Unformatted text preview: had winning records. The sum of the outdegrees of these vertices would then exceed the sum of the indegrees by at least n -2. If we add Alice's score to the sums, the sum of the outdegrees would exceed the sum of the indegrees by at least n (since Alice has a better record than the players tied for second). In any digraph, however, the outdegree and indegree sums are the same, so it follows that George's indegree must be at least n greater than his outdegree. This is impossible. A similar argument shows that the players tied for second cannot have losing records, so these players must have an equal number of wins and losses. Since each player has n -1 matches, it follows that n - 1 is even, so n is odd. 14. This is a type I scheduling problem. The shortest time is 20 hours. There are two critical paths: St, V, Q, R, S, T, U and St, V, Q, R, S, X, U. St( -,0) N(Y, 12) T(S, 15) or U(T, 20)...
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- Summer '10
- Graph Theory, Vertex, Summation, Shortest path problem, players, The Players