Counterexamples to an Extremal Conjecture for Random Cycle-Factors

TL;DR

This paper disproves a conjecture about the maximum expected number of cycles in random cycle-factors of directed regular graphs for degrees three and higher, while confirming it for degree two.

Contribution

It provides explicit counterexamples for degrees three and above, challenging the previously believed extremal configuration in random cycle-factors.

Findings

Counterexamples for all degrees d ≥ 3 and multiples n=kd with k ≥ 2.

Confirms the conjecture holds true for degree d=2.

Shows a sharp dichotomy between degree two and higher degrees.

Abstract

Christoph, Dragani\'{c}, Gir\~{a}o, Hurley, Michel, and M\"{u}yesser 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.