Counterexamples to an Extremal Conjecture for Random Cycle-Factors

2026-04-28

AI Summary

This study addresses an extremal conjecture by Christoph et al. concerning the maximum expected number of cycles in a random cycle factor of directed $d$-regular graphs. Through a combination of combinatorial constructions and probabilistic analysis, the authors construct explicit counterexamples that strictly outperform the conjectured extremal configuration for all $d \geq 3$, thereby disproving the universality of the conjecture in higher degrees. In contrast, they confirm that the conjecture holds when $d = 2$. These findings reveal a fundamental distinction between the quadratic and higher-degree cases, establishing a sharp threshold at $d = 2$ and delineating a clear boundary in the behavior of random cycle factors across different regularities.

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.

Problem

• cycle-factor
• directed regular graph
• expected number of cycles
• extremal conjecture
• random cycle-factor

Innovation

• cycle-factors
• extremal conjecture
• directed regular graphs
• expected cycle count
• counterexamples