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Math Help - Directed Hamiltonian cycle

  1. #1
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    Directed Hamiltonian cycle

    G is a simple directed graph on n vertices and each verticle is the head of at least n/2 arcs and each vertice is the tail of at least n/2 arcs. Prove that G contains a directed Hamiltonian cycle.

    I would be grateful, if you could help me!
    Last edited by doug; October 14th 2010 at 06:45 AM.
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  2. #2
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    I don't know how to prove this, but this looks similar to the Ghouila-Houiri theorem.
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  3. #3
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    Quote Originally Posted by emakarov View Post
    I don't know how to prove this, but this looks similar to the Ghouila-Houiri theorem.
    Thanks, but unfortunately I don't find the prove of Ghouila-Houiri theorem, the only reference to this theorem is wikipedia and there is no prove at wikipedia.
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  4. #4
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    Consider the largest directed cycle C possible in G. If It is not Hamiltonian, then consider a vertex v not in C. Since both the in-degree and out-degree of v is more than half the size of C, there will be v' and v" in C such that the edge (v',v") is in C, and the edges (v',v) and (v,v") are in G. Can you fill in the details ?
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