For every d >= 3 and n = k d with k >= 2, there exist directed d-regular graphs on n vertices whose random cycle-factors have expected cycle count strictly larger than k H_d, disproving the conjecture that the disjoint union of K_d^circ maximizes it.
Counting 1-factors in regular bipartite graphs.Journal of Combinatorial Theory, Series B, 72(1):122–135
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Counterexamples to an Extremal Conjecture for Random Cycle-Factors
For every d >= 3 and n = k d with k >= 2, there exist directed d-regular graphs on n vertices whose random cycle-factors have expected cycle count strictly larger than k H_d, disproving the conjecture that the disjoint union of K_d^circ maximizes it.