Let T be a transitive tournament with n > 1 vertices.
Prove (by contradiction or otherwise) that T is not strongly connected.
This one has had me stumped for a little while.
As the question suggests proof by contradiction that's my aim but I haven't really progressed.
I can see (from looking at and drawing several examples of transitive tournaments) that there will be a vertex with an in-degree of zero, which would result in there being no way for the graph to be strongly connected, so I feel it's somewhere down this line of thinking, although I doubt that could be used in a proof by contradiction.
Any help or pointing in some direction would be greatly appreciated. Thanks!
Best Answer
Take two vertices $x$ and $y$. Assuming $T$ is strongly connected, there is a directed path from $x$ to $y$. As $T$ is transitive, there is an edge $x\to y$. Inversely, there is a directed path from $y$ to $x$, and therefore an edge from $y\to x$. But now there are two edges connecting $x$ and $y$.