1. No category

Johdatus diskreettiin matematiikkaan
Harjoitus 6, 21.10.2015
1. Are the following two graphs
isomorphic? (Explain why!)
2. Are the following two graphs
isomorphic? (Explain why!)
3. If n teams play a match against every other team, how many matches will be
played?
4. If a house has one outside door, show that, it has at least one room that has
odd number of doors.
5. (a) Does there exist a graph of six vertices with degrees {2, 3, 3, 3, 3, 3}?
(b) Does there exist a graph of five vertices with degrees {0, 1, 2, 3, 4}?
(c) How many graphs of four vertices there is with degrees of vertices being
{1, 1, 2, 2}? (isomorphic graphs are considered the same)
(d) How many graphs of 10 vertices there exists with degrees of vertices being
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1}? (isomorphic graphs are considered the same)
6. Define the vertices of the directed graph (Kuva 1) that have most 4-walks
between them (4-walk is a walk of length 4 steps). Between which vertices there is
least amount of 4-walks?
7. How many q-paths there is on a complete graph? (path is a walk that visits any
vertex at most once)
Kuva 1. which vertices have most 4-walks between them?
Johdatus diskreettiin matematiikkaan
Ohjaus 6, 21.10.2015
1. Can you measure exactly four liters of water if you have one full eight liter
bucket, and empty five and three liter buckets?1
2. Olkoon V = {a, b, c, d, e, f, g} ja E = {a, d}, {b, e}, {c, e}, {d, g}, {e, f } . Esitä
verkko (V, E) geometrisena kaaviona ja määritä sen naapurimatriisi. Esitä myös
kärkijoukon V 0 = {b, c, d, e} virittämä aliverkko geometrisena kaaviona sekä määritä sen naapurimatriisi.
3. Draw the graph whose adjacency

0
0

0
N =
1
1
1
matrix is
0
0
0
1
1
1
0
0
0
1
1
1
1
1
1
0
0
0
1
1
1
0
0
0

1
1

1
,
0
0
0
4. Prove that
(a) if a graph has atleast two vertices, then there exists two vertices that have
a same degree,
(b) if the edges of a complete graph of six vertices are colored with two colors,
so that every edge is colored with one color, there exists a triangle of a
single color.
1Think
the solution on a graph that has vertices tripples XY Z, where X is the amount of
water on eightliter bucket, Y in five liter bucket, and Z in three liter bucket.