MATH 310, FALL 2003 (Combinatorial Problem Solving) Lecture 3, Friday, September 5

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MATH 310, FALL 2003(Combinatorial Problem
Solving)Lecture 3, Friday, September 5
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Complete graph Kn.

• A graph on n vertices in which each vertex is
  adjacent to all other vertices is called a
  complete graph on n vertices, denoted by Kn.

K20
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Some complete graphs

• Here are some complete graphs.
• For each one determine the number of vertices,
  edges, and the degree of each vertex.
• Every graph on n vertices is a subgraph of Kn.

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Example 2 Isomorphism in Symmetric Graphs

• The two graphs on the left are isomorphic.
• Top graph vertices clockwise a,b,c,d,e,f,g
• Bottom graph vertices clockwise 1,2,3,4,5,6,7
• Possible isomorphisma-1,b-5,c-2,d-6,e-3,f-7,g-4.

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Example 3 Isomorphism ofDirected Graphs

• Some hints how to prove non-isomorphism
• If two graphs are not isomorphic as undirected
  graphs, they cannot be isomorphic as directed
  graphs.
• (p,q) label on a vertex indegree p, outdegree
  q.
• Look at the directed edges and their (p,q,r,s)
  labels!

(2,3)
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(p,q)
(r,s)
(p,q,r,s)
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3
e
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1.3. Edge Counting

• Homework (MATH 3101F)
• Read 1.4. Write down a list of all newly
  introduced terms (printed in boldface)
• Do Exercises1.3 4,6,8,12,13
• Volunteers
• ____________
• ____________
• Problem 13.
• News
• Please always bring your updated list of terms to
  class meeting.
• Homework in now labeled for easier
  identification
• (MATH 310, , Day-MWF)

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Theorem 1

• In any graph, the sum of the degrees of all
  vertices is equal to twice the number of edges.

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Corollary

• In any graph, the number of vertices of odd
  degree is even.

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Example 2 Edges in a Complete Graph

• The degree of each vertex of Kn is n-1. There are
  n vertices. The total sum is n(n-1) twice the
  number of edges.
• Kn has n(n-1)/2 edges.
• On the left K15 has 105 edges.

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Example 3 Impossible graph

• Is it possible to have a group of seven people
  such that each person knows exactly three other
  people in the group?

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Bipartite Graphs

• A graph G is bipartite if its vertices can be
  partitioned into two sets VL and VR and every
  edge joins a vertex in VL with a vertex in VR
• Graph on the left is biparite.

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Theorem 2

• A graph G is bipartite if and only if every
  circuit in G has even length.

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Example 5 Testing for a Bipartite Graph

• Is the graph on the left bipartite?