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    Euler Digraph: Characterization

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    Page 90, Theorem 4.6 ( To show that a digraph is hamiltonian if and only if for each vertex v, indegree (v) =outdegree (v)

    Page 92 Exercise problem 4.1 ( A modified version of konigsberg problem where, two extra bridges are built. )

    Page 93, Figure 4.5 need to show as Hamilton. (Show that dodecahedron is hamiltonian)

    See attached file for full problem description.

    © BrainMass Inc. brainmass.com October 9, 2019, 7:34 pm ad1c9bdddf
    https://brainmass.com/math/graphs-and-functions/euler-digraph-characterization-122201

    Attachments

    Solution Preview

    Problem 4.1 and 4.5, I have attached figures.

    Problem 4.1:
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    We construct a graph of the present day Konigsberg bridges and see that there are exactly two vertices (A and B) have odd degree. We know that every connected graph with exactly 2 odd vertices have an Eulerian tour starting and ending at these vertices. Hence we can device such a route passing through all the edges. The attached figure shows such a route.

    Problem of figure 4.5
    --------
    We need to show the graph of dodecahedron is Hamiltonian. We do this by providing such a cycle explicitly. In the attached figure, follow the vertex sequence 1,2,3,...,19,20,1 which is such a Hamiltonian cycle.

    Theorem 4.6
    -------------
    Theorem: Let D be a connected non-trivial digraph. D is Eulerian if and
    only if id(v) = od(v) for every vertex. (id:in degree, od:out degree).

    Proof:
    A trail in a graph is a path without repeated edges.

    Suppose D is Eulerian. Consider any ...

    Solution Summary

    Here we prove the Necessary and sufficient condition for a digraph to be Eulerian. It also solves three problems from Graphs and Digraphs concerning a modified Konigsberg problem and provide a Hamiltonian cycle for dodecahedron.

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