In a round-robin tournament, every two distinct players play against each other just once. For around-robin tournament with no tied games, a record of who beat whom can be described with a tournament digraph, where the vertices correspond to players and there is an edge $\langle x\to y\rangle$ iff $x$ beat $y$ in their game.

A ranking is a path that includes all the players. So in a ranking, each player won the game against the next ranked player, but may very well have lost their games against players ranked later—who ever does the ranking may have a lot of room to play favorites.

(a)Give an example of a tournament digraph with more than one ranking.

(b)Prove that every finite tournament digraph has a ranking.

(a)图为

(b)只要证明存在一个经过每个点的path即可，假设图中共有$n$个节点，设最长路径为$r$，并且设长度为$r$的全体路径为