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Math Help - Cycle in Graph Theory.

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    Cycle in Graph Theory.

    G = (V,E) has m edges and n vertices.
    Proof that if m >= n, G has a cycle.
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  2. #2
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    Quote Originally Posted by le_su14 View Post
    G = (V,E) has m edges and n vertices.
    Proof that if m >= n, G has a cycle.
    Definition: A connected graph that contains no cycles is called a TREE.

    Theorem: (This is a theorem that I assume you know). Given a graph that is a tree the # of edges is 1 less the # of vertices.

    Now given any (finite) graph we can break it up into connected peices. Those peices are called CONNECTED COMPONENTS. So for example an ordinary tree has 1 connected components.

    Definition: A graph where each connected component is a tree is called a FOREST.

    Using the theorem that was mentioned try to prove the following stronger result. Given a graph with c connected components with e edges and v vertices then e = v - c.

    Now it is pretty obvious that if e\geq v then there must exist a cycle.
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