Cyclic Hamilton Cycle Decompositions of Carousel Tournaments of Order $pq$

Kelly’s conjecture asks whether every regular tournament admits a Hamilton cycle decomposition. Motivated by its symmetry-preserving extension, we study cyclic Hamilton decompositions of carousel tournaments. For an odd integer $n$, let $$T_n=\text{Cay}({\mathbb{Z}}_n,\{1,2,\dots ,\frac{n-1}{2}\})$$ be the carousel tournament. We ask whether $T_n$ has a Hamilton decomposition invariant under translation by $1$. Although the answer is immediate when $n$ is prime, composite orders introduce a genuine obstruction: nonunit differences generate short cycles rather than Hamilton cycles. We resolve a general composite-order family by proving that, whenever $n=pq$ for primes $7\le p<q$, the tournament $T_n$ admits a cyclic Hamilton cycle decomposition. The proof combines Hamiltonian difference sequences over prime fields with a matching argument that constructs two base paths with disjoint difference sets in ${\mathbb{Z}}_{pq}$. Thus our result gives an infinite family supporting the cyclic, symmetry-preserving extension of the Hamilton decomposition problem for regular tournaments.