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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- Fall '09
- Graph Theory, vertices, transitive tournament, 2Tr-1 vertices