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MAT 311 Discrete Math Week 5 Homework_Complete_Answer

MAT 311 Discrete Math Week 5 Homework_Complete_Answer

MAT 311 Discrete Math Week 5 Homework_Complete_Answer

MAT 311 Discrete Math Week 5 Homework_Complete_Answer

MAT 311 Discrete Math Week 5 Homework_Complete_Answer

MAT 311 Discrete Math Week 5 Homework_Complete_Answer


MAT 311 Discrete Math Week 5 Homework_Complete_Answer

MAT 311 Week 5 Homework Answer

1) A league of 13 teams is playing a “round-robin” style tournament, where each team plays every other team exactly once. How many games total need to be played?
.

Justify your answer using a graph model—say what the vertices and edges of your graph represent, and what (if any) theorems you use. (Upload your file below.)

2) …The complete bipartite graph Km, n is the simple undirected graph with m + n vertices split into two sets V1 and V2 (|V1| = m, |V2| = n)such that vertices x, y share an edge if and only if x V1 and y V2. For example, K3, 4 is the following graph, where V1 is the top row of vertices and V2 is the bottom row.

What is the fewest number of colors needed to color the vertices of Km, n such that no two vertices of the same color are joined by an edge?

3) …A truncated icosidodecahedron (also known as a “great rhombicosidodecahedron” or “omnitruncated icosidodecahedron”) is a polyhedron with 120 vertices. Each vertex looks the same: a square, a hexagon, and a decagon come together at each vertex. How many edges does a truncated icosidodecahedron have?

Explain how you arrive at your answer. (Note: the picture in the figure doesn’t show the vertices or edges on the back of the polyhedron.)

4) …The following graph has 45 vertices.

Does this graph have an Euler circuit? Why or why not?

Yes. There are exactly two even-degree vertices.
Yes. There are exactly two odd-degree vertices.
No. There is an odd-degree vertex.
No. There is an even-degree vertex.
No. There are 2 even degree vertices.

5) …The following graph has 45 vertices.

Does this graph have an Euler path? Why or why not?

Yes. There are exactly two odd-degree vertices.
Yes. There are exactly two even-degree vertices.
No. There is an odd-degree vertex.
No. There is an even-degree vertex.
No. There are 2 even degree vertices.

6)…The following graph has 45 vertices.

The graph below is a copy of the above graph, but with some additional edges added so that all of the vertices in the resulting graph have degree four.

How many edges does this new graph have? Explain how you can use a theorem from this section to make counting the edges easier.

7) …Let S = {1, 2, 3, 5, 10, 15, 20}.It is a fact that (S, |) is a poset. Draw its Hasse diagram.

8) …Let X = {5, 10, 15, 20, 25, 30, 35, 40}.Define a relation

on X as follows. For a, b X,a

bif b can be obtained from a by adding a (possibly empty) collection of dimes (10 cents) and quarters (25 cents). So, for example, 25

35, but 25 30.

Give a pair of incomparable elements in (X, ).(Enter your answers as a comma-separated list.)

9)…Explain why the relation R on {1, 2, 4, 6} given by
R = {(1, 1), (2, 2), (4, 4), (6, 6), (1, 2), (2, 4), (1, 4), (4, 2)} is not a partial ordering on {1, 2, 4, 6}. Be specific.
1 —Select— Reflexivity Transitivity Antisymmetry . fails. For example, 2 R 2 . and 3 . R 2, but 2 4 ? = ≠ . 5 ..

10)…Let X = {5, 10, 15, 20, 25, 30, 35, 40}.Define a relation

on X as follows. For a, b X,a

bif b can be obtained from a by adding a (possibly empty) collection of dimes (10 cents) and quarters (25 cents). So, for example, 25 35, but 25 30.

List all minimal elements of (X, ).(Enter your answers as a comma-separated list.)

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MAT 311 Discrete Math Week 5 Homework_Complete_Answer