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I need help witha  graph theory questionLet T be a tree with more than one vertex....

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ashifs | Student, Undergraduate | Honors

Posted November 9, 2011 at 2:32 AM via web

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I need help witha  graph theory question

Let T be a tree with more than one vertex. Prove that
T must have atleast two verticies of degree 1.

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nathanshields | High School Teacher | (Level 1) Associate Educator

Posted January 20, 2012 at 7:25 AM (Answer #1)

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Proof by induction.

I.  Initial condition.  Let T be a tree with 2 vertices.  Then T has two vertices of degree 1, by definition of tree (connected graph without cycles).

II: Rule of induction.  Let b U be a tree with n, and addume that U has at least 2 vertices of degree 1.  Adding another vertex will either (i) maintain the number of vertices of degree 1 (by connecting to a vertex of degree one) or (ii) increase the number of vertices of degree 1 (by connecting to a vertex of degree greater than 1).

Taking I and II together means that any tree with 2 or more vertices will contain at least two vertices of degree 1.  QED

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