MC 302 Graph Theory Tuesday, 10504

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MC 302 Graph TheoryTuesday, 10/5/04

• Todays reading exercises
• Still 3.1, Exercises 2,7,11
• Today (Questions?)
• Return, go over, HW 2
• Questions for Exam 1, Thursday, 10/7
• More on Trees

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Longest Paths and Degree-1 Vertices in a Tree

• From last week A graph is a tree IFF there is a
  unique path between each pair of vertices.
• Theorem 2.2 Let T be a tree with at least two
  vertices, and let P be a path in T of longest
  possible length. If u and v are the first and
  last vertices of P, then both u and v have degree
  1.
• Proof Because the length of P is at least 1
  (why?), u and v have one neighbor on P. If
  either of them, WLOG u, has another neighbor w,
  then w is either on or off P. But in both cases
  this leads to a contradiction...

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Proof technique review Mathematical Induction

• Principle of Mathematical Induction Given some
  statement S that involves an integer n, suppose
  that
• 1. S is true for some particular integer n0.
• 2. IF S is true for some integer k ? n0, THEN
  S is true for the next integer k 1.
• Then S is true for all integers n ? n0.

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Using Mathematical Induction

• Step 1. Prove the base case
• Prove S is true when n n0.
• Step 2. Prove the inductive case as follows
• 2a. Begin by saying Suppose that S is true for
  some value k ? n0, which means that
• 2b. Now say We must show that S is true for n
  k 1, which would mean that
• 2c. Examine what you want to prove in 2b to see
  how it follows from your assumption in 2b.

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Example Prove that n2 n is divisible by 2 for
all n ? 1.

• Base case.
• Inductive case.
• 2a. Suppose that the statement is true for some
  value k ? n0, which means that
• 2b. We must show that S is true for n k 1,
  which would mean that
• How can we use our assumption in 2a to prove what
  we want in 2b?

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Degree-1 Vertices, and of edges, in a Tree

• Cor. 2.3 Every tree with at least 2 vertices has
  at least 2 vertices of degree 1.
• Theorem 2.4 If T is a tree with n vertices, then
  it has precisely n-1 edges.
• Proof is by induction on the number n of
  vertices.
• Base case When n 1, T is
• Inductive case
• Suppose it's true if T is a tree with k vertices
  
• Show it's true if T is a tree with k1 vertices...

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Spanning Trees

• A spanning tree of a graph G is a subgraph T
  that (a) contains all the vertices of G, and (b)
  is a tree (connected and acyclic).
• Theorem. Every connected graph G has at least one
  spanning tree.
• Proof?
• Hint If G has a cycle, we can remove an edge
  from it without disconnecting G.

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Some Characterizations of Trees

• Theorem For a graph G with n vertices, the
  following statements are equivalent (often
  abbreviated FSAE)
• 1. G is a tree.
• 2. G is connected and acyclic.
• 3. G is connected and has n-1 edges.
• 4. G is acyclic and has n-1 edges.
• 5. There is a unique path between any two
  vertices of G.
• Proof How do we prove an FSAE Theorem?

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Tree statement implications

• Prove 3 ? 1?
• Prove 4 ? 1?