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    Forests and Eulerian Graphs

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    Let F be a forest. Add a vertex x to F and join x to each vertex of odd degree in F. Prove that the graph obtained in this way is randomly Eulerian from x, and every graph randomly Eulerian from x can be obtained in this way.

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    Solution Preview

    (1)
    Proof that a graph G obtained by adding a vertex V to a forest F and connecting V with each vertex of odd degree in F is Eulerian and V is a randomly Eulerian vertex in it.

    A forest is a disjoint union of trees, and a tree is a a graph in which any two vertices are ...

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