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 p
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
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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
bloc
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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
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
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.
Propo
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
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 spe
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.
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
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 c
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-chroma
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
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