Beneš and Shuffle-Exchange Counterexamples

Przemek Chojecki

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.