双向完全图的游牧分解
Nomadic Decompositions of Bidirected Complete Graphs
摘要：
我们用$K^*_n$来表示$n$顶点上的双向完全图。$K^*_n$的游牧哈密尔顿分解是哈密尔顿分解，具有“游牧者”沿着哈密顿圈行走（每个时间步移动一个顶点）而不碰撞的附加性质。类似地定义了游牧近汉密尔顿分解，除了分解中的循环具有长度$n-1$而不是长度$n$。J.A. Bondy询问是否所有$n$都存在这些$K^*_n$分解。我们发现，$K^*_n$承认游牧近哈密顿分解时，$n\not\equiv 2\bmod 4$。
英文摘要：
We use $K^*_n$ to denote the bidirected complete graph on $n$ vertices. A nomadic Hamiltonian decomposition of $K^*_n$ is a Hamiltonian decomposition, with the additional property that ``nomads'' walk along the Hamiltonian cycles (moving one vertex per time step) without colliding. A nomadic near-Hamiltonian decomposition is defined similarly, except that the cycles in the decomposition have length $n-1$, rather than length $n$. J.A. Bondy asked whether these decompositions of $K^*_n$ exist for all $n$. We show that $K^*_n$ admits a nomadic near-Hamiltonian decomposition when $n\not\equiv 2\bmod 4$.
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