For $n\ge3$ let $a(n)$ be the minimum number of Hamiltonian paths in a strong (i.e., strongly connected) tournament of order $n.$

Where is $a(n)$ discussed in the literature? Is the exact value known? If not, what nontrivial bounds are known?

**My attempt:** (It is assumed throughout that $n\ge3.$)

**(I)** $a(n)$ is always odd.

This is because of Rédei's theorem, that a tournament has an odd number of Hamiltonian paths.

**(II)** $a(n+1)\ge a(n)+n-1.$

This follows easily from the fact that every strong tournament of order $n+1$ contains a cycle of length $n.$

**(III)** $\lfloor(n-1)^2/2\rfloor+1\le a(n)\le3\cdot2^{n-4}+3.$

These are the **trivial bounds**. The lower bound (which happens to be OEIS sequence A099392) follows from **(II)** by induction. For $n\gt4,$ the upper bound (OEIS sequence A060013) is achieved by taking the transitive tournament with vertices $v_1,\dots,v_n$ and edges $v_iv_j\ (i\lt j),$ and reversing the edges $v_1v_3$ and $v_2v_n.$

**(IV)** $a(3)=3,\ a(4)=5,\ a(5)=9,\ a(6)\in\{13,15\}.$

That's all I know. I already asked this question on math.stackexchange.com without result.