Graph Theory Practice Sheet for Midterm 2 · Graph Theory Practice Sheet for Midterm 2 This sheet...
Transcript of Graph Theory Practice Sheet for Midterm 2 · Graph Theory Practice Sheet for Midterm 2 This sheet...
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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
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