MGF 1107 Section 15.2 Euler Paths and Euler Circuits Puzzle books ...

Objective 2: The learner will use Euler’s Theorem.
Euler’s Theorem:
Number of Odd Vertices in a Connected Graph
(Must be an even number)
Exactly two
•At least one
Euler path but
no Euler circuit
•Each Euler path
must start at
one of the odd
vertices and
end at the
other one
Zero
•At least one
Euler circuit
(which is also
an Euler path)
•It can start and
end at any
vertex.
More than two
•No Euler paths
and No Euler
circuits
Example 2: Given the number of odd and even vertices, determine if the graph has an Euler path
(but no Euler circuits), an Euler circuit, or neither.
a) The graph has 24 even vertices and no odd vertices.
b) The graph has 84 even vertices and two odd vertices.
c) The graph has 55 even vertices and six odd vertices.
Example 3: Explain why the graph in problem 7 on page 837 has at least one Euler path. Use trial
and error to find one such path.
Solution: A (3) – odd, B (3) – odd, C (2) – even, D (4) – even, E (2) even.
The graph has exactly two odd vertices so it has at least one Euler path.
One path : A, C, D, A, B, D, E, B