飞跃之旅
Leaper Tours
摘要：
设$p$和$q$为正整数。$(p, q)$跳跃器$L$是一个广义的骑士，它沿着一个坐标轴跳跃$p$单位，沿着另一个坐标轴跳跃$q$单位。考虑一个自由$L$，这意味着$p + q$是奇数，而$p$和$q$是互素。我们证明了$L$旅游板的大小$4pq \times n$的所有充分大的正整数$n$。结合最近建立的猜想Willcocks，其中指出，$L$旅游方板的一面$2(p + q)$，我们得出结论，此外，$L$旅游所有板的双方都是均匀和足够大。这，特别是，完全解决了问题的哈密顿跳跃图上足够大的正方形板。
英文摘要：
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.
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