Introduction to

1
Section 7.1

• Introduction to
• Graph Theory

2
Euler trails and circuits

• Can you draw the following picture without
  lifting your pencil?

3
Euler trails and circuits

• Can you draw the following picture without
  lifting your pencil?

4
Euler trails and circuits

• Can you draw the following picture without
  lifting your pencil?

3, 2, 1, 5, 4, 6, 5, 2, 6, 3, 4
5
The geographic context

• Can you tour the following city using every
  bridge exactly once?

6
The graph model

• Can you tour the following city using every
  bridge exactly once?

7
The graph model

• Can you tour the following city using every
  bridge exactly once?

8
Definitions

• A graph consists of vertices (or nodes) and edges
  connecting pairs of vertices.
• A walk is a list v1, e1, v2, e2, , vn, where the
  edges (es) connect the vertices they fall
  between them. When there are no multiple
  (parallel) edges, we do not need to list the
  edges in our description of a walk. A walk is
  closed if v1 vn.

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Definitions

• A trail is a walk with no repeated edges, and a
  closed trail is called a circuit. An Euler trail
  (or Euler circuit) is one that uses every edge in
  the graph.
• A cycle is a nontrivial circuit in which the only
  repeated node is the first/last one.

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Example

• The walk C,1,D,5,B is a trail.
• The walk A,2,C,3,A is a circuit.
• Since edges 2 and 3 have the same endpoints (A
  and C), we call them multiple edges or
  parallel edges.
• The walk A,7,B,5,D,5,B,6,A is a closed walk that
  is not a circuit.

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More terminology

• A loop is an edge that has the same vertex at
  each end.
• The degree of a vertex is the number of edges
  coming out of the vertex.
• A simple graph is a graph with no loops or
  multiple edges.
• A graph is connected if between every pair of
  vertices there is a walk.

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Examples

• Deg(1) ____ Deg(2) ____
• Deg(3) ____ Deg(4) ____
• Deg(5) ____ Deg(6) ____
• This is a simple graph because it has no loops or
  multiple edges.
• The walk 1,2,3,4,6,2,5 is a trail.
• The walk 1, 2, 6, 5, 1 is a circuit.
• The walk 3, 2, 1, 5, 4, 6, 5, 2, 6, 3, 4 is an
  Euler trail.

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Practice

• Find the number of nodes and the number of edges
  in G.
• Find the degree of each node.
• Compare the sum of the degrees and the number of
  edges.

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Eulers result

• Theorem. If every vertex of a connected graph has
  even degree, then the graph has an Euler circuit.
• Corollary. If there are exactly two vertices of
  odd degree in a connected graph, then the graph
  has an Euler trail.

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Practice

• Find an Euler trail or Euler circuit or explain
  why none exists.

16
Practice

• Find an Euler trail or Euler circuit or explain
  why none exists.

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Exercises from Section 7.1

• 1-4, 6-11, 13-16