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notes-100225 - MATH 682 Notes Combinatorics and Graph...

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Unformatted text preview: MATH 682 Notes Combinatorics and Graph Theory II 1 Tournaments Definition 1. An orientation of the complete undirected graph is called a tournament . The definition above is of a natural sort of real-world structure: if there are n competitors in a tournament conducted via a round-robin (i.e. a structure such that each competitor plays each other competitor exactly once), then this structure is a handy combinatorial way to record the wins and losses, directing each edge from the winner towards the loser. The rather fuzzily-described real-world problem which naturally follows from this is: is there a sensible mapping from a tournament on n vertices to an ordering on an n-element set? Ideally we would have a total ordering, but doing so would be in many circumstances unfeasible: there is no total ordering we could in good faith associate with the directed cycle C 3 , or in a symmetric structure on any odd number of vertices, so it behooves us to relax the question to a partial ordering. What we mean by “sensible” above is admittedly rather vague, which takes this problem outside the realm of formal combinatorics, although it’s a question still of interest at mathematics’s intersection with the social sciences: economics and voting theory. However, we note that cycles are really the only problem with a sensible ordering, so we might pay specific attention to the question of what sort of tournament has no cycles and thus an unambiguous ordering of the participants: Proposition 1. The acyclic tournament on n vertices is uniquely defined up to permutation of vertices. Proof. We shall prove this by induction on n . The case n = 1 is trivial. By a result seen yesterday, an acyclic digraph has a vertex of zero indegree. Thus an acyclic tournament T on n vertices has a vertex v such that d- ( v ) = 0. Since the edges among V ( T )- v are orientations of a complete graph on n- 1 vertices, T- v is a tournament on n- 1 vertices, and it is acyclic, since it is a subgraph of an acyclic graph. Since T is an orientation of a complete graph, there are n- 1 edges incident on v , and since d- ( v ) = 0, they are all outbound edges. Thus, T can be specifically characterized as an acyclic tournament on n- 1 vertices, together with a vertex v , and edges from v to every other vertex of the graph....
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notes-100225 - MATH 682 Notes Combinatorics and Graph...

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