September 23, 2013
Warm-up.
1. Please sit in a seat that you will stay in for a while.
2. Sketch one of your pictures from question 2 from
the worksheet that you got on Thursday in class. Do
not state whether or not it can be drawn without
lifting your pencil (no back-tracking.)
3. What is the next term in the sequence?
a. 4, 11, 18, 25, ______
b. 1, 4, 9, 16, ______
graph - a finite set of dots (vertices)
and connecting links (edges)
Graphs can represent city road
maps, air routes, etc.
vertex - (pl. vertices) a point (dot) in a
graph where the edges meet
edge - a link that joins two vertices in
a graph
B
C
A
D
G
E
F
path - a connected sequence of edges
showing a route, described by
naming the vertices traveled
circuit - a path that starts and ends at
the same vertex
B
G
E
C
A
D
F
September 23, 2013
valence - of a vertex is the number of
edges touching the vertex
Connected? or Not?
a.
b.
B
C
A
E
D
G
F
connectedness every vertex can be reached from any
other vertex by traversing the edges
B
of the graph
E
G
C
d.
c.
A
D
F
These are Pentammies.
1.
http://www.brainbashers.com/showpuzzles.asp?formpost=Y&gotcha=Y&page=1&field=cgot
2. How can you build pig pens so that you can put nine
These are not Pentammies.
pigs in four pens such that each pen has an odd
number of pigs?
What makes something a Pentammie?
Is the figure to the right one?
A graph that has an Euler Circuit1. has even valent vertices
2. is connected
A graph that has an Euler Path1. has only 2 odd valent vertices
2. is connected
bridge - an edge of a connected graph
that if removed from the graph
will cause the graph to become
disconnected
B
G
E
C
A
D
F
September 23, 2013
Fleury's Algorithm
1. is the graph connected?
2. do all the vertices have even
valence?
a. graph is connected
b. graph has all vertices with even valence
Steps:
1. Pick any vertex as a starting point.
2. Mark your path as you move from vertex to vertex, travel along
any edge that you wish, except, DO NOT travel along an edge
that is a bridge for the graph formed by the edges that have not
yet been traveled, unless it is the only edge that can be traveled.
3. Continue until you meet your starting point.
If a connected graph has 2 odd vertices, you can
use Fleury's Algorithm, but start at one odd
vertex and end at the other.
September 23, 2013
Warm-up
Please write today's date in blue book (9/9/13) and
then complete the problems below.
What is the next term in the sequence?
1. o,t,t,f,f,s,s,e,_______
2. 77, 49, 36, 18, ________
Euler Circuit vs Euler Path
Both cover every edge of the graph without backtracking.
Euler Circuit
Euler Path
1. graph is connected
2. graph has all vertices with
even valence
1. graph is connected
2. graph has only two vertices
of odd valence
cover every edge of the graph
starting and ending in the
same place
cover every edge of the graph
starting at one odd vertex and
ending at the other
Eulerize!!!!!
September 23, 2013
Warm-up
Please write today's date 9/10/13 in the blue book
and then complete this problem.
Sketch an optimal Eulerization of the graph.
*remember
to use color
or dot your
duplicate
edges
September 23, 2013
Warm-up:
Show an optimal Eulerization of the graph using
color or dotted edges.
googlemaps.com
googlemaps.com
googlemaps.com
September 23, 2013
Warm-up. Complete this Latin Calc./ Ken-Ken Puzzle.
Use digits 1-4 only once in each row and column. (9+
means those 4 numbers add to 9, 16X means those 3
numbers multiply to 16 ...)
Sketch a graph that is described. IF IT IS NOT POSSIBLE, WRITE
NOT POSSIBLE. Your answers will vary. There is more than one
correct answer. You just need to sketch one correct answer.
a graph with 4 vertices, two with valence of 2 and two with valence of 3.
a graph with 4 vertices, one with valence of 3 and two with valence of 2.
Sketch a graph that is described. IF IT IS NOT POSSIBLE, WRITE
NOT POSSIBLE. Your answers will vary. There is more than one
correct answer. You just need to sketch one correct answer.
A graph with 2 odd vertices that can be Eulerized with one duplicate edge
Sketch a graph that is described. IF IT IS NOT POSSIBLE, WRITE
NOT POSSIBLE. Your answers will vary. There is more than one
correct answer. You just need to sketch one correct answer.
a graph that is not connected
a graph with 2 odd vertices that cannot be Eulerized with one duplicate edge
(it will take more that one duplicate edge)
a graph with five vertices of valence 1, 1, 2, 2, and 3.
cannot have an odd number of even valences
September 23, 2013
Sketch a graph that is described. IF IT IS NOT POSSIBLE, WRITE
NOT POSSIBLE. Your answers will vary. There is more than one
correct answer. You just need to sketch one correct answer.
Sketch a graph that is described. IF IT IS NOT POSSIBLE, WRITE
NOT POSSIBLE. Your answers will vary. There is more than one
correct answer. You just need to sketch one correct answer.
Warm-up:
Warm-up:
a graph with a vertex of valence 0
a graph that has an odd number of vertices and an Euler circuit
a graph with a vertex of valence 0
a graph that has an odd number of vertices and an Euler circuit
a graph that has a bridge
a graph that has all vertices with even valance, but does not have an Euler circuit
a graph that has a bridge
a graph that has all vertices with even valance, but does not have an Euler circuit