Counterexamples to an Extremal Conjecture for Random Cycle-Factors

Abstract

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.