Problem 1.
Let G = (V, E ) be a bipartite graph.
Let V = AB be the bipartition of the vertex set.
of a bipartite graph we have
A
E =
2
B
2
and
By the definition
E = .
It follows from the pigeonhole principle that |A| 3 or |B | 3.
without loss of generali
Exam 1
21-484 Graph Theory
Name (andrewid) - X
1. (16 points) Let T be a tree with exactly one vertex of degree i for each i = 2, 3, . . . 100 and all other
vertices of degree 1. Determine, with proof, the number of vertices of T .
Let l be the number of
Last Name:
First Name:
Graph Theory
Exam 2
March 20, 2013
Problem Points Score
1
16
2
16
3
16
4
16
5
12
6
12
7
12
Total
100
1. (16 points) Show that a connected graph with at least two edges is a
block if and only if any two adjacent edges lie on a cycle.
Last Name:
First Name:
Graph Theory
Exam 3
April 20, 2013
Problem Points Score
1
16
2
16
3
16
4
16
5
12
6
12
7
12
Total
100
1. (16 points) Prove that the chromatic number of a graph is the maximum
of the chromatic numbers of its blocks.
It is easy to see
HW1
21-484 Graph Theory
SOLUTIONS (hbovik)
Diestel 1.2: Let d N and V := cfw_0, 1d ; thus, V is the set of all 01 sequences of length d. The
graph on V in which two such sequences form an edge if and only if they dier in exactly one position
is called the
HW2
21-484 Graph Theory
SOLUTIONS (hbovick) - Q
1, Diestel 1.27: Prove or disprove that a graph is bipartite if and only if no two adjacent vertices
have the same distance from any other vertex.
Proposition 1.6.1 in Diestel states that a graph is bipartit
HW3
21-484 Graph Theory
SOLUTIONS (hbovik) - Q
1: Suppose that 13 people are each dealt 4 cards from a standard 52-card deck. Show that it is possible
for each of them to select one of their cards so that no two people have selected a card of the same
ran
HW4
21-484 Graph Theory
Name (andrewid) - X
1, Diestel 3.5: Deduce the k = 2 case of Mengers theorem (3.3.1) from Proposition 3.1.1.
Let G be 2-connected, and let A and B be 2-sets.
We handle some special cases (thus later in the induction if these occur
HW5
21-484 Graph Theory
SOLUTIONS (hbovik) - Q
1, Diestel 3.8: Let G be a k -connected graph, and let xy be an edge of G. Show that G/xy is
k -connected if and only if G cfw_x, y is (k 1)-connected.
Let G be a k -connected graph and let xy be an edge in
21-484 Graph Theory
HW6
Name (andrewid) - X
1: Draw K 7 on a torus with no edge crossings.
A quick calculation reveals that an embedding of K 7 on the torus is a 2-cell embedding. At that point,
it is hard to go wrong if you start drawing C 3 faces, altho
HW7
21-484 Graph Theory
Name (andrewid) - X
1: Given k and a k -coloring of a k -chromatic graph, prove that for any color c there is a vertex of color
c which is adjacent to vertices of every other color.
Let a k -chromatic graph have a k -coloring given
21-484 Graph Theory
HW8
SOLUTIONS (hbovik) - Q
1: Determine, with proof, the edge-chromatic number of the Petersen graph.
The Petersen graph has maximum degree 3, so by Vizings theorem its edge-chromatic number is 3 or 4.
We will prove that in fact the ed
HW8
21-484 Graph Theory
Name (andrewid) - X
1, Diestel 7.16: Prove the Erds-Ss conjecture for the case when the tree considered is a path.
oo
(Hint. Use Exercise 8 of Chapter 1.)
We seek to prove that the maximal number of edges in a graph with n vertices
HW10
21-484 Graph Theory
SOLUTIONS (hbovik) - Q
1, Diestel 9.3: An arithmetic progression is an increasing sequence of numbers of the form a, a + d, a +
2d, a + 3d . . . Van der Waerdens theorem says that no matter how we partition the natural numbers
int