完全图和3-一致超图的循环关联排序
Cyclic Incidence Orderings of Complete Graphs and 3-Uniform Hypergraphs
中文总结 AI 辅助
本文研究完全k-一致超图边的循环排序，其中顶点关联序列为公共词的循环移位，证明了k=2和k=3时排序存在的充要条件，并利用反射卷积和配对相交计数完成必要性证明。
中文摘要 AI 辅助
我们研究n个顶点上的完全k-一致超图的所有边的循环排序，其中每个顶点的二元关联序列是某个公共词的循环移位。这些移位是独立选择的，对顶点没有规定的作用。对于2≤k<n，已知当gcd(n,k)=1时，即使要求相邻边必须通过单个顶点交换来区分，该条件也充分。我们回顾一个简短的轨道构造，并在没有任何邻接要求的情况下，证明前两个非平凡一致性的逆命题。对于k=2，排序存在当且仅当n=2或n为奇数；对于k=3，排序存在当且仅当n=3或3不整除n。必要性证明使用反射卷积恒等式和配对相交计数来将顶点移位限制在挠陪集内。对于三元组情形，通过保持重数的膨胀和条件素数幂容量界完成论证。
英文摘要
We study cyclic orderings of all edges of a complete $k$-uniform hypergraph on $n$ vertices in which the binary incidence sequences of the vertices are cyclic shifts of a common word. The shifts are chosen independently, with no prescribed action on the vertices. For $2\leq k<n$, coprimality $\gcd(n,k)=1$ is known to suffice even when consecutive edges must differ by a single vertex exchange. We recall a short orbit construction and prove the converse for the first two nontrivial uniformities without any adjacency requirement. For $k=2$, an ordering exists exactly when $n=2$ or $n$ is odd; for $k=3$, exactly when $n=3$ or $3\nmid n$. The necessity proofs use reflected convolution identities and pair-intersection counts to constrain the vertex shifts to a torsion coset. For triples, multiplicity-preserving dilation and conditional prime-power capacity bounds complete the argument.
