthermore $L$ tours all boards both of whose sides are even and sufficiently large. This, in particular, completely resolves the question of the Hamiltonicity of leaper graphs on sufficiently large square boards.