Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2011: Test 3
Name:
Problem
1
2
3
Total
Points Score
40
40
20
100
1
Q1: (40pts)
(a) Show that a graph G with minimum degree at least contains a path of
length .
(b) Suppose tha
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2010: Test 4
Name:
Problem
1
2
3
Total
Points Score
40
40
20
100
1
Q1: (40pts)
How many ways are there of k -coloring the squares of the above picture if
the group acting is e
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2010: Test 3
Name:
Problem
1
2
3
Total
Points Score
40
40
20
100
1
Q1: (40pts)
Let (1), (2), . . . , (n) be an arbitrary permutation of [n]. Show that there
exists i such that
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2010: Test 1
Name:
Problem
1
2
3
Total
Points Score
40
40
20
100
1
Q1: (40pts)
The sequence a0 , a1 , . . . , an , . . . satises the following:
a0 = 1 and
an 6an1 = 5n
for n 1
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2009: Test 4
Name:
Problem
1
2
3
Total
Points
40
40
20
100
1
Score
Q1: (40pts)
How many ways are there of k -coloring the squares of the above picture if
the group acting is e
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2009: Test 3
Name:
Problem
1
2
3
Total
Points
40
40
20
100
1
Score
Q1: (40pts)
Prove that if n > 0 and a1 , a2 , . . . , an+1 are distinct positive integers then
there is a pa
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2009: Test 2
Name:
Problem
1
2
3
Total
Points
40
40
20
100
1
Score
Q1: (40pts)
The sequence a0 , a1 , . . . , an , . . . satises the following:
a0 = 1 and
an 5an1 = 4n
for n 1
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2009: Test 1
Name:
Problem
1
2
3
Total
Points
40
40
20
100
1
Score
Q1: (40pts)
k as, k bs and n 2k cs are placed on the vertices of an n vertex polygon
so that each a is follo
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2008: Test 4
Name:
Problem
1
2
3
Total
Points
33
33
34
100
1
Score
Q1: (33pts)
(a) Show that if you Red-Blue color the edges of K9 then either there is a
vertex with Red degre
Department of Mathematics
Carnegie Mellon University
21-301 Combinatorics, Fall 2008: Test 3
Name:
Problem
1
2
3
Total
Points
33
33
34
100
1
Score
Q1: (33pts)
Let G = K2,n denote the bipartite graph (X, Y, E ) where |X | = 2, |Y | = n
and an edge (x, y )