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10302009 Advanced Graph Theory HW - -2Tr 1 vertices So the...

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Advanced Graph Theory HW 10-30-2009 HW 3.1 If we begin with a complete graph Kr , and assign directions to every edge, the resulting object is called a tournament of order r. If a, b are vertices of a tournament, and the edge ab is directed from a to b, then we write a → b. A tournament is called transitive if whenever a → b and b → c , then a → c also. Define the transitivity Ramsey number T(r) to be the smallest n such that every tournament on n vertices contains a transitive tournament on r vertices. Prove T(r) ≤ - 2r 1 . Proof: The assertion is trivial for r ≤ 1 (obviously, T(0)=0 and T(1)=1). We assume that r ≥ 2. And it is easy to know T(2)=2. Part 1: proof of - ≥ ( ) 2Tr 1 T r . Suppose we have a tournament on
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Unformatted text preview: -2Tr 1 vertices. So the degree of every vertex is - -2Tr 1 1 . Then we choose the vertex, say v , with the biggest number of edges go out, say x . So ≥ (- - )= ( - ) x 12 2Tr 1 1 T r 1 Thus v point to at least -Tr 1 vertices, which can make sure of a transitive tournament on -r 1 vertices. So this transitive tournament plus v is a transitive tournament on r vertices. That is, in any tournament on -2Tr 1 vertices, we can find a transitive tournament on r vertices. Because T(r) is the smallest number such that every tournament on T(r) vertices contains a transitive tournament on r vertices, we know- ≥ ( ) 2Tr 1 T r . Thus we have: ≤- ≤- ≤⋯≤ -= - × = -Tr 2Tr 1 22Tr 2 2r 2T2 2r 2 2 2r 1 Page 1...
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