Section 7.1
Introduction to
Graph Theory
Euler trails and circuits
Can you draw the following picture without
lifting your pencil?
Euler trails and circuits
Can you draw the following picture without
lifting your pencil?
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
The “geographic” context
Can you tour the following city using every
bridge exactly once?
The graph model
Can you tour the following city using every
bridge exactly once?
The graph model
Can you tour the following city using every
bridge exactly once?
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 (“e”s) 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.
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.
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.
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.
Examples
Deg(1) = ____
Deg(3) = ____
Deg(5) = ____
Deg(2) = ____
Deg(4) = ____
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.
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.
Euler’s 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.
Practice
Find an Euler trail or Euler circuit or
explain why none exists.
Practice
Find an Euler trail or Euler circuit or
explain why none exists.
Exercises from Section 7.1
1-4, 6-11, 13-16