AI 中文总结
本文给出混洗型网络可重排性猜想的反例。构造\(N\)正则图反驳贝内什不等式，证明标准有向混洗交换网络中\(d(k,3)=6(k\geq3)\)，说明混洗交换猜想不成立，建议用\(d(k,n)=3n - 3\)替代原猜想。
AI 中文摘要
我们给出了关于混洗型网络的两个可重排性猜想的明确反例。首先，对于每个\(N\geq2\)，我们构造了一个简单的\(N\)正则有序两阶段图\(L_N\)，其\(F(L_N)=2\)且\(R(L_N)\geq N\)，反驳了图论中的贝内什不等式\(R(L)\leq2F(L)\)及其在开放问题园地上陈述的分区稳定器形式。其次，对于标准有向混洗交换网络，我们证明对于每个\(k\geq3\)，\(d(k,3)=6\)，而已知二进制值为\(d(2,3)=5\)。因此，混洗交换猜想\(d(k,n)=2n - 1\)在\((k,n)=(3,3)\)时就不成立了，对于\(k\geq3\)，剩余的上界\(d(k,n)\leq3n - 3\)表明\(d(k,n)=3n - 3\)是一个自然的替代问题。
英文摘要
We give explicit counterexamples to two rearrangeability conjectures for shuffle-type networks. First, for every $N\ge2$ we construct a simple $N$-regular ordered two-stage graph $L_N$ with $F(L_N)=2$ and $R(L_N)\ge N$, refuting the graph-theoretic Beneš inequality $R(L)\le2F(L)$ and its partition-stabilizer form as stated on Open Problem Garden. We retain the sharp cut obstruction, exact mask-composition identity, exact middle criterion, first nontrivial-level result, and balanced-middle sufficient condition that explain which extra hypotheses can replace mere external connectivity. Second, for the standard directed shuffle-exchange network, we prove $d(k,3)=6$ for every $k\ge3$, while the known binary value is $d(2,3)=5$. Hence the shuffle-exchange conjecture $d(k,n)=2n-1$ fails already at $(k,n)=(3,3)$, and the remaining upper bound $d(k,n)\le3n-3$ for $k\ge3$ suggests $d(k,n)=3n-3$ as a natural replacement problem.
