Every [tournament](https://en.wikipedia.org/wiki/Tournament_(graph_theory)) contains a directed Hamiltonian path (a path visiting every vertex exactly once).


Suppose that $T$ is a tournament on $[n]:=\{1,\ldots,n\}$ for some integer $n\geq 2$ with exactly one directed Hamiltonian path. Does this imply that $T$ is isomorphic to the tournament $([n], E)$ where $E = \{(i,j)\in [n]: i<j\}$?



Tournaments with exactly one directed Hamiltonian path
It does. Suppose, wlog that the unique Hamiltonian path is $1\to 2\to\cdots\to n$. If $i\to n$ for all $i<n$ then you are done by using the inductive hypothesis on $[n-1]$, otherwise there exists an $i$ such that $n\to i$. 


Pick the smallest such $i$ and notice that if $i=1$ then $n\to 1\to 2\to\cdots \to n-1$ is a new Hamiltonian path, and finally if $i>1$ then $1\to 2\to \cdots \to i-1\to n\to i\to \cdots \to n-1$ is a new Hamiltonian path.


