Christoph, Draganić, Girão, Hurley, Michel, and Müyesser conjectured that, when $d\mid n$, the expected number of cycles in a uniformly random cycle-factor of a directed $d$-regular graph on $n$ vertices is uniquely maximised by the disjoint union of $n/d$ copies of the complete looped digraph $K_d^\circ$, with value $(n/d)H_d$ [FOCS 2025]. We disprove this conjecture in the strongest possible range. For every $d\ge 3$ and every multiple $n=kd$ with $k\ge 2$, we construct a directed $d$-regular graph on $n$ vertices whose uniformly random cycle-factor has expected cycle count strictly larger than $kH_d$. We also show that the conjectured extremal picture is correct in degree $d=2$, giving a sharp dichotomy between degree two and all higher degrees.

翻译：克里斯托夫、德拉加尼奇、吉朗、赫尔利、米歇尔和米耶塞尔猜想：当 $d\mid n$ 时，$n$ 个顶点上的有向 $d$-正则图的均匀随机环因子中环数的期望值，由 $n/d$ 个完全带环有向图 $K_d^\circ$ 的不交并唯一最大化，其值为 $(n/d)H_d$ [FOCS 2025]。我们在最强可能的范围内否证了这一猜想。对于每个 $d\ge 3$ 和每个满足 $k\ge 2$ 的倍数 $n=kd$，我们构造了一个 $n$ 个顶点上的有向 $d$-正则图，其均匀随机环因子的期望环数严格大于 $kH_d$。我们还证明了在度 $d=2$ 的情形下，猜想的极值图论图像是正确的，从而在度二与所有更高度之间给出了一个清晰的分界。