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I require help to understand this question, which is in my exercise list:

A group of n $ \ge $ 2 badminton players play a round-robin tournament (i.e. everyone plays against everyone else once). Each game ends in either a win or a loss. Show that it is possible to label the players $ P_1 $, $ P_2 \cdots $, $ P_n $ in such a way that $ P_1 $ defeated $ P_2 $, $ P_2 $ defeated $ P_3 \cdots $, and $ P_{n-1} $ defeated $ P_n $.

Supposedly, it is supposed to be proven using mathematical induction. But I don't see how induction can be used to solve this. Could someone please advise me on how to go about proving this problem?

Donald
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    This is finding a Hamiltonian path in a tournament (complete directed graph). And the inductive proof can be found here. – Parcly Taxel Sep 22 '16 at 16:00
  • I don't understand why we need to have $ V_{in} $ and $ V_{out} $ What is their purpose? – Donald Sep 22 '16 at 16:08
  • In the last step of the proof over there they are used to construct an explicit Hamiltonian path in the enlarged tournament. – Parcly Taxel Sep 22 '16 at 16:10
  • Oh. I see. Let me try to summarise it: If there is a set of 2 players, there is a clear winner and loser. Hence, it is trivial to calculate the edge information. If there are $ >2 $ players, then we partition them into different sets such that there are two players playing against each other in each set and we can use the inductive hypothesis to determine their ordering. Since we can determine the ordering for any $ n > 2 $, then we have proven the problem. Did I understand correctly? – Donald Sep 22 '16 at 16:21
  • Aha, yes.${}{}$ – Parcly Taxel Sep 22 '16 at 16:23
  • Actually, what do $ V_{in} $ and $ V_{out} $ mean? The set of winners and losers? – Donald Sep 22 '16 at 16:29
  • For a particular player $v$, $V_{in}$ is the set of players who defeated $v$. $V_{out}$ is the set of players $v$ defeated. – Parcly Taxel Sep 22 '16 at 16:31
  • I understand now. Thanks for your help! – Donald Sep 22 '16 at 16:37

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