MATH 581
Exam #1
Name:
1. (10 points) Using Dijkstras algorithm, nd a shortest path on this graph between A
and G, showing your work:
A
4
2
1
B
D
4
10
5
C
E
5
7
G
4
F
6
The following table shows the progression of Dijkstras algorithm; subscripts, which
yo
MATH 581
Exam #3 Solutions
1. (10 points) Find three distinct acyclic orientations of the following graph.
There are several such based on dierent possible choices of vertex-orderings.
2. (10 points) Describe (either in words or a diagram) a K5 -free grap
MATH 581
Final Exam
Please work on this independently. You are welcome to use any written resources (i.e.
books, e-texts, etc.), but do not copy information verbatim or consult with people either in
or out of this class.
1. (50 points) Draw a graph satisf
MATH 581
Problem Set #1 Solutions
1. Let G be a simple graph with n vertices and m edges. Show that G is a complete graph
if and only if m = n .
2
First, we shall prove that a complete graph on n vertices has n edges. In a complete
2
graph every vertex is
MATH 581
Problem Set #2 Solutions
1. Let GK be a graph with 64 vertices, one for each square of an 8 8 chessboard, and let
two vertices be adjacent if a chess king (capable of moving a single square orthogonally
or diagonally) could move from one to the o
MATH 581
Problem Set #3 Solutions
1. (10 points) Let B1 and B2 be two blocks of a connected simple graph G. Prove that
if B1 and B2 have a common vertex v , the graph G v produced by removing v from
G is disconnected.
If two blocks overlap in a single ver
MATH 581
Problem Set #4
1. (10 points) Recall that Gc is the graph in which all non-adjacent vertices of G are
adjacent and vice versa. Prove that if G has 11 or more vertices, G and Gc cannot
both be planar.
For brevity, let us denote the number of verti
MATH 581
Problem Set #5
Due on Wednesday, April 1.
1. (15 points) Let f (r) = R(3, 3, . . . , 3); that is, let f (r) be the least value of n such
r terms
that coloring a Kn with r colors guarantees a monochromatic K3 . For example, using
results seen in c
MATH 581
Practice Exam #3
1. (10 points) Describe (either in words or a diagram) a K6 -free graph on 11 vertices
with as many edges as possible. How many edges does it have?
This is a complete 5-partite (quintapartite) graph in which four parts have 2 ver