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：Christoph、Draganić、Girão、Hurley、Michel与Müyesser猜想：当$d\mid n$时，$n$个顶点上$d$-正则有向图的均匀随机循环因子中循环数的期望值，由$n/d$个完全环状有向图$K_d^\circ$的不交并唯一达到最大，其值为$(n/d)H_d$ [FOCS 2025]。我们在最强可能的范围内否定了这一猜想。对于每个$d\ge 3$及每个满足$k\ge 2$的倍数$n=kd$，我们构造了一个$n$个顶点上的$d$-正则有向图，其均匀随机循环因子的期望循环数严格大于$kH_d$。我们还证明了该猜想在度数为$d=2$时给出了正确的极值图景，这揭示了度数为二与所有更高度数之间的鲜明二分性。