1

Graph Theory Practice Sheet for Midterm 2

This sheet is not meant to be exhaustive, but rather as a supplement to theproblems from the homework since the last exam.

1. This is a problem in the direction of Vizing's Theorem. Show that for a graph ,you can always color it using at most colors. As a hint, you should thinkabout the simple vertex coloring algorithm and how it worked.

2. This problem is the direction of Brook's Theorem. Suppose that is a graph, is a cut vertex and are the components of . Show that if

is less than for each , then we will also have $\chi(G) \leq\Delta + 1.

3. Can you draw a graph with and with the graph containing notriangles? If you can, do it. If not, say why not.

4. Draw a graph with and (or show no suchgraph exists).

5. In the graph shown below, exhibit a minimum vertex cut and a minimal vertex cut which isn't minimum. How can you tell that your minimum

vertex cut is actually minimum?

6. Find all cut vertices and blocks in the graph below:

G

2Δ − 1

G v

G , G , … , G1 2 k G − v

χ(G) Δ(G) + 1 I

ξ(G) = 4

κ(G) = 2, λ(G) = 2 δ(G) = 3

u − v

u − v

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