## MACM 201 – D100 AND D200 ASSIGNMENT #7

加拿大数学代写 Answer all questions on paper or a tablet using your own handwriting. Put your name, student ID number and page number at the top of each page.

**Instructions **

Answer all questions on paper or a tablet using your own handwriting. Put your name, student ID number and page number at the top of each page. If you use paper make a photo of each page and upload your solutions to crowdmark. If you use a tablet, export your assignment to .pdf and upload the .pdf to crowdmark.

**Textbook Reading **

**Exercises **

**A.Textbook Questions **

11.4 Exercises 2, 14ad, 26abc.

11.5 Exercises 1, 6.

12.1 Exercises 4, 6, 10

**B.Instructor Questions 加拿大数学代写**

Questions on 11.4

1.Find a subgraph of the graph *G *below that is subdivision of *K*3*,*3. Conclude that *G *is not planar.

2.Let *G *= (*V, E*) be a connected simple graph with *|**V **| ≥ *11.

Show that *G *or its complementis not planar.

3.Draw a planar embedding of the tetrahedron *T*. Draw *T*^{∗}* *the dual of *T*.

### Questions on 11.5

4.Recall that *K*_{m,n}* *denotes the complete bipartite graph with *m *+ *n *vertices.

(a) Does *K*_{2}_{,}_{3} have a Hamiltonian cycle? If yes draw one. If not explain.

(b) Does *K _{2}_{,}_{3}* have a Hamiltonian path? If yes draw one. If not explain.

(c) Find the *K_{m,n}*

*with the fewest vertexes which has a Hamiltonian cycle.*

(d) Find the *K_{m,n}*

*with the fewest vertexes which has a Hamiltonian path.*

5.Below is a non-planar drawing of the cube graph. 加拿大数学代写

Draw a planar embedding of the cube graph.

Draw all Hamiltonian cycles that include the edge *{*1*, *2*}*. I found four.

6.What is the converse of Theorem 11.8?

Give a counter example to the converse of Theorem 11.8.

### Questions on 12.1 加拿大数学代写

7.If a tree has four vertices of degree 2, four of degree 4, and two of degree 5, how many pendant vertices does it have?

8.In class we proved the following theorem:

If *T *= (*V, E*) is a tree and *u, v **∈ **V *are distinct, there is a unique path in *T *from *u *to *v*.

Prove that the converse of the theorem is also true, namely

Let *G *= (*V, E*) be a simple graph. If for every pair of vertices *u, v **∈ **V *there is a unique path in *G *from *u *to *v *then *G *is a tree.

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