A tournament with n players/teams is an orientation of the complete graph Kn. The graph models wins and losses in a round robin competition. Every player/team plays one game against each of the...

A tournament with n players/teams is an orientation of the complete graph Kn. The graph models wins and losses in a round robin competition. Every player/team plays one game against each of the others, and the edge between vertex a and vertex b is directed from a to b if b defeats a. Assume no ties are allowed.

(a) Does every tournament have a sink? (A champion who defeats all the others?)

(b) Orient K₅
so every player/team wins 2 games and loses 2 games.

(c) Can K₆
be oriented so every player/team wins the same number of games?

(d) Prove that if a has more wins than losses, then there must be a b who has more losses than wins.

(e) Prove that every tournament contains a directed Hamilton Path.