The maximum number of Hamiltonian cycles in tournaments

Source abstract

Let $C(n)$ denote the maximum number of directed Hamiltonian cycles in a tournament on $n$ vertices. We prove that, as $n\to\infty$, $$C(n) = \left(\frac{\cosh 1}{\cos 1}\pm o(1)\right)\frac{(n-1)!}{2^n}.$$ The bound is asymptotically attained by the regular tournament with vertices $\mathbb Z/n\mathbb Z$ where $(i,j)$ is a directed edge if $j - i \pmod n< n/2$. This answers a question of Alon and proves a conjecture of Wormald. Among the other implications of these results is asymptotically determining the maximum number of Hamiltonian paths in a tournament on $n$ vertices. We provide our own exposition of these proofs, which were originally found by GPT-5.6.