Download Euler and Hamilton Paths/Circuits

Euler and Hamilton Paths/Circuits
A ____________ is a figure made up of points_________ connected by
non-intersecting edges.
Look at the graph below:
A ____________is the ______________ of
two edges.
 A _____________ is ______ if it is
connected to an __________
_______________________

A ____________ is _______ if it is
connected to an _________
_________________________
An ________ joins any two _________. It
can be __________ or ______________
Euler Path: a graph is an Euler path if it can be________________
_______________________________________________________
_______________________________________________________
 Vertices may be passed through ________________________
 The starting and ending points _________________________
Euler Circuit: a circuit is similar to a Euler path, except
_________________________________________________
_________________________________________________
Examples of Euler paths and circuits:
Trace each graph to determine if it is an Euler Path or an Euler Circuit, or
neither state why.
Ex 1:
Ex 2:
Ex 3:
What is the relationship between the nature of the vertices (odd or even)
and the kind of graph (path or circuit)?
Complete the table below using the following graphs.
Graph 1
Graph 4
Graph #
Graph 3
Graph 2
Graph 5
number of odd
vertices
Graph 6
number of even
vertices
Is it a…
Euler path = P
Euler circuit = C
Neither = N
1
2
3
4
5
6
Conclusions: Based on the observations of your table above:
1) A graph with all vertices being even contains an Euler _________
2) A graph with ______ odd vertices and _____________ contains an
Euler ________.
3) A graph with more than 2 _______ vertices does not contain an
Euler _______________
To name a path or circuit you list the vertices ________________________.
Example 1a: Name an Euler circuit
B
a) One possible solutions is
C
D, E, F, A, D, C, A, B, D
A
b) Can you find another one?
D
________________________
F
E
Example 1b:
Given A, B, E, F, B, C, D, F, E, D is this Euler path or circuit or neither? _______________________
How can you tell? Explain your answer _____________________________________________________
____________________________________________________________________________________
Ex 2: Find an Euler circuit if possible, if not list an Euler path. _____________________________________
C
B
F
E
A
D
Hamilton Paths and Circuits
A __________________________ is a continuous path that passes through every _______________once and
only once.
A __________________________ is a Hamilton path that begins and ends at the same vertex.
(the starting /end vertex will be the only vertex touched twice.
How is a Hamilton Path different from a Euler path or Circuit?__________________________________
____________________________________________________________________________________
Example 3:
Find a Hamilton Path __________________________________________
(you name them like an Euler path or a Euler Circuit.)
M
S
R
K
A
O
T
H