Exam 4 Math 375 Spring 2014
Name:
Instructions: Use the techniques discussed in class.
100
1. The
complete
graph
on
100
vertices,
K
= 4950 edges and
100 , has
2
100
= 75287520 copies of K5 in it. Prove that no matter which 494 edges
5
you delete from K100 , the resulting graph has at least one copy of K5 in it.
In other words, prove that every graph with 100 vertices and 4456 edges has
at least one copy of K5 in it. (Kuratowski’s theorem does not apply here.)
2. The following matrix represents a tournament on 6 vertices. There is a
directed edge of the form a → b in the tournament if and only if there is a
1 in row a, column b in the matrix. Find a directed Hamilton path in the
tournament (a path of the form a → b → c → d → e → f where abcdef is a
permutation of 123456).

0
1

0

0

1
1
0
0
1
1
0
1
1
0
0
1
1
0
1
0
0
0
0
1
1
0
1
0
1
0
0

0
0

1

0

1
0
3. In class we proved that there are nn doubly-rooted trees on vertex set [n]
by exhibiting a one-to-one correspondence between doubly-rooted trees and
functions from [n] to [n].
(a) Let T be the doubly-rooted tree depicted below. Produce the corresponding function in two-line format.
6
11
1
5
8
10
2
9
4
7
3
1 2 3 4 5 6 7 8
(b) Let f : [8] → [8] be the function f =
. Produce
2 6 2 5 2 5 6 2
the corresponding doubly-rooted tree.
4. Are the two graphs G and H below isomorphic or not? If so, prove it. If
not, identify a structure in G that cannot exist in H or vice versa. Restrict
yourself to rigorous graph properties like vertex degrees, paths, cycles, etc
and avoid vague notions like “left” or “right” or “diagonal” or “sideways” or
“can be rotated into” or “on the boundary” etc.
G
H
2