Name __________________________________
MAT 102 – Survey of Contemporary Mathematical Topics
Professor Pestieau
Assignment 3 – Graph Theory (Part I)
Due in class on Monday, December 5
Show and justify your work neatly on all the following problems to receive full
credit.
Problem 1 – Car Logos
[35 pts]
Consider the six car logos shown below.
Using the table provided on the next page analyze each car logo by doing the following:

In the 1st column, draw a graph representing the logo. For simplicity, assume the lines
or curves you see in the logo have no width and always meet at a vertex. Label all
vertices in the graph with letters (A, B, C, …)

In the 2nd column, find out if the graph of the logo is Eulerian. If it is, provide an
Euler circuit for the graph. If it is not, explain why.

In the 3rd column, find out if the graph of the logo is pen-traceable (i.e. it can be
drawn without lifting your pen from the paper and without tracing over the same
segment twice). If it is, show a tracing that satisfies this condition. If it is not, explain
why.

In the 4th column, find out if the graph is Hamiltonian.
Hamiltonian circuit for the graph.
Car
a) AUDI
b) HONDA
c) VW
d) MERCEDES
Graph of Logo
Is the Graph
Eulerian?
If it is, provide a
Is the Graph
Pen-Traceable?
Is the Graph
Hamiltonian?
e) TOYOTA
f) RENAULT
Problem 2 – Crossing Bridges
[20 pts]
The figure below depicts a system of bridges and land areas that a mathematician designed for
the garden of his East Hampton mansion.
a)
Draw a graph that corresponds to this system of bridges and land areas.
b)
Can the mathematician take a walk and cross each bridge exactly once? If yes, shown
such a walk that he/she could take. If no, explain why he/she can’t do the walk. Justify
your answers from the graph above.
c)
Suppose our mathematician likes to play with dynamite and suddenly decides to blow up
the bridge from H to I. Answer the same question now.
Problem 3 – Traveling in South America
Exercise # 46 on page 588
[25 pts]
Problem 4 – Route Planning
Exercise # 22 on page 604
[20 pts]
Bonus Problem – The Knight’s Move in Chess
[10 pts]
In chess, a knight can move two squares either vertically or horizontally and then one square in
a perpendicular direction.
Depending on where the knight is situated on an 8 x 8 chessboard, he has a minimal mobility of
2 moves when he’s in a corner (check this!) and a maximal mobility of 8 moves when he’s at
least two squares away from the edge (as shown in the figure below).
Let C be the graph with 64 vertices, all corresponding to the squares of a chessboard.
Two vertices of C are joined by an edge whenever a knight can go from one of the
corresponding squares to the other in one move.
Does C have a Euler walk? Justify your answer.
Note: You don’t have to draw C to answer this question!