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arXiv · 2604.26101

Counterexamples to an Extremal Conjecture for Random Cycle-Factors

Rishikesh Gajjala
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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.

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BibTeXRIS

Rishikesh Gajjala. 2026-04-28. Counterexamples to an Extremal Conjecture for Random Cycle-Factors. https://arxiv.org/abs/2604.26101

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