Problem 1.

Let $G$ be the graph below.

Determine each of the following:
a. What is the chromatic number of $G$ ? Explain your answer.
b. Is $G$ planar? Why or why not?

Problem 2.

a. Draw an acyclic graph with 7 vertices and 4 components.
b. Prove that if $G$ is an acyclic graph with $v$ vertices and 4 components, each of which is a tree, that $G$ has $v-4$ edges.
c. Draw a graph with 7 vertices and 6 edges that is not a tree.

Problem 3.

Let $A={1,2,3,4}$ and $B={x, y, z, w, v}$. Let $A_{1}={3}$ and let $h: A_{1} \rightarrow B$ be the function defined by $h(3)=w$.
a. How many extensions of $h$ to $A$ are there?
b. How many extensions of $h$ to $A$ are one-to-one?
c. How many extensions of $h$ to $A$ are onto?
Do not simplify any binomial coefficients, exponents, or factorials.

Problem 4.

a. Show that $K_{6,7}$ has a path containing all vertices in the graph.
b. Explain why $K_{6,7}$ is not Hamiltonian.

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